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Solo Hermelin
Solo Hermelin

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1 Introduction to Quantum Mechanics SOLO HERMELIN Updated: 11.11.13 10.10.14
Introduction to Quantum MechanicsSOLO Table of Content 2 Introduction to Quantum Mechanics Classical Mechanics Gravity Optics Electromagnetism Quantum Weirdness History Physical Laws of Radiometry Zeeman Effect, 1896 Discovery of the Electron, 1897 Planck’s Law 1900 Einstein in 1905 Bohr Quantum Model of the Atom 1913. Einstein’s General Theory of Relativity 1915 Quantum Mechanics History
Introduction to Quantum MechanicsSOLO Table of Content (continue – 1) 3 De Broglie Particle-Wave Law 1924 Wolfgang Pauli states the “Quantum Exclusion Principle” 1924 Heisenberg, Born, Jordan “Quantum Matrix Mechanics”, 1925 Wave Packet and Schrödinger Equation, 1926 Operators in Quantum Mechanics Hilbert Space and Quantum Mechanics Von Neumann - Postulates of Quantum Mechanics Conservation of Probability Expectations Value and Operators The Expansion Theorem or Superposition Principle Matrix Representation of Wave Functions and Operators Commutator of two Operators A and B Time Evolution Operator of the Schrödinger Equation Heisenberg Uncertainty Relations
Introduction to Quantum MechanicsSOLO Table of Content (Continue -2) 4 Time Independent Hamiltonian The Schrödinger and Heisenberg Pictures Transition from Quantum Mechanics to Classical Mechanics. Pauli Exclusion Principle Klein-Gordon Equation for a Spinless Particle Non-relativistic Schrödinger Equation in an Electromagnetic Field Pauli Equation Dirac Equation Light Polarization and Quantum Theory Copenhagen Interpretation of Quantum Mechanics Measurement in Quantum Mechanics Schrödinger’s Cat Solvay Conferences Bohr–Einstein Debates Feynman Path Integral Representation of Time Evolution Amplitudes
Introduction to Quantum MechanicsSOLO Table of Content (Continue -3) 5 Quantum Field Theories References Aharonov–Bohm Effect Wheeler's delayed choice experiment Zero-Point Energy Quantum Foam De Broglie–Bohm Theory in Quantum Mechanics Bell's Theorem Bell Test Experiments Wheeler's delayed choice experiment Hidden Variables
Physics The Presentation is my attempt to study and cover the fascinating subject of Quantum Mechanics. The completion of this presentation does not make me an expert on the subject, since I never worked in the field. I thing that I reached a good coverage of the subject and I want to share it. Comments and suggestions for improvements are more than welcomed. 6 SOLO Introduction to Quantum Mechanics
Physics NEWTON's MECHANICS ! ANALYTIC MECHANICS FLUID & GAS DYNAMICS THERMODYNAMICS MAXWELL ELECTRODYNAMICS CLASSICAL THEORIES NEWTON's GRAVITY OPTICS 1900 At the end of the 19th century, physics had evolved to the point at which classical mechanics could cope with highly complex problems involving macroscopic situations; thermodynamics and kinetic theory were well established; geometrical and physical optics could be understood in terms of electromagnetic waves; and the conservation laws for energy and momentum (and mass) were widely accepted. So profound were these and other developments that it was generally accepted that all the important laws of physics had been discovered and that, henceforth, research would be concerned with clearing up minor problems and particularly with improvements of method and measurement. "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" - Lord Kelvin 1900: This was just before Relativity and Quantum Mechanics appeared on the scene and opened up new realms for exploration. Completeness of a Theory 7 Return to Table of Content SOLO
8 Classical TheoriesSOLO 1.1 Newton’s Laws of Motion “The Mathematical Principles of Natural Philosophy” 1687 First Law Every body continues in its state of rest or of uniform motion in straight line unless it is compelled to change that state by forces impressed upon it. Second Law The rate of change of momentum is proportional to the force impressed and in the same direction as that force. Third Law To every action there is always opposed an equal reaction. td rd constF   ==→= → :vv0 ( )vm td d p td d F  == 2112 FF  −= vmp  = td pd F  = 12F  1 2 21F  r  - Position v:  mp = - Momentum
9 SOLO 1.2 Work and Energy The work W of a force acting on a particle m that moves as a result of this along a curve s from to is defined by: F  1r  2r  ∫∫ ⋅      =⋅= ⋅∆ 2 1 2 1 12 r r r r rdrm dt d rdFW      r  1r  2r  rd  rdr  + 1 2 F  m s rd  is the displacement on a real path. ⋅⋅∆ ⋅= rrmT  2 1 The kinetic energy T is defined as: 1212 2 1 2 1 2 1 2 TTrrd m dtrr dt d mrdrm dt d W r r r r r r −=      ⋅=⋅      =⋅      = ∫∫∫ ⋅ ⋅ ⋅⋅⋅⋅⋅        For a constant mass m Classical Theories
10 SOLO Work and Energy (continue) When the force depends on the position alone, i.e. , and the quantity is a perfect differential ( )rFF  = rdF  ⋅ ( ) ( )rdVrdrF  −=⋅ The force field is said to be conservative and the function is known as the Potential Energy. In this case: ( )rV  ( ) ( ) ( ) 212112 2 1 2 1 VVrVrVrdVrdFW r r r r −=−=−=⋅= ∫∫ ∆      The work does not depend on the path from to . It is clear that in a conservative field, the integral of over a closed path is zero. 12W 1r  2r  rdF  ⋅ ( ) ( ) 01221 21 1 2 2 1 =−+−=⋅+⋅=⋅ ∫∫∫ VVVVrdFrdFrdF path r r path r rC         Using Stoke’s Theorem it means that∫∫∫ ⋅×∇=⋅ SC sdFrdF  0=×∇= FFrot  Therefore is the gradient of some scalar functionF  ( ) rdrVdVrdF  ⋅−∇=−=⋅ ( )rVF  −∇= Classical Theories
11 SOLO Work and Energy (continue) and ⋅ →∆→∆ ⋅−=⋅−= ∆ ∆ = rF dt rd F t V dt dV tt  00 limlim But also for a constant mass system ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅=⋅=      ⋅+⋅=      ⋅= rFrrmrrrr m rrm dt d dt dT  22 1 Therefore for a constant mass in a conservative field ( ) .0 constEnergyTotalVTVT dt d ==+⇒=+ Classical Theories
SOLO 1.5 Rotations and Angular Momentum Classical Theories md td rd mdpd   == v md td rd pd td d Fd 2 2   == md td rd pdHd CG   ×=×= ρρ: ∫∫ == M md td rd pdP   ∫∫ == M md td rd FdF 2 2  - Angular Rotation Rate of the Body (B) relative to Inertia (I) - Force ∫∫ ×== M CGCG md td rd HdH   ρ - Angular Momentum Relative to C.G. BBBBBBIIIIII zzyyxxzzyyxxr 111111 ++=++=  BIBBBIBBBIBB III zz td d yy td d xx td d z td d y td d x td d 111111 0111 ×=×=×= === ←←← ωωω   IB←ω  - Momentum 12
SOLO 1.6 Lagrange, Hamilton, Jacobi Classical Theories Carl Gustav Jacob Jacobi (1804-1851) William Rowan Hamilton 1805-1865 Joseph Louis Lagrange 1736-1813 Lagrangiams Lagrange’s Equations: nicQ q L q L dt d m k k ikin ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ mkcqc k t n i i k i ,,2,10 1  ==+∑= ( ) ( ) ( )qVtqqTtqqL  −= ,,:,, ni cQ q H p p H q m j j iji i i i i ,,2,1 1    =        ++ ∂ ∂ −= ∂ ∂ = ∑= λ mkcqc k t n i i k i ,,2,10 1  ==+∑= Extended Hamilton’s Equations Constrained Differential Equations Hamiltonian ( )tqqTqpH n i ii ,,: 1   −= ∑= ni q T p i i ,,2,1   = ∂ ∂ = Hamilton-Jacobi Equation 0,, =      ∂ ∂ + ∂ ∂ k k q S qtH t S       ∂ ∂ = k kjj q S qtq ,,φ kk q S p ∂ ∂ = 13
14 SOLO 1.4 Basic Definitions Given a System of N particles. The System is completely defined by Particles coordinates and moments: ( ) ( ) ( ) ( ) ( ) ( ) ( ) Nl ktpjtpitp td rd mp ktzjtyitxzyxrr zlylxl l ll lllkkkll ,,2,1 ,,    =      ++== ++== where are the unit vectors defining any Inertial Coordinate Systemkji  ,, r  1r  2r  rd  rdr  + 1 2 F  m s The path of the Particles are defined by Newton Second Law NlF td rd m td pd l l l l ,,2,12 2   === ∑ Given , the Path of the Particle is completely defined and is Deterministic (if we repeat the experiment, we obtain every time the same result). ( ) ( ) ( )tFandtptr lll ∑== 0,0  In Classical Mechanics: •Time and Space are two Independent Entities. •No limit in Particle Velocity •Since every thing is Deterministic we can Measure all quantities simultaneously. The outcome of all measurements are repeatable and depends only on the accuracy of the measurement device. •Causality: Every Effect hase a Cause that preceed it. Classical Theories Return to Table of Content
GRAVITY Classical Theories GF  GF M m      EQPOISSON G GU r MG UU r GM g gm r MG mr r mM GF ρπ4&& 1 2 2 =∇=−∇=      −∇= −=      ∇=−= Newton’s Law of Universal Gravity Any two body attract one another with a Force Proportional to the Product of the Masses and inversely Proportional to the Square of the Distance between them. G = 6.67 x 10-8 dyne cm2 /gm2 the Universal Gravitational Constant Instantaneous Propagation of the Force along the direction between the Masses (“Action at a Distance”). 15
Newton was fully aware of the conceptual difficulties of his action-at-a-distance theory of gravity. In a letter to Richard Bentley Newton wrote: "It is inconceivable, that inanimate brute matter should, without the mediation of something else, which is not material, operate upon, and affect other matter without mutual contact; as it must do, if gravitation, ...., be essential and inherent in it. And this is one reason, why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another, at a distance through vacuum, without the mediation of anything else, by and through their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it." GRAVITY Classical Theories 16 Return to Table of Content
17 SOLO Newton published “Opticks”1704 Newton threw the weight of his authority on the corpuscular theory. This conviction was due to the fact that light travels in straight lines, and none of the waves that he knew possessed this property. Newton’s authority lasted for one hundred years, and diffraction results of Grimaldi (1665) and Hooke (1672), and the view of Huygens (1678) were overlooked. Optics Every point on a primary wavefront serves the source of spherical secondary wavelets such that the primary wavefront at some later time is the envelope o these wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wave at each point in space. Christiaan Huygens 1629-1695 Huygens Principle 1678 Light: Waves or Particles Classical Theories
18 SOLO In 1801 Thomas Young uses constructive and destructive interference of waves to explain the Newton’s rings. Thomas Young 1773-1829 1801 - 1803 In 1803 Thomas Young explains the fringes at the edges of shadows using the wave theory of light. But, the fact that was belived that the light waves are longitudinal, mad difficult the explanation of double refraction in certain crystals. Optics Run This Young Double Slit Experiment Classical Theories
19 POLARIZATION Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that two rays polarized at right angles to each other never interface. SOLO Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young 1773-1829 1816-1817 longitudinal waves transversal waves Classical Theories Run This
20 SOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. 1818Diffraction - History Classical Theories
21 Diffraction SOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. P 0P Q 1x 0x 1y 0y η ξ Fr  Sr  ρ  r  O 'θ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen From a source P0 at a distance from a aperture a spherical wavelet propagates toward the aperture: ( ) ( )Srktj S source Q e r A tU − = ' ' ω According to Huygens Principle second wavelets will start at the aperture and will add at the image point P. ( ) ( ) ( )( ) ( ) ( )( ) ∫∫ Σ ++− Σ +−− == dre rr A Kdre r U KtU rrktj S sourcerkttjQ P S 2/2/' ',', πωπω θθθθ where: ( )',θθK obliquity or inclination factor ( ) ( )SSS nrnr 11cos&11cos' 11 ⋅=⋅= −− θθ ( ) ( )   === === 0',0 max0',0 πθθ θθ K K Obliquity factor and π/2 phase were introduced by Fresnel to explain experiences results. Fresnel Diffraction Formula Fresnel took in consideration the phase of each wavelet to obtain: Run This Return to Table of Content Classical Theories
22 MAXWELL’s EQUATIONS SOLO Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Electric Current DensityeJ  [ ]2− ⋅mA Free Electric Charge Distributioneρ [ ]3− ⋅⋅ msA 1. AMPÈRE’S CIRCUIT LAW (A) 1821 eJ t D H    + ∂ ∂ =×∇ 2. FARADAY’S INDUCTION LAW (F) 1831 t B E ∂ ∂ −=×∇   3. GAUSS’ LAW – ELECTRIC (GE) ~ 1830 eD ρ=⋅∇  4. GAUSS’ LAW – MAGNETIC (GM) 0=⋅∇ B  André-Marie Ampère 1775-1836 Michael Faraday 1791-1867 Karl Friederich Gauss 1777-1855 James Clerk Maxwell (1831-1879) 1865 Electromagnetism MAXWELL UNIFIED ELECTRICITY AND MAGNETISM Classical Theories
23 SOLO ELECTROMGNETIC WAVE EQUATIONS For Homogeneous, Linear and Isotropic Medium ED  ε= HB  µ= where are constant scalars, we haveµε, t E t D H t t H t B E ED HB ∂ ∂ = ∂ ∂ =×∇ ∂ ∂ ∂ ∂ −= ∂ ∂ −=×∇×∇ = =       εµ µ ε µ Since we have also tt ∂ ∂ ×∇=∇× ∂ ∂ ( ) ( ) ( )                   =⋅∇= ∇−⋅∇∇=×∇×∇ = ∂ ∂ +×∇×∇ 0& 0 2 2 2 DED EEE t E E     ε µε t D H ∂ ∂ =×∇   t B E ∂ ∂ −=×∇   For Source less Medium 02 2 2 = ∂ ∂ −∇ t E E   µε Define meme KK c KK v === ∆ 00 11 εµµε where ( ) smc /103 10 36 1 104 11 8 9700 ×=       ×× == −− ∆ π π εµ c is the velocity of light in free space. Electromagnetism Run This Return to Table of Content Classical Theories
Completeness of a Theory SOLO At the end of the 19th century, physics had evolved to the point at which classical mechanics could cope with highly complex problems involving macroscopic situations; thermodynamics and kinetic theory were well established; geometrical and physical optics could be understood in terms of electromagnetic waves; and the conservation laws for energy and momentum (and mass) were widely accepted. So profound were these and other developments that it was generally accepted that all the important laws of physics had been discovered and that, henceforth, research would be concerned with clearing up minor problems and particularly with improvements of method and measurement. "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" - Lord Kelvin 1900: 1894: "The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote.... Our future discoveries must be looked for in the sixth place of decimals." - Albert. A. Michelson, speech at the dedication of Ryerson Physics Lab, U. of Chicago 1894 This was just before Relativity and Quantum Mechanics appeared on the scene and opened up new realms for exploration. 24 Classical Theories
25 QUANTUM THEORIES
Many classical particles, both slits are open http://www.mathematik.uni- muenchen.de/~bohmmech/Poster/post/postE.htmlThe Double Slit Experiment A single particle, both slits are open Many particles, one slit is open. Many atomic particles, both slits are open http://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics#Schr.C3.B6ding er_wave_equation SOLO Run This 26 QUANTUM THEORIES https://www.youtube.com/watch?v=Q1YqgPAtzho&src_vid=4C5pq7W5yRM&feature=iv&annotation_id=an
According to the results of the double slit experiment, if experimenters do something to learn which slit the photon goes through, they change the outcome of the experiment and the behavior of the photon. If the experimenters know which slit it goes through, the photon will behave as a particle. If they do not know which slit it goes through, the photon will behave as if it were a wave when it is given an opportunity to interfere with itself. The double-slit experiment is meant to observe phenomena that indicate whether light has a particle nature or a wave nature. Richard Feynman observed that if you wish to confront all of the mysteries of quantum mechanics, you have only to study quantum interference in the two-slit experiment The Double Slit Experiment SOLO Run This 27 QUANTUM THEORIES
QUANTUM THEORIES Some trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (A–B) and quantum mechanics (C–H). In quantum mechanics (C–H), the ball has a wave function, which is shown with real part in blue and imaginary part in red. The trajectories C,D,E,F, (but not G or H) are examples of standing waves, (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy 28
SPECIAL RELATIVITY GENERAL RELATIVITY COSMOLOGICAL THEORIES MODERN THEORIES Modern Physics NONRELATIVISTIC QUANTUM MECHANICS QUANTUM THEORIES NEWTON's MECHANICS ! ANALYTIC MECHANICS FLUID & GAS DYNAMICS THERMODYNAMICS MAXWELL ELECTRODYNAMICS CLASSICAL THEORIES NEWTON's GRAVITY OPTICS 1900 29
http://www.bubblews.com/news/401138-what-is-quantum-theory 30 Return to Table of Content QUANTUM THEORIES
31 SOLO http://thespectroscopynet.com/educational/Kirchhoff.htm Spectroscopy 1868 A.J. Ångström published a compilation of all visible lines in the solar spectrum. 1869 A.J. Ångström made the first reflection grating. Anders Jonas Angström a physicist in Sweden, in 1853 had presented theories about gases having spectra in his work: Optiska Undersökningar to the Royal Academy of Sciences pointing out that the electric spark yields two superposed spectra. Angström also postulated that an incandescent gas emits luminous rays of the same refrangibility as those which it can absorb. This statement contains a fundamental principle of spectrum analysis. http://en.wikipedia.org/wiki/Spectrum_analysis
32 ParticlesSOLO 1874 George Johnstone Stoney 1826 - 1911 As early as 1874 George Stoney had calculated the magnitude of his electron from data obtained from the electrolysis of water and the kinetic theory of gases. The value obtained later became known as a coulomb. Stoney proposed the particle or atom of electricity to be one of three fundamental units on which a whole system of physical units could be established. The other two proposed were the constant universal gravitation and the maximum velocity of light and other electromagnetic radiations. No other scientist dared conceive such an idea using the available data. Stoney's work set the ball rolling for other great scientists such as Larmor and Thomas Preston who investigated the splitting of spectral lines in a magnetic field. Stoney partially anticipated Balmer's law on the hydrogen spectral series of lines and he discovered a relationship between three of the four lines in the visible spectrum of hydrogen. Balmer later found a formula to relate all four. George Johnstone Stoney was acknowledged for his contribution to developing the theory of electrons by H.A. Lorentz , in his Nobel Lecture in 1902. George Stoney estimates the charge of the then unknown electron to be about 10-20 coulomb, close to the modern value of 1.6021892 x 10-19 coulomb. (He used the Faraday constant (total electric charge per mole of univalent atoms) divided by Avogadro's Number. Return to Table of Content
33 Physical Laws of RadiometrySOLO Stefan-Boltzmann Law Stefan – 1879 Empirical - fourth power law Boltzmann – 1884 Theoretical - fourth power law For a blackbody: ( ) ( ) ( ) ( )       ⋅ ⋅==       = − == − ∞∞ ∫∫ 42 12 32 45 2 4 0 2 5 1 0 10670.5 15 2 : 1/exp 1 Kcm W hc k cm W Td Tc c dMM BBBB  π σ σλ λλ λλ LUDWIG BOLTZMANN (1844 - 1906) Stefan-Boltzmann Law JOSEF STEFAN (1835 – 1893) 1879 1884 1893 Wien’s Displacement Law 0= λ λ d Md BB Wien 1893 from which: The wavelength for which the spectral emittance of a blackbody reaches the maximum is given by: mλ KmTm  ⋅= µλ 2898 Wien’s Displacement Law WILHELM WIEN (1864 - 1928) Nobel Prize 1911
34 SOLO Johan Jakob Balmer presented an empirical formula describing the position of the emission lines in the visible part of the hydrogen spectrum. Spectroscopy 1885 Johan Jakob Balmer 1825 - 1898 Balmer Formula ( )222 / nmmB −=λ ,6,5,4,3,106.3654,2 8 =×== − mcmBn δH violet blue - green red 1=n 2=n 3=n 4=n 5=n ∞=n Lyman serie Balmer serie Paschen serie Brackett serie 0=E Energy         −= 2232 0 4 11 8 1 nnhc em f ελ 1=fn 2=f n 3=fn 4=f n Balmer was a mathematical teacher who, in his spare time, was obsessed with formulae for numbers. He once said that, given any four numbers, he could find a mathematical formula that connected them. Luckily for physics, someone gave him the wavelengths of the first four lines in the hydrogen spectrum.
35 SOLO Spectroscopy 1887 Johannes Robert Rydberg 1854 - 1919 Rydberg Formula for Hydrogen 2 2 1 1 1 H i f R n nλ   = − ÷ ÷   1=n 2=n 3=n 4=n 5=n ∞=n Lyman serie Balmer serie Paschen serie Brackett serie 0=E Energy         −= 2232 0 4 11 8 1 nnhc em f ελ 1=fn 2=fn 3=f n 4=fn 34 6.62606876 10h J s− = × gPlank constant 31 9.10938188 10em kg− = ×Electron mass 19 1.602176452 10e C− = ×Electron charge 12 0 8.854187817 10 /F mε − = ×Permittivity of vacuum Rydberg generalized Balmer’s hydrogen spectral lines formula. Theodore Lyman 1874 - 1954 2in = Balmer series (1885) Johan Jakob Balmer 1825 - 1898 Friedric Paschen 1865 - 1947 3in = Paschen series (1908) 4in = Brackett series (1922) Lyman series (1906)1in = Rydberg Constant for Hydrogen 17 x105395687310973.1 − = mRH 4 2 3 08 e H m e R h cε = Later in the Bohr Model was fund that Frederick Sumner Brackett 1896 - 1988
36 PhotoelectricitySOLO In 1887 Heinrich Hertz, accidentally discovered the photoelectric effect. Hertz conducted his experiments that produced radio waves. By chance he noted that a piece of zinc illuminated by ultraviolet light became electrically charged. Without knowing he discovered the Photoelectric Effect. 1887 Heinrich Rudolf Hertz 1857-1894 - - - - - - - - -- - - - - metallic surface ejected electrons incoming E.M. waves http://en.wikipedia/wiki/Photoelectric_effect http://en.wikipedia/wiki/Heinrich_Hertz Return to Table of Content
37 SpectroscopySOLO Zeeman Effect Pieter Zeeman observed that the spectral lines emitted by an atomic source splited when the source is placed in a magnetic field. In most atoms, there exists several electron configurations that have the same energy, so that transitions between different configuration correspond to a single line. 1896 Because the magnetic field interacts with the electrons, this degeneracy is broken giving rice to very close spectral lines. no magnetic field B = 0 cba ,, fed ,, a b c d e f magnetic field B 0≠ http://en.wikipedia.org/wiki/Zeeman_effect Pieter Zeeman 1865 - 1943 Nobel Prize 1902 Return to Table of Content
38 Physical Laws of RadiometrySOLO Wien Approximation to Black Body Radiation Wien's Approximation (also sometimes called Wien's Law or the Wien Distribution Law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in 1896. The equation does accurately describe the short wavelength (high frequency) spectrum of thermal emission from objects, but it fails to accurately fit the experimental data for long wavelengths (low frequency) emission. WILHELM WIEN (1864 - 1928) Comparison of Wien's Distribution law with the Rayleigh–Jeans Law and Planck's law, for a body of 8 mK temperature The Wien ‘s Law may be written as where • I(ν,T) is the amount of energy per unit surface area per unit time per unit solid angle per unit frequency emitted at a frequency ν. • T is the temperature of the black body. • h is Planck's constant. • c is the speed of light. • k is Boltzmann's constant 1896 Return to Table of Content
39 SOLO Particles J.J. Thomson showed in 1897 that the cathode rays are composed of electrons and he measured the ratio of charge to mass for the electron. Discovery of the Electron 1897 Joseph John Thomson 1856 – 1940 Nobel Prize 1922 The total charge on the collector (assuming all electrons are stick to the cathode collector and no secondary emissions is: e qnQ ⋅= The energy of the particles reaching the cathode is: 2/2 vmnE ⋅⋅= Uvm q E Q e 12 2 = ⋅ = U v m qe 2 2 = Thomson Atom Model Wavelike Behavior for Electrons Return to Table of Content
40 Physical Laws of RadiometrySOLO Rayleigh–Jeans Law Comparison of Rayleigh–Jeans law with Wien approximation and Planck's law, for a body of 8 mK temperature In 1900, the British physicist Lord Rayleigh derived the λ−4 dependence of the Rayleigh–Jeans law based on classical physical arguments.[3] A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir James Jeans in 1905. The Rayleigh–Jeans law revealed an important error in physics theory of the time. The law predicted an energy output that diverges towards infinity as wavelength approaches zero (as frequency tends to infinity) and measurements of energy output at short wavelengths disagreed with this prediction. John William Strutt, 3rd Baron Rayleigh 1842- 1919 James Hopwood Jeans 1877 - 1946 Rayleigh considered the radiation inside a cubic cavity of length L and temperature T whose walls are perfect reflectors as a series of standing electromagnetic waves. At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The three wavelengths λ1, λ2 and λ3, in the three directions orthogonal to the walls can be: ,2,1,, 2 === i i i nzyxi n Lλ 1900 1905
41 Physical Laws of RadiometrySOLO Rayleigh–Jeans Law (continue ) The Rayleigh–Jeans law agrees with experimental results at large wavelengths (or, equivalently, low frequencies) but strongly disagrees at short wavelengths (or high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe. Comparison of Rayleigh–Jeans law and Planck's law The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, although the concept goes back to 1900 with the first derivation of the λ − 4 dependence of the Rayleigh–Jeans law; Solution Max Planck solved the problem by postulating that electromagnetic energy did not follow the classical description, but could only oscillate or be emitted in discrete packets of energy proportional to the frequency, as given by Planck's law. This has the effect of reducing the number of possible modes with a given energy at high frequencies in the cavity described above, and thus the average energy at those frequencies by application of the equipartition theorem. The radiated power eventually goes to zero at infinite frequencies, and the total predicted power is finite. The formula for the radiated power for the idealized system (black body) was in line with known experiments, and came to be called Planck's law of black body radiation. Based on past experiments, Planck was also able to determine the value of its parameter, now called Planck's constant. The packets of energy later came to be called photons, and played a key role in the quantum description of electromagnetism. ( ) ( ) λ λ π λ λ λλ d Tk d V N Tkdu 4 8 == Rayleigh–Jeans Law Return to Table of Content
42 Physical Laws of RadiometrySOLO MAX PLANCK (1858 - 1947) 4242( ) ν ννπ νν ν d e h c du kT h 1 8 3 2 − = ( ) ν νπ νν ν de c h du kT h − = 3 3 8 WILHELM WIEN (1864 - 1928) Wien’s Law 1896 ( ) ν νπ νν dTk c du 3 2 8 = Rayleigh–Jeans Law 1900 - 1905 John William Strutt, 3rd Baron Rayleigh 1842- 1919 James Hopwood Jeans 1877 - 1946 Comparison of Rayleigh–Jeans law with Wien approximation and Planck's law, for a body of 8 mK temperature Tkh <<ν Tkh >>ν
43 Physical Laws of RadiometrySOLO MAX PLANCK (1858 - 1947) Planck’s Law 1900 ( ) ν ννπ νν ν d e h c du kT h 1 8 3 2 − = Planck derived empirically, by fitting the observed black body distribution to a high degree of accuracy, the relation By comparing this empirical correlation with the Rayleigh-Jeans formula Planck concluded that the error in classical theory must be in the identification of the average oscillator energy as kT and therefore in the assumption that the oscillator energy is distributed continuously. He then posed the following question: If the average energy is defined as how is the actual oscillator energies distributed? ( ) ν νπ νν dTk c du 3 2 8 = 1/ − = kTh e h E ν ν KT KWk Wh   ineTemperaturAbsolute- constantBoltzmannsec/103806.1 constantPlanksec106260.6 23 234 −⋅⋅= −⋅⋅= − −
44 Physical Laws of RadiometrySOLO MAX PLANCK (1858 - 1947) If the average energy is defined as how is the actual oscillator energies distributed? 1/ − = kTh e h E ν ν Planck deviated appreciable from the concepts of classical physics by assuming that the energy of the oscillators, instead of varying continuously, can assume only certain discrete values νε hnn = Let determine the average energy ( ) ( )  +++ ++ === −− −− ∞ = − ∞ = − ∞ = − ∞ = − ∑ ∑ ∑ ∑ kThkTh kThkTh n kTnh n kTnh n kTE n kTE n ee eeh e enh e eE E n n /2/ /2/ 0 / 0 / 0 / 0 / 1 2 νν νν ν ν ν ν From Statistical Mechanics we know that the probability of a system assuming energy between ε and ε+dε is proportional to exp (-ε/kT) dε x ee kTh ex kThkTh − =+++ − = −− 1 1 1 / /2/ ν νν  ( ) ( )2 0 /2/ 0 / 11 1 2 / x x h xxd d xhxn xd d xheehenh n n ex kThkTh n kTnh kTh − =      − ==++= ∑∑ ∞ = = −− ∞ = − − ννννν ν ννν  where n is an integer (n = 0, 1, 2, …), and h =6.6260. 10-14 W. sec2 is a constant introduced empirically by Planck , the Planck’s Constant.
45 Physical Laws of RadiometrySOLO MAX PLANCK (1858 - 1947) Planck’s Postulate: The energy of the oscillators, instead of varying continuously, can assume only certain discrete values νε hnn = where n is an integer (n = 0, 1, 2, …). We say that the oscillators energy is Quantized. ( ) 11 1 1 1 // / / 2/ / 0 / 0 / 0 / 0 / − = − = − − === − − − − − ∞ = − ∞ = − ∞ = − ∞ = − ∑ ∑ ∑ ∑ kThkTh kTh kTh kTh kTh n kTnh n kTnh n kTE n kTE n e h e e h e e e h e enh e eE E n n νν ν ν ν ν ν ν ν ν νν The average energy is
46 Physical Laws of RadiometrySOLO Plank’s Law ( ) 1/exp 1 2 5 1 − = Tc c M BB λλ λ Plank’s Law applies to blackbodies; i.e. perfect radiators. The spectral radial emittance of a blackbody is given by: ( ) KT KWk Wh kmc Kmkhcc mcmWchc    ineTemperaturAbsolute- constantBoltzmannsec/103806.1 constantPlanksec106260.6 lightofspeedsec/458.299792 10439.1/ 107418.32 23 234 4 2 4242 1 −⋅⋅= −⋅⋅= −= ⋅⋅== ⋅⋅⋅== − − − µ µπ Plank’s Law 1900 MAX PLANCK 1858 - 1947 Nobel Prize 1918 Return to Table of Content
47 SOLO Particles J.J. Thomson showed in 1897 that the cathode rays are composed of electrons and he measured the ratio of charge to mass for the electron. In 1904 he suggested a model of the atom as a sphere of positive matter in which electrons are positioned by electrostatic forces. Thomson Atom Model 1904 -- -- -- -- -- -- -- -- -- -- Joseph John Thomson 1856 – 1940 Nobel Prize 1922 Plum Pudding Model Return to Table of Content
48 PhotoelectricitySOLO Einstein and Photoelectricity Albert Einstein explained the photoelectric effect discovered by Hertz in 1887 by assuming that the light is quantized (using Plank results) in quantities that later become known as photons. 1905 - - - - - - - - -- - - - - metallic surface ejected electrons incoming E.M. waves k E 0 ν ν 0 2 2 1 νν hhvmE ek −== The kinetic energy Ek of the ejected electron is: where: functionworksec frequencylight constantPlanksec106260.6 0 234 −⋅ − −⋅⋅= − Wh Hz Wh ν ν Albert Einstein 1879 - 1955 Nobel Prize 1921 To eject an electron the frequency of the incoming EM wave v must be above a threshold v0 (depends on metallic surface). Increasing the Intensity of the EM Wave will increase the number of electrons ejected, but not their energy. Return to Table of Content
1905 EINSTEIN’S SPECIAL THEORY OF RELATIVITY Special Relativity Theory 49
EINSTEIN’s SPECIAL THEORY OF RELATIVITY (Continue) First Postulate: It is impossible to measure or detect the Unaccelerated Translation Motion of a System through Free Space or through any Aether-like Medium. Second Postulate: Velocity of Light in Free Space, c, is the same for all Observers, independent of the Relative Velocity of the Source of Light and the Observers. Second Postulate (Advanced): Speed of Light represents the Maximum Speed of transmission of any Conventional Signal. Special Relativity Theory 50
51 SOLO x z y 'x 'z 'y v  'u 'OO 'u− A B Consequence of Special Theory of Relativity The relation between the mass m of a particle having a velocity u and its rest mass m0 is: 2 2 0 1 c u m m − = Special Relativity Theory EINSTEIN’s SPECIAL THEORY OF RELATIVITY (Continue) The Kinetic Energy of a free moving particle having a momentum p = m u, a velocity u and its rest mass m0 is: 42 0 222 cmcpT += The velocity of a photon is u = c, therefore, from the first equation, it has a rest mass 00 =photonm And has a Kinetic Energy and Total Energy of νhEcpT VTE V === += =0 Therefore if v is the photon Frequency and λ is photon Wavelength, we have cm h p hc cmph cp = = === ν ν λ
Locality and NonlocalitySOLO Event inside Light Cone EVENT HERE AND NOW Simultaneous Event at different place A Light Cone is the path that a flash of light, emanating from a single Event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through space-time. The Light Cone Equation is ( ) 022222 =−++ tczyx Events Inside the Light Cone ( ) 022222 <−++ tczyx Events Outside the Light Cone ( ) 022222 >−++ tczyx Einstein’s Theory of Special Relativity Postulates that no Signal can travel with a speed higher than the Speed of Light c. Thousands of experiments performed with Particles (Photons, Electrons. Neutrons,…) complied to this Postulate. However no experiments could be performed with Sub-particles, so, in my opinion the confirmation of this Postulate is still an open issue. Light Cone 52
Locality and NonlocalitySOLO Event inside Light Cone EVENT HERE AND NOW Simultaneous Event at different place According to Einstein only Events within Light Cone (shown in the Figure) can communicate with an event at the Origin, since only those Space-time points can be connected by a Signal traveling with the Speed of Light c or less. We call those Events “Local” although they may be separated in Space-time. Locality The Postulates of Relativity require that all frames of reference to be equivalent. So, if the Events are “Local” in any realizable frame of reference, they must be “Local” in all equivalent Frame of Reference. Two Space-time Points within Light Cone are called “timelike”. Nonlocality Two Space-time Points outside Light Cone are said to have “Spacelike Separation”. “Nonlocality” connected Points outside the Light Cone. They have Space-time separation. Simultaneously Events (Time = 0), in any given Reference Frame , cannot be causally connected unless the signal between them travels at superluminal speed. Some physicists use the term “Holistic” instead of “Nonlocal”. “Holistic” = “Nonlocal” 53 Return to Table of Content
54 SOLO 1908 Geiger-Marsden Experiment. Ernest Rutherford 1871 - 1937 Nobel Prize 1908 Chemistry Hans Wilhelm Geiger 1882 – 1945 Nazi Physicist Sir Ernest Marsden 1889 – 1970 Geiger-Marsden working with Ernest Rutherford performed in 1908 the alpha-particle scattering experiment. H. Geiger and E. Marsden (1909), “On a Diffuse Reflection of the α- particle”, Proceedings of the Royal Society Series A 82:495- 500 A small beam of α-particles was directed at a thin gold foil. According to J.J. Thomson atom-model it was anticipated that most of the α-particles would go straight through the gold foil, while the remainder would at most suffer only slight deflections. Geiger-Marsden were surprised to find out that, while most of the α-particles were not deviated, some were scattered through very large angles after passing the foil. QUANTUM THEORIES
55 ParticlesSOLO Electron Charge R.A. Millikan measured the charge of the electron by equalizing the weight m g of a charged oil drop with an electric field E. 1909 Robert Andrews Millikan 1868 – 1953 Nobel Prize 1923
56 SOLO Rutherford Atom Model 1911 Ernest Rutherford finds the first evidence of protons. To explain the Geiger-Marsden Experiment of 1908 he suggested in 1911 that the positively charged atomic nucleus contain protons. Ernest Rutherford 1871 - 1937 Nobel Prize 1908 Chemistry Hans Wilhelm Geiger 1882 – 1945 Nazi Physicist Sir Ernest Marsden 1889 – 1970 -- -- -- -- -- -- -- -- -- -- +2 +2 +2 Rutherford assumed that the atom model consists of a small nucleus, of positive charge, concentrated at the center, surrounded by a cloud of negative electrons. The positive α-particles that passed close to the positive nucleus were scattered because of the electrical repealing force between the positive charged α-particle and the nucleus . QUANTUM MECHANICS Return to Table of Content
57 1913 SOLO Niels Bohr presents his quantum model of the atom. Niels Bohr 1885 - 1962 Nobel Prize 1922 QUANTUM MECHANICS Bohr Quantum Model of the Atom.
58 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model In 1911, Bohr travelled to England. He met with J. J. Thomson of the Cavendish Laboratory and Trinity College, Cambridge, and New Zealand's Ernest Rutherford, whose 1911 Rutherford model of the atom had challenged Thomson's 1904 Plum Pudding Model.[ Bohr received an invitation from Rutherford to conduct post-doctoral work at Victoria University of Manchester. He adapted Rutherford's nuclear structure to Max Planck's quantum theory and so created his Bohr model of the atom.[ In 1885, Johan Balmer had come up with his Balmer series to describe the visible spectral lines of a hydrogen atoms: that was extended by Rydberg in 1887, to Additional series by Lyman (1906), Paschen (1908) ( )222 / nmmB −=λ 2 2 1 1 1 H i f R n nλ   = − ÷ ÷   Bohr Model of the Hydrogen Atom consists on a electron, of negative charge, orbiting a positive charge nucleus. The Forces acting on the orbiting electron are AttractionofForceticElectrosta r e F ForcelCentripeta r m F e c 2 0 2 2 4 v επ = = m – electron mass v – electron orbital velocity r – orbit radius e – electron charge ( )229 0 /109 4 1 coulombmN ⋅×= επ QUANTUM MECHANICS
59 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 1) The Conditions for Orbit Stability are 2 0 22 4 v r e r m FF ec επ = = rm e 04 v επ = The Total Energy E, of the Electron, is the sum of the Kinetic Energy T and the Potential Energy V r e r e r e r em VTE 0 2 0 2 0 2 0 22 84842 v επεπεπεπ −=−=−=+= To get some quantitative filing let use the fact that to separate the electron from the atom we need 13.6 eV (this is an experimental result), then E = -13.6 eV = 2.2x10-18 joule. Therefore ( ) ( ) ( ) m joule coulombmN coulomb E e r 11 18 229 219 0 2 103.5 102.2 /109 2 106.1 8 − − − ×= ×− ⋅× × −=−= επ QUANTUM MECHANICS
1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 2) The problem with this Model is, since the electron accelerates with a =v2 /r, according to Electromagnetic Theory it will radiate energy given by Larmor Formula (1897) ( ) sec/109.2sec/106.4 43 2 43 2 109 4233 0 6 3 0 22 evjoule rmc e c ae P ×=×=== − επεπ As the electron loses energy the Total Energy becomes more negative and the radius decreases, and since P is proportional to 1/r4 , the electron radiates energy faster and faster as it spirals toward the nucleus. Bohr had to add something to explain the stability of the orbits. He knew the results of the discrete Hydrogen Spectrum lines and the quantization of energy that Planck introduced in 1900 to obtain the Black Body Radiation Equation. Sir Joseph Larmor FRS (1857 – 1942) QUANTUM MECHANICS 60
1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 3) To understand Bohr novelty let look at an Elastic Wire that vibrates transversally. At Steady State the Wavelengths always fit an integral number of times into the Wire Length. This is true if we bend the Wire and even if we obtain a Closed Loop Wire. If the Wire is perfectly elastic the vibration will continue indefinitely. This is Resonance. Bohr noted that the Angular Momentum of the Orbiting electron in the Atom Hydrogen Model had the same dimensions as the Planck’s Constant. This led him to postulate that the Angular Momentum of the Orbiting Electrons must be multiple of Planck’s Constant divided by 2 π. ,3,2,1 24 v 0 === n h nr rm e mrm n n n πεπ ,3,2,12 0 22 == n em hn rn π ε therefore QUANTUM MECHANICS 61
1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 4) Energy Levels and Spectra We obtained ,3,2,1 1 88 222 0 4 0 2 =      −=−= n nh em r e E n n εεπ ,3,2,12 0 22 == n em hn rn π ε and Energy Levels: The Energy Levels are all negative signifying that the electron does not have enough energy to escape from the atom. The lowest energy level E1 is called the Ground State. The higher levels E2, E3, E4,…, are called Excited States. In the limit n →∞, E∞=0 and the electron is no longer bound to the nucleus to form an atom. QUANTUM MECHANICS 62
63 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 5) According to the Bohr Hydrogen Model when the electron is excited he drops to a lower state, and a single photon of light is emitted Initial Energy – Final Energy = Photon Energy vh nnh em nh em nh em EE iffi fi =         −=         +        −=− 2222 0 4 222 0 4 222 0 4 11 8 1 8 1 8 εεε where v is the photon frequency. If λ is the Wavelength of the photon we have         −=         −= − == 222232 0 4 1111 8 1 if H if fi nn R nnch em ch EE c v ελ 2in = Balmer series (1885) 3in = Paschen series (1908) 4in = Brackett series (1922) Lyman series (1906)1in = We recovered the Rydberg Formula (1887) ( ) ( ) ( ) 17 3348212 41931 32 0 4 10097.1 sec1063.6/103/1085.88 106.1101.9 8 − −− −− ×= −×××××× ××× = m joulesmmfarad coulombkg ch em ε QUANTUM MECHANICS
64 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 6) 2. The Bohr model treats the electron as if it were a miniature planet, with definite radius and momentum. This is in direct violation of the uncertainty principle (formulated by Werner Heisenberg in 1927) which dictates that position and momentum cannot be simultaneously determined. 1. It fails to provide any understanding of why certain spectral lines are brighter than others. There is no mechanism for the calculation of transition probabilities. While the Bohr model was a major step toward understanding the quantum theory of the atom, it is not in fact a correct description of the nature of electron orbits. Some of the shortcomings of the model are: The electrons in free atoms can will be found in only certain discrete energy states. These sharp energy states are associated with the orbits or shells of electrons in an atom, e.g., a hydrogen atom. One of the implications of these quantized energy states is that only certain photon energies are allowed when electrons jump down from higher levels to lower levels, producing the hydrogen spectrum. The electron must jump instantaneously because if he moves gradually it will radiate and lose energy in the process. The Bohr model successfully predicted the energies for the hydrogen atom, but had significant failures. Quantized Energy States QUANTUM MECHANICS Return to Table of Content
1915Einstein’s General Theory of Relativity The “General” Theory of Relativity takes in consideration the action of Gravity and does not assume Unaccelerated Observer like “Special” Theory of Relativity. Principle of Equivalence – The Inertial Mass and the Gravitational Mass of the same body are always equal. (checked by experiments first performed by Eötvos in 1890) Principle of Covariance -- The General Laws of Physics can be expressed in a form that is independent of the choise of the coordinate system. Principle of Mach -- The Gravitation Field and Metric (Space Curvature) depend on the distribution of Matter and Energy. SOLO GENERAL RELATIVITY Dissatisfied with the Nonlocality (Action at a Distance) of Newton’s Law of GravityEinstein developed the General Theory of Gravity. Albert Einstein 1879 - 1955 Nobel Prize 1921 65
GENERAL RELATIVITY Einstein’s General Theory Equation  TENSOR MOMENTUMENERGY CURVATURETIMESPACE TG c RgR − − =− µνµνµν π 2 8 2 1  The Matter – Energy Distribution produces the Bending (Curvature) of the Space-Time. All Masses are moving on the Shortest Path (Geodesic) of the Curved Space-Time. In the limit (Weak Gravitation Fields) this Equation reduce to the Poisson’s Equation of Newton’s Gravitation Law SOLO 66
SOLO GENERAL RELATIVITY 67
GENERAL RELATIVITY Einstein’s General Theory of Relativity (Summary) • Gravity is Geometry • Mass Curves Space – Time • Free Mass moves on the Shortest Path in Curved Space – Time. SOLO Newton’s Gravity The Earth travels around the Sun because it is pulled by the Gravitational Force exerted by the Mass of the Sun. Mass (somehow) causes a Gravitational Force which propagates instantaneously (Action at a Distance) and causes True Acceleration. Einstein’s Gravity The Earth travels around the Sun because is the Shortest Path in the Curved Space – Time produced by the Mass of the Sun. Mass (somehow) causes a Warping, which propagates with the Speed of Light, and results in Apparent Acceleration. 68 Return to Table of Content
69 Photons EmissionSOLO Theory of Light Emission. Concept of Stimulated Emission 1916 Albert Einstein 1879 - 1955 Nobel Prize 1921 http://members.aol.com/WSRNet/tut/ut4.htm Spontaneous Emission & Absorption Stimulated Emission & Absorption “On the Quantum Mechanics of Radiation” Run This Einstein’s work laid the foundation of the Theory of LASER (Light Amplification by Stimulated Emission) Return to Table of Content
E. RUTHERFORD OTTO STERN W. GERLACH A. COMPTON L. de BROGLIE W. PAULI QUANTUM MECHANICS 1919: ERNEST RUTHERFORD FINDS THE FIRST EVIDENCE OF PROTONS. HE SUGGESTED IN 1914 THAT THE POSITIVELY CHARGED ATOMIC NUCLEUS CONTAINS PROTONS. 1922: OTTO STERN AND WALTER GERLACH SHOW “SPACE QUANTIZATION” 1923: ARTHUR COMPTON DISCOVERS THE QUANTUM NATURE OF X RAYS, THUS CONFIRMS PHOTONS AS PARTICLES. 1924: LOUIS DE BROGLIE PROPOSES THAT MATTER HAS WAVE PROPERTIES. 1924: WOLFGANG PAULI STATES THE QUANTUM EXCLUSION PRINCIPLE. 70
W. HEISENBERG MAX BORN P. JORDAN S. GOUDSMITH G. UHLENBECK E. SCHRODINGER QUANTUM MECHANICS 1925: WERNER HEISENBERG, MAX BORN, AND PASCAL JORDAN FORMULATE QUANTUM MATRIX MECHANICS. 1925: SAMUEL A. GOUDSMITH AND GEORGE E. UHLENBECK POSTULATE THE ELECTRON SPIN 1926: ERWIN SCHRODINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 71
W. HEISENBERGMAX BORN PAUL DIRAC J. von NEUMANN QUANTUM MECHANICS 1926: MAX BORN GIVES A PROBABILISTIC INTERPRATATION OF THE WAVE FUNCTION. 1927: WERNER HEISENBERG STATES THE QUANTUM UNCERTAINTY PRINCIPLE. 1928: PAUL DIRAC STATES HIS RELATIVISTIC QUANTUM WAVE EQUATION. HE PREDICTS THE EXISTENCE OF THE POSITRON. 1932: JHON von NEUMANN WROTE “THE FOUNDATION OF QUANTUM MECHANICS” 72 Return to Table of Content
73 SOLO 1922 Otto Stern and Walter Gerlach show “Space Quantization” Walter Gerlach 1889 - 1979 They designed the Stern-Gerlach Experiment that determine if a particle has angular momentum. http://en.wikipedia.org/wiki/Stern-Gerlach_experiment Otto Stern 1888 – 1969 Nobel Prize 1943 They directed a beam of neutral silver atoms from an oven trough a set of collimating slits into an inhomogeneous magnetic field. A photographic plate recorded the configuration of the beam. They found that the beam split into two parts, corresponding to the two opposite spin orientations, that are permitted by space quantization. Run This QUANTUM MECHANICS
74 SOLO 1923 Arthur Compton discovers the quantum nature of X rays, thus confirms photons as particles. Arthur Holly Compton 1892 - 1962 Nobel Prize 1927 incident photon ( ) ( ) chp hE photon photon /ν ν =− =− ( ) ( ) 0 2 0 =− =− electron electron p cmE target electron Compton effect consists of a X ray (incident photons) colliding with rest electrons incident photon scatteredphoton ( ) ( ) chp hE photon photon /ν ν =− =− ( ) ( ) 0 2 0 =− =− electron electron p cmE ( ) ( ) chp hE photon photon /' ' ν ν =+ =+ ( ) ( ) ( ) ( )' 2 2 0 2 2242 0 νν −= +=+ −+=+ hT TcmTp cpcmE electron photonelectron ϕ θ ( )ϕ νν λλ cos1 ' ' 0 −=−=− cm hcc scatteredelectron target electron is scattered in the φ direction (detected by an X-ray spectrometer) and the electrons in the θ direction. Run This QUANTUM MECHANICS Return to Table of Content
75 SOLO 1924 Louis de Broglie proposes that matter has wave properties and using the relation between Wavelength and Photon mass: Louis de Broglie 1892 - 1987 Nobel Prize 1929 cm h p hc cmph cp = = === ν ν λ He postulate that any Particle of mass m and velocity v has an associate Wave with a Wavelength λ. QUANTUM MECHANICS
SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Model using de Broglie Relation To understand Bohr novelty let look at an Elastic Wire that vibrates transversally. At Steady State the Wavelengths always fit an integral number of times into the Wire Length. This is true if we bend the Wire and even if we obtain a Closed Loop Wire. If the Wire is perfectly elastic the vibration will continue indefinitely. This is Resonance. ,3,2,12 4 0 === nr m r e h nn n n π επ λ ,3,2,12 0 22 == n em hn rn π ε We found the Electron Orbital Velocity Return to Bohr Hydrogen Model using de Broglie Relation Louis de Broglie 1892 - 1987 Nobel Prize 1929 rm e 04 v επ = Using de Broglie Relation m r e h m h 04 v επ λ == At Steady State the Wavelengths always fit an integral number of times into the Wire Length. We obtain the same relation as Bohr for the Orbit radius: QUANTUM MECHANICS 76 Return to Table of Content
77 SOLO 1924 Wolfgang Pauli states the “Quantum Exclusion Principle” Wolfgang Pauli 1900 - 1958 Nobel Prize 1945 QUANTUM MECHANICS Return to Table of Content
QUANTUM THEORIES Werner Heisenberg, Max Born, and Pascal Jordan formulate Quantum Matrix Mechanics. QUANTUM MATRIX MECHANICS. Werner Karl Heisenberg (1901 – 1976) Nobel Price 1932 Max Born (1882–1970) Nobel Price 1954 Ernst Pascual Jordan (1902 – 1980) Nazi Physicist http://en.wikipedia.org/wiki/Matrix_mechanics 1925 Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps occur. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, and is the basis of Dirac's bra-ket notation for the wave function. SOLO In 1928, Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics, but Heisenberg alone won the 1932 Prize "for the creation of quantum mechanics, the application of which has led to the discovery of the allotropic forms of hydrogen",[47] while Schrödinger and Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory".[47] On 25 November 1933, Born received a letter from Heisenberg in which he said he had been delayed in writing due to a "bad conscience" that he alone had received the Prize "for work done in Gottingen in collaboration — you, Jordan and I."[48] Heisenberg went on to say that Born and Jordan's contribution to quantum mechanics cannot be changed by "a wrong decision from the outside." 78 Return to Table of Content
1925 SAMUEL A. GOUDSMITH AND GEORGE E. UHLENBECK POSTULATE THE ELECTRON SPIN George Eugene Uhlenbeck (1900 – 1988) Samuel Abraham Goudsmit (1902 – 1978) Two types of experimental evidence which arose in the 1920s suggested an additional property of the electron. One was the closely spaced splitting of the hydrogen spectral lines, called fine structure. The other was the Stern-Gerlach experiment which showed in 1922 that a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into two beams. Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. Classically this could occur if the electron were a spinning ball of charge, and this property was called electron spin. In 1925, the Dutch Physicists S.A. Goudsmith and G.E. Uhlenbeck realized that the experiments can be explained if the electron has an magnetic property of Rotation or Spin. They work actually showed that the electron has a quantum-mechanical notion of spin that is similar to the mechanical rotation of particles. http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html no magnetic field B = 0 cba ,, fed ,, a b c d e f magnetic field B 0≠ Zeeman’s Effect QUANTUM MECHANICS 79
Spin In quantum mechanics and particle physics, Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. Spin is a solely quantum-mechanical phenomenon; it does not have a counterpart in classical mechanics (despite the term spin being reminiscent of classical phenomena such as a planet spinning on its axis).[ Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. Orbital angular momentum is the quantum- mechanical counterpart to the classical notion of angular momentum: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus).The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.[ http://en.wikipedia.org/wiki/Spin_(physics) In some ways, spin is like a vector quantity; it has a definite “magnitude”; and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number.[2] However, in a technical sense, spins are not strictly vectors, and they are instead described as a related quantity: a Spinor. In particular, unlike a Euclidean vector, a spin when rotated by 360 degrees can have its sign reversed QUANTUM MECHANICS 80
ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 In January 1926, Schrödinger published in Annalen der Physik the paper "Quantisierung als Eigenwertproblem" [“Quantization as an Eigenvalue Problem”] on wave mechanics and presented what is now known as the Schrödinger equation. In this paper, he gave a "derivation" of the wave equation for time-independent systems and showed that it gave the correct energy eigenvalues for a hydrogen-like atom. This paper has been universally celebrated as one of the most important achievements of the twentieth century and created a revolution in quantum mechanics and indeed of all physics and chemistry. A second paper was submitted just four weeks later that solved the quantum harmonic oscillator, rigid rotor, and diatomic molecule problems and gave a new derivation of the Schrödinger equation. A third paper in May showed the equivalence of his approach to that of Heisenberg. http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 SOLO 81
MAX BORN GIVES A PROBABILISTIC INTERPRATATION OF THE WAVE FUNCTION. 1926 Max Born (1882–1970) Nobel Price 1954 Max Born wrote in 1926 a short paper on collisions between particles, about the same time as Schrödinger paper “Quantization as an Eigenvalue Problem”. Born rejected the Schrödinger Wave Field approach. He had been influenced by a suggestion made by Einstein that, for photons, the Wave Field acts as strange kind of ‘phantom’ Field ‘guiding’ the photon particles on paths which could therefore be determined by Wave Interference Effects. Max Born reasoned that the Square of the Amplitude of the Waveform in some specific region of configuration space is related to the Probability of finding the associated quantum particle in that region of configuration space. Since Probability is a real number, and the integral of all Probabilities over all regions of configuration space, the Wave Function must satisfy 1* =∫ +∞ ∞− dVψψ Condition of Normalization of the Wave Function Therefore the probability of finding the particle between a and b is given by [ ] ( ) ( )∫=≤≤ b a xdxxbXaP ψψ * Einstein rejected this interpretation. In a 1926 letter to Max Born, Einstein wrote: "I, at any rate, am convinced that He [God] does not throw dice."[ QUANTUM MECHANICS SOLO 82 Return to Table of Content
QUANTUM MECHANICS In December 1926 Einstein wrote a letter to Bohr which contains a phrase that has since become symbolic of Einstein’s lasting dislike of the element of chance implied by the quantum theory: J. Baggott, “The Meaning of Quantum Theory”, Oxford University Press, 1992, pp. 28-29 SOLO 1926 http://en.wikipedia.org/wiki/Max_Born “Quantum mechanics is very impressive. But an inner voice tells me that it is not the real thing. The theory produce a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.” 83
84 SOLO Wavelike Behavior for Electrons In 1927, the wavelike behavior of the electrons was demonstrated by Davisson and Germer in USA and by G.P. Thomson in Scotland. Quantum 1927 Clinton Joseph Davisson 1881 – 1958 Nobel Prize 1937 Lester Halbert Germer 1896 - 1971
85 SOLO Wavelike Behavior for Electrons Quantum 1927 G.P. Thomson carried a series of experiments using an apparatus called an electron diffraction camera. With it he bombarded very thin metal and celluloid foils with a narrow electron beam. The beam then was scattered into a series of rings. George Paget Thomson 1892 – 1975 Nobel Prize 1937 Using these results G.P. Thomson proved mathematically that the electron particles acted like waves, for which he received the Nobel Prize in 1937. J.J. Thomson the father of G.P. proved that the electron is a particle in 1897, for which he received the Nobel Prize in 1906. Discovery of the Electron Results of a double-slit- experiment performed by Dr. Tonomura showing the build-up of an interference pattern of single electrons. Numbers of electrons are 11 (a), 200 (b), 6000 (c), 40000 (d), 140000(e).
86 SOLO Optics HistoryRaman Effect 1928 http://en.wikipedia.org/wiki/Raman_scattering http://en.wikipedia.org/wiki/Chandrasekhara_Venkata_Raman Nobel Prize 1930 Chandrasekhara Venkata Raman 1888 – 1970 Raman Effect was discovered in 1928 by C.V. Raman in collaboration with K.S. Krishnan and independently by Grigory Landsberg and Leonid Mandelstam. Monochromatic light is scattered when hitting molecules. The spectral analysis of the scattered light shows an intense spectral line matching the wavelength of the light source (Rayleigh or elastic scattering). Additional, weaker lines are observed at wavelength which are shifted compared to the wavelength of the light source. These are the Raman lines. Virtual Energy States IR Absorbance Excitation Energy Rayleigh Scattering Stokes - Raman Scattering Anti-Stokes - Raman Scattering
87 SOLO Stimulated Emission and Negative Absorption 1928 Rudolph W. Landenburg confirmed existence of stimulated emission and Negative Absorption Lasers History Rudolf Walter Ladenburg (June 6, 1882 – April 6, 1952) was a German atomic physicist. He emigrated from Germany as early as 1932 and became a Brackett Research Professor at Princeton University. When the wave of German emigration began in 1933, he was the principal coordinator for job placement of exiled physicist in the United States. Albert Einstein and Rudolf Ladenburg, Princeton Symposium, on the occasion of Ladenburg's retirement, May 28, 1950. Hedwig Kohn is in the background on the left. Photo courtesy of AIP Emilio Segrè Visual Archives. Return to Table of Content
QUANTUM MECHANICS SOLO Wave Packet A wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere Depending on the evolution equation, the wave packet's envelope may remain constant (no dispersion, see figure) or it may change (dispersion) while propagating. As an example of propagation without dispersion, consider wave solutions to the following wave equation: ψ ψ 2 2 2 2 v 1 ∇= ∂ ∂ t where v is the speed of the wave's propagation in a given medium. The wave equation has plane-wave solutions ( ) ( )trki eAtr ω ψ −⋅ =   , ( ) v,/1111 2222 kkkkknPropagatioWaveofDirectionkzkykxkk zyxzyx =++=++= ω A wave packet without dispersion A wave packet with dispersion ( ) ( ) ( )tcxiktcx etx −+−− = 0 2 ,ψ 88 Run This
QUANTUM MECHANICS SOLO Wave Packet The wave equation has plane-wave solutions ( ) ( )rkti eAtr   ⋅−− = ω ψ , ( ) v,/1111 2222 kkkkknPropagatioWaveofDirectionkzkykxkk zyxzyx =++=++= ω ( )rptE h r k k ktE h rkt hv    ⋅− / =⋅−=⋅− = = 122 /E π ω νπω ( ) p h p h v k hhphv / ===== =/== 122 v 2 v 2/://v πλλ π λ ππω ( ) ( ) ( ) ( )rptEhirkti eAeAtr   ⋅−/−⋅−− == / , ω ψ where v is the velocity , v is the frequency, λ is the Wavelength of the Wave Packet. The Energy E and Momentum p of the Particle are ( ) ( ) λ π λ νπν ππ hh phhE hhhh / ==/== =/=/ 2 &2 2/:2/: de Broglie RelationEinstein Relation The wave packet travels to the direction for ω = kv and to direction for ω = - kv.k1 k1− 89
QUANTUM MECHANICS SOLO Wave Packet A wave packet is a localized disturbance that results from the sum of many different wave forms. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. From the basic solutions in one dimension, a general form of a wave packet can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) ( ) tEhirptEhi erepApd h tr //− +∞ ∞− ⋅−//− = / = ∫ //3 3 2 1 ,   ψ π ψ ( ) ( ) ( ) ( ) ( )perrd h pA rphi   Φ= / = ∫ +∞ ∞− ⋅/− :0, 2 1 /3 3 ψ π The factor comes from Fourier Transform conventions. The amplitude contains the coefficients of the linear superposition of the plane-wave solutions. Using the Inverse Fourier Transform we obtain: ( )3 2/1 π ( )pA  ( ) ( ) ( ) ( ) ∫ +∞ ∞− ⋅// / = rphi epApd h r  /3 3 2 1 π ψwhere zyx pdpdpdpd =3 dzdydxrd =3 Define ( ) ( ) ( ) ( ) ∫ +∞ ∞− ⋅//− / =Φ rphi etrrd h tp  /3 3 , 2 1 :, ψ π Wave Function in Momentum Space 90 Return to Table of Content
ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 Following Max Planck's quantization of light (see black body radiation), Albert Einstein interpreted Planck's quanta to be photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, one of the first signs of wave–particle duality. Since energy and momentum are related in the same way as frequency and wavenumber in special relativity, it followed that the momentum p of a photon is proportional to its wavenumber k. c k h hwherekh h p c πν λ π πλ νλ 22 :, 2 : /= ===//== Louis de Broglie hypothesized that this is true for all particles, even particles such as electrons. He showed that, assuming that the matter waves propagate along with their particle counterparts, electrons form standing waves, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.[7] These quantized orbits correspond to discrete energy levels, and de Broglie reproduced the Bohr model formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum: hn h nL /== π2 According to de Broglie the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit: n λ = 2 π r http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Historical Background and Development QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 SOLO 91
ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 ( ) ( ) ( )λνπ νπω νλ ω ψ /2 2 /v v/ , xtixti eAeAtx −− = = −− == http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Historical Background and Development (continue – 1) Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by William R. Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system — the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action. For a general form of a Progressive Wave Function in + x direction with velocity v and frequency v: The Energy E and Momentum p of the Particle are λ π λ νπν hh phhE / ==/== 2 2 ( ) ( ) ( )xptEhi eAtx −/− = / ,ψTherefore QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 SOLO 92
ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Historical Background and Development (continue – 2) We want to find the Differential Equation yielding the Wave Function . We have Wave Function: ( ) ( ) ψ ψ 2 2 / 2 2 2 2 h p eA h p x xptEhi / −= / −= ∂ ∂ −/− At particle speeds small compared to speed of light c, the Total Energy E is the sum of the Kinetic Energy p2 /2m and the Potential Energy V (function of position and time): ψψψ ψ V m p EV m p E +=⇒+= × 22 22 2 2 22 x hp ∂ ∂ /−= ψ ψ ti h E ∂ ∂/ −= ψ ψ cV xm h ti h <<− ∂ ∂/ = ∂ ∂/ v 2 2 22 ψ ψψ QUANTUM MECHANICS SOLO ( ) ( ) ( )xptEhi eAtx −/− = / ,ψ 93
ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Historical Background and Development (continue – 3) cV xm h ti h <<− ∂ ∂/ = ∂ ∂/ v 2 2 22 ψ ψψ Non-Relativistic One-Dimensional Time Dependent Schrödinger Equation In the same way cV m h ti h <<−∇ / = ∂ ∂/ v 2 2 2 ψψ ψ Non-Relativistic Three-Dimensional Time Dependent Schrödinger Equation QUANTUM MECHANICS SOLO This is a Linear Partial Differential Equation. It is also a Diffusion Equation (with an Imaginary Diffusion Coefficient), but unlike the Heat Equation, this one is also a Wave Equation given the imaginary unit present in the transient term. 94
1926 Schrödinger Equation Time-dependent Schrödinger equation (single non-relativistic particle) A wave function that satisfies the non-relativistic Schrödinger equation with V=0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary". QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 SOLO 95
Schrödinger Equation: Steady State Form Using ti h E ∂ ∂/ −= ψ ψ and the Time-dependent Schrödinger equations cV xm h ti h <<− ∂ ∂/ = ∂ ∂/ v 2 2 22 ψ ψψ Non-Relativistic One-Dimensional Time Dependent Schrödinger Equation cV m h ti h <<−∇ / = ∂ ∂/ v 2 2 2 ψψ ψ Non-Relativistic Three-Dimensional Time Dependent Schrödinger Equationwe can write ( ) cVE h m x <<=− / + ∂ ∂ v0 2 22 2 ψ ψ Non-Relativistic One-Dimensional Steady-State Schrödinger Equation ( ) cVE h m <<− / +∇ v 2 2 2 ψψ Non-Relativistic Three-Dimensional Steady-State Schrödinger Equation QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 1926 SOLO 96 Return to Table of Content
Operators in Quantum Mechanics Since, according to Born, ψ*ψ represents Probability of finding the associated quantum particle in a region we can compute the Expectation (Mean) Value of the Total Energy E and of the Momentum p in that region using ( ) ( ) ( ) ( )∫ +∞ ∞− = xdtxtxEtxtE ,,,* ψψ ( ) ( ) ( ) ( )∫ +∞ ∞− = dxtxtxptxtp ,,,* ψψ But those integrals can not compute exactly, since p (x,t) is unknown if x is know, according to Uncertainty Principle. A way to find is by differentiating the Free-Particle Wave Function pandE ( ) ( )xptEhi eA −/− = / ψ ( ) ( ) ( ) ( ) ψ ψ ψ ψ E h i eAE h i t p h i eAp h i x xptEhi xptEhi / −= / −= ∂ ∂ / = / = ∂ ∂ −/− −/− / / Rearranging we obtain ψψ ψψ t hiE xi h p ∂ ∂ /= ∂ ∂/ = t hiE xi h p ∂ ∂ /= ∂ ∂/ = :ˆ :ˆ QUANTUM MECHANICS SOLO We can look at p and E as Operators on ψ (the symbol means “Operator”)∧ Note: One other way to arrive to this result by manipulating the integrals will be given in the following presentations. 97
Operators in Quantum Mechanics (continue – 1) We obtained Moment Operatorxi h p ∂ ∂/ =:ˆ t hiE ∂ ∂ /=:ˆ Total Energy Operator Although we derived those operators for free particles, they are entire general results, equivalent to Schrödinger Equation. To see this let write the Operator Equation    Operator Energy Potential Operator Energy Kinetic Operator Energy Total VTE ˆˆˆ +=  2 2222 22 1 2 ˆ xm h xi h mm p T Operator Energy Kinetic ∂ ∂/ −=      ∂ ∂/ ==We have  V xm h t hiE Operator Energy Total + ∂ ∂/ −= ∂ ∂ /= 2 22 2 ˆ Applying this Operator on Wave Function ψ we recover the Schrödinger Equation ψ ψψ V xm h t hi + ∂ ∂/ −= ∂ ∂ / 2 22 2 The two descriptions (Operator and Schrödinger’s) are equivalent. QUANTUM MECHANICS SOLO 98
QUANTUM MECHANICS Operators in Quantum Mechanics (continue – 3) We obtained Moment Operatorxi h p ∂ ∂/ =:ˆ t hiE ∂ ∂ /=:ˆ Total Energy Operator Because p and E can be replaced by their Operators in an equation, we can use those Operators to obtain Expectation Values for p and E. ∫∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− ∂ ∂/ =      ∂ ∂/ == dx xi h dx xi h dxpp ψ ψψψψψ *** ˆ ∫∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− ∂ ∂ /=      ∂ ∂ /== xd x hixd x hixdEE ψ ψψψψψ *** ˆ Let define the Hamiltonian Operator V xm h H ˆ 2 :ˆ 2 22 + ∂ ∂/ −= Schrödinger Equation in Operator form is ψψ EH ˆˆ = This Equation has a form of an Eigenvalue Equation of the Operator with Eigenvalue Ê and Eigenfunction as the Wavefunction ψ. Hˆ SOLO 99
Dirac bracket notation Paul Adrien Maurice Dirac ( 1902 –1984) A elegant shorthand notation for the integrals used to define Operators was introduced by Dirac in 1939 onWavefunctiket nn ψψ ⇔"" Instead of dealing with Wavefunctions ψn, we defined a related Quantum “State”, denoted |ψ› which is called a “ket”, “ket vector”, “state” or “state vector”. The complex conjugate of |ψ› is called the “bra” and is denoted by ‹ψ|. onWavefunctibra nn * "" ψψ ⇔   ket m bra n ψψ When a “bra” is combined with a “ket” we obtain a “bracket”. The following integrals are represented by “bra” and “ket” mnmn AdA ψψτψψ |ˆ|ˆ* ≡∫ mnmn d ψψτψψ |* ≡∫ nnnnnn aAaA ψψψψ =⇔= ˆˆ Operators in Quantum Mechanics (continue – 5) ( ) ( ) ( ) ( ) mnmnmnmnmn AAdAAdA ψψψψτψψψψτψψ |ˆ|ˆ|ˆ|ˆˆ *** ==== ∫∫ nnnnnn aAaA ψψψψ ****** ˆˆ =⇔= QUANTUM MECHANICS SOLO 100 Return to Table of Content
QUANTUM THEORIES HILBERT SPACE AND QUANTUM MECHANICS. Ernst Pascual Jordan (1902 – 1980) Nazi Physicist http://en.wikipedia.org/wiki/Matrix_mechanics Born had also learned Hilbert’s theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of Hilbert’s work “Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen” published in 1912. Jordan, too was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert’s book Methoden der mathematischen Physik I, which was published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of quantum mechanics. In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert Space to describe the algebra and analysis which were used in the development of quantum mechanics Max Born (1882–1970) Nobel Price 1954 John von Neumann (1903 –1957) David Hilbert (1862 –1943) Richard Courant (1888 –1972) SOLO 101
102 Functional AnalysisSOLO Vector (Linear) Space: E is a Linear Space if Addition and Scalar Multiplication are Defined Addition Scalar Multiplication From those equations follows: The null element 0 ∈ E is unique. The addition inverse |η› of |ψ›, (|ψ›+|η›= 0) is unique. E∈∀=⋅ ψψ 00 |η› = (-1) |ψ› is the multiplication inverse of |ψ›. αβ −= E∈∀+=+ χψψχχψ ,1 Commutativity ψψ +=+∈∃ 00..0 tsE3 Identity 0.. =+∈∃∈∀ χψχψ tsEE4 Inverse E∈∀=⋅ ψψψ15 Normalization ( ) ( ) βαψψβαψβα ,& ∀∈∀= E6 Associativity 8 ( ) αηψηαψαηψα ∀∈∀+=+ &, E Distributivity 7 ( ) βαψψβψαψβα ,& ∀∈∀+=+ E Distributivity 2 Associativity( ) ( ) E∈∀++=++ ηχψηψχηχψ ,, The same apply for “bra” ‹ψ| the “conjugate” of the “ket” |ψ›. See also “Functional Analysis ” Presentation for a detailed description
103 Functional AnalysisSOLO Vector (Linear) Space: E is a Linear Space if Addition and Scalar Multiplication are Defined Linear Independence, Dimensionality and Bases    ∈≠ = ⇒=∑= CsomefortrueifDependentLinear allifonlytrueiftIndependenLinear i in i ii 0 0 01 α α ψα A set of vectors |ψi› (i=1,…,n) that satisfy the relation Dimension of a Vector Space E , is the maximum number N of Linear Independent Vectors in this space. Thus, between any set of more that N Vectors |ψi› (i=1,2,…,n>N), there exist a relation of Linear Dependency. Any set of N Linearly Independent Vectors |ψi› (i=1,2,…,N), form a Basis of the Vector Space E ,of Dimension N, meaning that any vector |η› ∈ E can be written as a Linear Combination of those Vectors. Ci N i ii ∈≠= ∑= αψαη 01 In the case of an Infinite Dimensional Space (N→∞), the space will be defined by a “Complete Set” of Basis Vectors. This is a Set of Linearly Independent Vectors of the Space, such that if any other Vector of the Space is added to the set, there will exist a relation of Linear Dependency to the Basis Vectors.
SOLO Functional Analysis Use of bra-ket notation of Dirac for Vectors. ketbra TransposeConjugateComplexHfefeefef HH − =⋅==⋅= ,| operatorkete operatorbraf | | Paul Adrien Maurice Dirac (1902 – 1984) Assume the are a basis and the a reciprocal basis for the Hilbert space. The relation between the basis and the reciprocal basis is described, in part, by: je| |if ketbra ji ji efef jij H iji −    = ≠ === 1 0 | ,δ 104 The Inner Product of the Vectors f and e is defined as Inner Product Using Dirac Notation ( ) ( )** & ψψψψ == To every “ket” corresponds a “bra”.
105 Functional AnalysisSOLO Inner Product Using Dirac Notation If E is a complex Linear Space, for the Inner Product (bracket) < | > between the elements (a complex number) is defined by: E∈∀ 321 ,, ψψψ * 1221 || >>=<< ψψψψ1 Commutative Law Using to we can show that:1 4 If E is an Inner Product Space, than we can induce the Norm: [ ] 2/1 111 , ><= ψψψ 2 Distributive Law><+>>=<+< 3121321 ||| ψψψψψψψ 3 C∈><>=< αψψαψψα 2121 || 4 00|&0| 11111 =⇔>=<≥>< ψψψψψ ( ) ( ) ( ) ><+><=><+><=>+<=>+< 1312 1 * 31 * 21 2 * 321 1 132 |||||| ψψψψψψψψψψψψψψ ( ) ( ) ( ) ><=><=><=>< 21 * 1 * 12 * 3 * 12 1 21 |||| ψψαψψαψψαψαψ ( ) ( ) * 1 1 111 2 11 |000|0|0|00|0| ><=>=<⇒><+><=>+>=<< ψψψψψψ
106 Functional AnalysisSOLO Inner Product ηψηψ ≤>< | Cauchy, Bunyakovsky, Schwarz Inequality known as Schwarz Inequality Let |ψ›, |η› be the elements of an Inner Product space E, than : x y ><= >< y y x y yx , , y y y y xxy y yx x ><−= >< − , , 2 0||||| 2* ≥><+><+><+>>=<++< ηηλψηληψλψψηλψηλψ Assuming that , we choose:0| 2/1 ≠= ηηη >< >< −= ηη ηψ λ | | we have: 0| | | | || | || | 2 2* ≥>< >< >< + >< ><>< − >< ><>< −>< ηη ηη ηψ ηη ψηηψ ηη ηψηψ ψψ which reduce to: 0 | | | | | | | 222 ≥ >< >< + >< >< − >< >< −>< ηη ηψ ηη ηψ ηη ηψ ψψ or: ><≥⇔≥><−><>< ηψηψηψηηψψ |0||| 2 q.e.d. Augustin Louis Cauchy )1789-1857( Viktor Yakovlevich Bunyakovsky 1804 - 1889 Hermann Amandus Schwarz 1843 - 1921 Proof:
107 Functional Analysis SOLO Hilbert Space A Complete Space E is a Metric Space (in our case ) in which every Cauchy Sequence converge to a limit inside E. ( ) 2121, ψψψψρ −= David Hilbert 1862 - 1943 A Linear Space E is called a Hilbert Space if E is an Inner Product Space that is complete with respect to the Norm induced by the Inner Product ||ψ1||=[< ψ1, ψ1>]1/2 . Equivalently, a Hilbert Space is a Banach Space (Complete Metric Space) whose Norm is induced by the Inner Product ||ψ1||=[< ψ1, ψ1>]1/2 . Orthogonal Vectors in a Hilbert Space: Two Vectors |η› and |ψ› are Orthogonal if 0|| == ηψψη Theorem: Given a Set of Linearly Independent Vectors in a Hilbert Space |ψi› (i=1,…,n) and any Vector |ψm› Orthogonal to all |ψi›, than it is also Linearly Independent. Proof: Suppose that the Vector |ψm› is Linearly Dependent on |ψi› (i=1,…,n) ∑= =≠ n i iim 1 0 ψαψ But ∑= ==≠ n i imimm 1 0 00  ψψαψψ We obtain a inconsistency, therefore |ψm› is Linearly Independent on |ψi› (i=1,…,n) Therefore in a Hilbert Space, of Finite or Infinite Dimension, by finding the Maximum Set of Orthogonal Vectors we find a Basis that “Complete” covers the Space. q.e.d.
108 Functional Analysis SOLO Hilbert Space Orthonormal Sets Let |ψ1›, |ψ2›, ,…, |ψn›, denote a set of elements in the Hilbert Space H. ( )             ><><>< ><><>< ><><>< = nnnn n n nG ψψψψψψ ψψψψψψ ψψψψψψ ψψψ ,,, ,,, ,,, :,,, 21 22212 12111 21      Jorgen Gram 1850 - 1916 Define the Gram Matrix of the set: Theorem: A set of functions |ψ1›, |ψ2›, ,…, |ψn› of the Hilbert Space H is linearly dependent if and only if the Gram determinant of the set is zero. zeroequalallnot inn αψαψαψα 02211 =+++ Proof: Linearly Dependent Set: Multiplying (inner product) this equation consecutively by |ψ1›, |ψ2›, ,…, |ψn›, we obtain: ( ) 0,,,det 0 0 0 ,,, ,,, ,,, 21 2 1 21 22212 12111 =⇔             =                         ><><>< ><><>< ><><>< n Solution nontrivial nnnnn n n G ψψψ α α α ψψψψψψ ψψψψψψ ψψψψψψ       q.e.d.
109 Functional Analysis SOLO Hilbert Space Orthonormal Sets (continue – 2) Theorem: A set of functions |ψ1›, |ψ2›, ,…, |ψn›, of the Hilbert space H is linearly dependent if and only if the Gram Determinant of the Set is zero. Proof: The Gram Matrix of an Orthogonal Set has only nonzero diagonal; therefore Determinant G (|ψ1›, |ψ2›, ,…, |ψn› ).=║|ψ1 ›║2 ║ |ψ2 › ║2 … ║ |ψn › ║2 ≠ 0, and the Set is Linearly Independent. q.e.d. Corollary: The rank of the Gram Matrix equals the dimension of the Linear Manifold L (|ψ1›, |ψ2›, ,…, |ψn› ). If Determinant G (ψ1›, |ψ2›, ,…, |ψn›) is nonzero, the Gram Determinant of any other Subset is also nonzero. Definition 1: Two elements |ψ›,|η› of a Hilbert Space H are said to be orthogonal if <ψ|η>=0. Definition 2: Let S be a nonempty Subset of a Hilbert Space H. S is called an Orthogonal Set if |ψ›┴|η› for every pair |ψ›,|η› є S and |ψ› ≠ |η›. If in addition ║ |ψ›║=1 for every |ψ› є S, then S is called an Orthonormal Set. Lemma: Every Orthogonal Set is Linearly Independent. If |η› is Orthogonal to every element of the Set (|ψ1›, |ψ2›, ,…, |ψn› ), then |η› is Orthogonal to Manifold L (|ψ1›, |ψ2›, ,…, |ψn› ). If then for every we have:nii ,,2,10, =∀=>< ψη ( )n n i ii L ψψψαχ ,,1 1 ∈= ∑= 0,, 1 =><>=< ∑= n i ii  ψηαχη
110 Functional AnalysisSOLO Hilbert Space Orthonormal Sets (continue – 3) Gram-Schmidt Orthogonalization Process Jorgen Gram 1850 - 1916 Erhard Schmidt 1876 - 1959 Let Ψ=(|ψ1›, |ψ2›, ,…, |ψn› ) any finite Set of Linearly Independent Vectors and L (|ψ1›, |ψ2›, ,…, |ψn› ) the Manifold spanned by the Set Ψ. The Gram-Schmidt Orthogonalization Process derive a Set (|e1›, |e2›, ,…, |en› ) of Orthonormal Elements from the Set Ψ. 11 : ψη = 1 11 21 22 11 21 21 1121212112122 , , , , ,,,0: η ηη ψη ψη ηη ψη α ηηαψηηψαψη >< >< −=⇒ >< >< =⇒ ><−>>=<=<⇒−= y ∑ ∑∑ − = − = − = >< >< −=⇒ >< >< =⇒ ><−>>=<=<⇒−= 1 1 1 1 1 1 , , , , ,,,0: i j j ji ji ii kk ki ik i j jkkjikki i j jijii kj η ηη ηψ ψη ηη ηψ α ηηαψηηηηαψη δ  
111 Functional AnalysisSOLO Hilbert Space Orthonormal Sets (continue – 4) Gram-Schmidt Orthogonalization Process (continue) Jorgen Gram 1850 - 1916 Erhard Schmidt 1876 - 1959 11 : ψη = 1 11 21 22 , , : η ηη ψη ψη >< >< −= ∑ − = >< >< −= 1 1 , , : i j j ji ji ii η ηη ηψ ψη   2/1 11 1 1 , : >< = ηη η e   Orthogonalization Normalization ∑ − = >< >< −= 1 1 , , : n j j ji jn nn η ηη ηψ ψη 2/1 22 2 2 , : >< = ηη η e 2/1 , : >< = ii i ie ηη η 2/1 , : >< = nn n ne ηη η
112 Functional AnalysisSOLO Hilbert Space Discrete |ei› and Continuous |wα› Orthonormal Bases From those equations we obtain ijji ee δ=| The Orthonormalization Relation ( )'| ' ααδαα −=ww A Vector |ψ› will be represented by ( ) ψψψψ ∑∑∑∑ ==== ==== n i ii n i ii n i ii n i ii eeeeeeec 1111 || ( )ψαψαψααψ αααααααα ∫∫∫∫ ==== wwdwwdwwdwcd i n j jiji n j jj ceeceec ij ==⇒= ∑∑ == 11  δ ψψ ( ) α ααδ αααααα αψαψ cwwcdwwcd ==⇒= ∫∫ −  ' ''''' Therefore Iee n i ii =∑=1 Iwwd =∫ ααα The Closure Relations I – the Identity Operator (its action on any state leaves it unchanged). α- a real number or vector, not complex-valued The Vectors in Hilbert Space can be Countable (Discrete) or Uncountable (Continuous).
113 Functional AnalysisSOLO Hilbert Space Series Expansions of Arbitrary Functions Definitions: Approximation in the Mean by an Orthonormal Series |ψ1›, |ψ2›, ,…, |ψn› : Let |η› be any function. The numbers: are called the Expansions Coefficients or Components of |η› with respect to the given Orthonormal System nn ψηα ,:= From the relation we obtain or 2 1 2 , ηηηα =≤∑= n i i 0| 11 2 1 ≥      −      −=      − ∑∑∑ === n i ii n i ii n i ii ψαηψαηψαη 0|2| ||| 1 * 1 * 1 * 1 * 11 * ≥−=+−= +−− ∑∑∑ ∑∑∑ === === n i ii n i ii n i ii n i ii n i ii n i ii ααηηααααηη ααηψαψηαηη
114 Functional AnalysisSOLO Hilbert Space Series Expansions of Arbitrary Functions Approximation in the Mean by an Orthonormal Series |ψ1›, |ψ2›, |ψ3›,… : Let |η› be any function. The numbers: are called the Expansions Coefficients or Components of |η› with respect to the given Orthonormal System nnc ψη,:= 2 1 2 ηα ≤∑= n i i Since the sum on the right is independent on n, is true also for n →∞, we have 2 1 2 ηα ≤∑ ∞ =i i Bessel’s Inequality Bessel’s Inequality is true for every Orthonormal System. It proves that the sum of the square of the Expansion Coefficients always converges. Friedrich Wilhelm Bessel 1784 - 1846
115 Functional AnalysisSOLO Hilbert Space Series Expansions of Arbitrary Functions Approximation in the Mean by an Orthonormal Series |ψ1›, |ψ2›, |ψ3›,… : If for a given Orthonormal System |ψ1›, |ψ2›, |ψ3›,… any piecewise continuous function |η› can be approximated in the mean to any desired degree of accuracy ε by choosing a n large enough ( n>N (ε) ), i.e.: ( )εεψαη Nnfor n i ii >≤− ∑=1 then the Orthonormal System |ψ1›, |ψ2›, |ψ3›,… is said to be Complete. For a Complete Orthonormal System |ψ1›, |ψ2›, |ψ3›,… the Bessel’s Inequality becomes an Equality: 2 1 2 ηα =∑ ∞ =i i Parseval’s Equality applies for Complete Orthonormal Systems This relation is known as the “Completeness Relation”. ( )( ) ∑∑∑∑∑ ∞ = ∞ = ∞ = ∞ = ∞ = +++=++=+ ++=++=+ 1 * 1 * 1 * 1 * 1 ,2, i ii i ii i ii i ii i iiii dcdc βββαβαααχη χχηηχηχηχη ∑∑ ∞ = ∞ = += 1 * 1 * ,2 i ii i ii βαβαχη A more general form, for , can be derived as follows:∑∑ ∞ = ∞ = == 1 * 1 * & i ii i ii ββχααη Marc-Antoine Parseval des Chênes 1755 - 1836
Functional AnalysisSOLO Hilbert Space Linear Operators in Hilbert Space An Operator L in Hilbert Space acting on a Vector |ψ›, produces a Vector |η›. ψη L  = L  ψη = The arrow over L means that the Operator is acting on the Vector on the Left, ‹ψ|. An Operator L is Linear if it Satisfies ( ) CLLL ∈+=+ βαηβψαηβψα ,  Consider the quantities . They are in general not equal.( ) ( ) ψηψη || LandL  Eigenvalues and Eigenfunction of a Linear Operator are defined by CL ∈= λψλψ  The Eigenfunction |ψ› is transformed by the Operator L into multiple of itself, by the Eigenvalue λ. The conjugate equation is ( ) CLL ∈== λψλψψ **  The corresponding Operator which transforms the “bra” ‹ψ| , called the Adjoint Operator, is L  The arrow over means that the Operator is acting on the Vector on the Right, |ψ›. L  116
Functional AnalysisSOLO Hilbert Space Adjoint or Hermitian Conjugates Operators An Operator L1 in Hilbert Space acting on a Vector |ψ›, produces a Vector |η›. Let have another Operator in Hilbert Space acting on the Vector |η›, and produce a Vector |χ›. ( ) 1111 LLorLL  =⇔ 1L  Operator ψη 1L  = 1L  Adjoint Operator 1L  ψη = 22 & LL  ηχηχ == Therefore 2112 & LLLL  ψχψχ == ( ) 21122112 LLLLorLLLL  =⇔ The Adjoint of a Product of Operators is obtained by Reversing the order of the Product of Adjoint of Operators. 117
Functional AnalysisSOLO Hilbert Space ILLLLILLLL ==== −−−− 1111 &  Inverse Operator Given ψη L  = L  ψη = The Inverse Operator on is the Operator that will return .ψL  ψ1− L  ψψη == −− LLL  11 Therefore ηψη ==− LLL  1 The Inverse Operator on is the Operator that will return .L  ψ ψ1− L  ψψη == −− 11 LLL  In the same way ηψη ==− LLL  1 Not all Operators have an Inverse. 118
Functional AnalysisSOLO Hilbert Space ( ) ( ) ( ) ηψηψψηψη ,||| * ∀== LLL  Hermitian Operator In Quantum Mechanics the Operators for which are equal present a great importance. They are called Hermitian or Self-Adjoint Operators. ( ) ( ) ψηψη || LandL  Properties of Hermitian Operators From the definition we can see that the direction of the arrow is not important and we can write ( ) ( ) ηψηψψηψηψη ,||||:|| * ∀=== LLLL  1 2 All the Eigenvalues of a Hermitian Operator are Real ( ) ( ) ( ) ( ) ( ) ψηψηψηηψψη |||||| ******* LLLLL  ==== ( ) ψψλψψλψλψ || =⇒∈= LCL  ( ) ψψλψψλψλψ || ** =⇒∈= LCL  Hermitian Operator : ( ) ( ) ( ) * 0 * 0||| λλψψλλψψψψ =⇒=−⇒= >   LL An Operator is Hermitian if it is equal to its Adjoint: Hermitian or Self-Adjoint Operators ( ) LLL  == 119
Functional AnalysisSOLO Hilbert Space ( ) ( ) ( ) ηψηψψηψη ,||| * ∀== LLL  Hermitian Operator In Quantum Mechanics the Operators for which are equal present a great importance. They are called Hermitian Operators. ( ) ( ) ψηψη || LandL  Properties of Hermitian Operators If all the Eigenvalues of an Operator are Real the Operator is Hermitian3 iiii i iiiiiiii iiiiiiii iii iii LL L L i L L ii ψψψψ ψψλψψλψψ ψψλψλψψψ ψλψ ψλψ λλ || ||| ||| * * *      =⇒     == == ⇒∀     = = = ∀ Hermitian Operator 120
Functional AnalysisSOLO Hilbert Space ( ) ( ) ( ) ηψηψψηψη ,||| * ∀== LLL  Hermitian Operator In Quantum Mechanics the Operators for which are equal present a great importance. They are called Hermitian Operators. ( ) ( ) ψηψη || LandL  Properties of Hermitian Operators 4 All the Eigenfunctions of a Hermitian Operator corresponding to different Eigenvalues are Orthogonal, the others can be Orthogonalized using the Gram-Schmidt Procedure. Therefore for a Hermitian Operator we can obtain a “complete Set” of Orthogonal (and Linearly Independent) Eigenfunctions     == = ⇒     == == ** * ||| || nmmmnmmn nmnnm mmmmm nnnnn L L L L ψψλψψλψψ ψψλψψ λλψλψ λλψλψ If |ψn› and |ψm› are two Eigenfunctions of the Hermitian Operator L, with eigenvalues λn and λm, respectively Hermitian Operator: nmmnmnmnnm LL ψψλψψλψψψψ |||| =⇒= If λm ≠ λn this equality is possible only if ψn and ψm are Orthogonal 0| =nm ψψ If λm = λn we can use the Gram-Schmidt Procedure to obtain a new Eigenfunction Orthogonal to |ψn›. n nn mn mm ψ ψψ ψψ ψψ         −= | | :~ 0| | | |~| =        −= nn nn mn mnmn ψψ ψψ ψψ ψψψψ The Hermitian Operators have Real Eigenvalues and Orthogonal Eigenfunctions. λm ≠ λn 121
Functional AnalysisSOLO Hilbert Space 1− =UU  Unitary Operator Properties of Unitary Operators A Unitary Operator U is defined by the relation where I is the Identity Operator.IUUUU ==  A Unitary Matrix is such that it’s Adjoint is equal to it’s Inverse. All Eigenvalues of a Unitary Matrix have absolute values equal to 1. Suppose |ψi› is an Eigenfunction and λi is the corresponding Eigenvalue of a Unitary Operator.  iUU U U iiiiiii I i iii iii ∀=⇒=⇒     = = 1| ** * λλψψλλψψ ψλψ ψλψ    1 2  ψηψηψη ,| ∀= I UU  For all <η| and |ψ› the Inner Product of equals‹η|ψ›ψη UandU  3 ψψψ ∀=U   ψψψψψψψψ ∀=== 2/1 2/1 2/1 || I UUUUU  122
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5 introduction to quantum mechanics

  • 2. Introduction to Quantum MechanicsSOLO Table of Content 2 Introduction to Quantum Mechanics Classical Mechanics Gravity Optics Electromagnetism Quantum Weirdness History Physical Laws of Radiometry Zeeman Effect, 1896 Discovery of the Electron, 1897 Planck’s Law 1900 Einstein in 1905 Bohr Quantum Model of the Atom 1913. Einstein’s General Theory of Relativity 1915 Quantum Mechanics History
  • 3. Introduction to Quantum MechanicsSOLO Table of Content (continue – 1) 3 De Broglie Particle-Wave Law 1924 Wolfgang Pauli states the “Quantum Exclusion Principle” 1924 Heisenberg, Born, Jordan “Quantum Matrix Mechanics”, 1925 Wave Packet and Schrödinger Equation, 1926 Operators in Quantum Mechanics Hilbert Space and Quantum Mechanics Von Neumann - Postulates of Quantum Mechanics Conservation of Probability Expectations Value and Operators The Expansion Theorem or Superposition Principle Matrix Representation of Wave Functions and Operators Commutator of two Operators A and B Time Evolution Operator of the Schrödinger Equation Heisenberg Uncertainty Relations
  • 4. Introduction to Quantum MechanicsSOLO Table of Content (Continue -2) 4 Time Independent Hamiltonian The Schrödinger and Heisenberg Pictures Transition from Quantum Mechanics to Classical Mechanics. Pauli Exclusion Principle Klein-Gordon Equation for a Spinless Particle Non-relativistic Schrödinger Equation in an Electromagnetic Field Pauli Equation Dirac Equation Light Polarization and Quantum Theory Copenhagen Interpretation of Quantum Mechanics Measurement in Quantum Mechanics Schrödinger’s Cat Solvay Conferences Bohr–Einstein Debates Feynman Path Integral Representation of Time Evolution Amplitudes
  • 5. Introduction to Quantum MechanicsSOLO Table of Content (Continue -3) 5 Quantum Field Theories References Aharonov–Bohm Effect Wheeler's delayed choice experiment Zero-Point Energy Quantum Foam De Broglie–Bohm Theory in Quantum Mechanics Bell's Theorem Bell Test Experiments Wheeler's delayed choice experiment Hidden Variables
  • 6. Physics The Presentation is my attempt to study and cover the fascinating subject of Quantum Mechanics. The completion of this presentation does not make me an expert on the subject, since I never worked in the field. I thing that I reached a good coverage of the subject and I want to share it. Comments and suggestions for improvements are more than welcomed. 6 SOLO Introduction to Quantum Mechanics
  • 7. Physics NEWTON's MECHANICS ! ANALYTIC MECHANICS FLUID & GAS DYNAMICS THERMODYNAMICS MAXWELL ELECTRODYNAMICS CLASSICAL THEORIES NEWTON's GRAVITY OPTICS 1900 At the end of the 19th century, physics had evolved to the point at which classical mechanics could cope with highly complex problems involving macroscopic situations; thermodynamics and kinetic theory were well established; geometrical and physical optics could be understood in terms of electromagnetic waves; and the conservation laws for energy and momentum (and mass) were widely accepted. So profound were these and other developments that it was generally accepted that all the important laws of physics had been discovered and that, henceforth, research would be concerned with clearing up minor problems and particularly with improvements of method and measurement. "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" - Lord Kelvin 1900: This was just before Relativity and Quantum Mechanics appeared on the scene and opened up new realms for exploration. Completeness of a Theory 7 Return to Table of Content SOLO
  • 8. 8 Classical TheoriesSOLO 1.1 Newton’s Laws of Motion “The Mathematical Principles of Natural Philosophy” 1687 First Law Every body continues in its state of rest or of uniform motion in straight line unless it is compelled to change that state by forces impressed upon it. Second Law The rate of change of momentum is proportional to the force impressed and in the same direction as that force. Third Law To every action there is always opposed an equal reaction. td rd constF   ==→= → :vv0 ( )vm td d p td d F  == 2112 FF  −= vmp  = td pd F  = 12F  1 2 21F  r  - Position v:  mp = - Momentum
  • 9. 9 SOLO 1.2 Work and Energy The work W of a force acting on a particle m that moves as a result of this along a curve s from to is defined by: F  1r  2r  ∫∫ ⋅      =⋅= ⋅∆ 2 1 2 1 12 r r r r rdrm dt d rdFW      r  1r  2r  rd  rdr  + 1 2 F  m s rd  is the displacement on a real path. ⋅⋅∆ ⋅= rrmT  2 1 The kinetic energy T is defined as: 1212 2 1 2 1 2 1 2 TTrrd m dtrr dt d mrdrm dt d W r r r r r r −=      ⋅=⋅      =⋅      = ∫∫∫ ⋅ ⋅ ⋅⋅⋅⋅⋅        For a constant mass m Classical Theories
  • 10. 10 SOLO Work and Energy (continue) When the force depends on the position alone, i.e. , and the quantity is a perfect differential ( )rFF  = rdF  ⋅ ( ) ( )rdVrdrF  −=⋅ The force field is said to be conservative and the function is known as the Potential Energy. In this case: ( )rV  ( ) ( ) ( ) 212112 2 1 2 1 VVrVrVrdVrdFW r r r r −=−=−=⋅= ∫∫ ∆      The work does not depend on the path from to . It is clear that in a conservative field, the integral of over a closed path is zero. 12W 1r  2r  rdF  ⋅ ( ) ( ) 01221 21 1 2 2 1 =−+−=⋅+⋅=⋅ ∫∫∫ VVVVrdFrdFrdF path r r path r rC         Using Stoke’s Theorem it means that∫∫∫ ⋅×∇=⋅ SC sdFrdF  0=×∇= FFrot  Therefore is the gradient of some scalar functionF  ( ) rdrVdVrdF  ⋅−∇=−=⋅ ( )rVF  −∇= Classical Theories
  • 11. 11 SOLO Work and Energy (continue) and ⋅ →∆→∆ ⋅−=⋅−= ∆ ∆ = rF dt rd F t V dt dV tt  00 limlim But also for a constant mass system ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅=⋅=      ⋅+⋅=      ⋅= rFrrmrrrr m rrm dt d dt dT  22 1 Therefore for a constant mass in a conservative field ( ) .0 constEnergyTotalVTVT dt d ==+⇒=+ Classical Theories
  • 12. SOLO 1.5 Rotations and Angular Momentum Classical Theories md td rd mdpd   == v md td rd pd td d Fd 2 2   == md td rd pdHd CG   ×=×= ρρ: ∫∫ == M md td rd pdP   ∫∫ == M md td rd FdF 2 2  - Angular Rotation Rate of the Body (B) relative to Inertia (I) - Force ∫∫ ×== M CGCG md td rd HdH   ρ - Angular Momentum Relative to C.G. BBBBBBIIIIII zzyyxxzzyyxxr 111111 ++=++=  BIBBBIBBBIBB III zz td d yy td d xx td d z td d y td d x td d 111111 0111 ×=×=×= === ←←← ωωω   IB←ω  - Momentum 12
  • 13. SOLO 1.6 Lagrange, Hamilton, Jacobi Classical Theories Carl Gustav Jacob Jacobi (1804-1851) William Rowan Hamilton 1805-1865 Joseph Louis Lagrange 1736-1813 Lagrangiams Lagrange’s Equations: nicQ q L q L dt d m k k ikin ii ,,2,1 1   =+= ∂ ∂ −      ∂ ∂ ∑= λ mkcqc k t n i i k i ,,2,10 1  ==+∑= ( ) ( ) ( )qVtqqTtqqL  −= ,,:,, ni cQ q H p p H q m j j iji i i i i ,,2,1 1    =        ++ ∂ ∂ −= ∂ ∂ = ∑= λ mkcqc k t n i i k i ,,2,10 1  ==+∑= Extended Hamilton’s Equations Constrained Differential Equations Hamiltonian ( )tqqTqpH n i ii ,,: 1   −= ∑= ni q T p i i ,,2,1   = ∂ ∂ = Hamilton-Jacobi Equation 0,, =      ∂ ∂ + ∂ ∂ k k q S qtH t S       ∂ ∂ = k kjj q S qtq ,,φ kk q S p ∂ ∂ = 13
  • 14. 14 SOLO 1.4 Basic Definitions Given a System of N particles. The System is completely defined by Particles coordinates and moments: ( ) ( ) ( ) ( ) ( ) ( ) ( ) Nl ktpjtpitp td rd mp ktzjtyitxzyxrr zlylxl l ll lllkkkll ,,2,1 ,,    =      ++== ++== where are the unit vectors defining any Inertial Coordinate Systemkji  ,, r  1r  2r  rd  rdr  + 1 2 F  m s The path of the Particles are defined by Newton Second Law NlF td rd m td pd l l l l ,,2,12 2   === ∑ Given , the Path of the Particle is completely defined and is Deterministic (if we repeat the experiment, we obtain every time the same result). ( ) ( ) ( )tFandtptr lll ∑== 0,0  In Classical Mechanics: •Time and Space are two Independent Entities. •No limit in Particle Velocity •Since every thing is Deterministic we can Measure all quantities simultaneously. The outcome of all measurements are repeatable and depends only on the accuracy of the measurement device. •Causality: Every Effect hase a Cause that preceed it. Classical Theories Return to Table of Content
  • 15. GRAVITY Classical Theories GF  GF M m      EQPOISSON G GU r MG UU r GM g gm r MG mr r mM GF ρπ4&& 1 2 2 =∇=−∇=      −∇= −=      ∇=−= Newton’s Law of Universal Gravity Any two body attract one another with a Force Proportional to the Product of the Masses and inversely Proportional to the Square of the Distance between them. G = 6.67 x 10-8 dyne cm2 /gm2 the Universal Gravitational Constant Instantaneous Propagation of the Force along the direction between the Masses (“Action at a Distance”). 15
  • 16. Newton was fully aware of the conceptual difficulties of his action-at-a-distance theory of gravity. In a letter to Richard Bentley Newton wrote: "It is inconceivable, that inanimate brute matter should, without the mediation of something else, which is not material, operate upon, and affect other matter without mutual contact; as it must do, if gravitation, ...., be essential and inherent in it. And this is one reason, why I desired you would not ascribe innate gravity to me. That gravity should be innate, inherent, and essential to matter, so that one body may act upon another, at a distance through vacuum, without the mediation of anything else, by and through their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it." GRAVITY Classical Theories 16 Return to Table of Content
  • 17. 17 SOLO Newton published “Opticks”1704 Newton threw the weight of his authority on the corpuscular theory. This conviction was due to the fact that light travels in straight lines, and none of the waves that he knew possessed this property. Newton’s authority lasted for one hundred years, and diffraction results of Grimaldi (1665) and Hooke (1672), and the view of Huygens (1678) were overlooked. Optics Every point on a primary wavefront serves the source of spherical secondary wavelets such that the primary wavefront at some later time is the envelope o these wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wave at each point in space. Christiaan Huygens 1629-1695 Huygens Principle 1678 Light: Waves or Particles Classical Theories
  • 18. 18 SOLO In 1801 Thomas Young uses constructive and destructive interference of waves to explain the Newton’s rings. Thomas Young 1773-1829 1801 - 1803 In 1803 Thomas Young explains the fringes at the edges of shadows using the wave theory of light. But, the fact that was belived that the light waves are longitudinal, mad difficult the explanation of double refraction in certain crystals. Optics Run This Young Double Slit Experiment Classical Theories
  • 19. 19 POLARIZATION Arago and Fresnel investigated the interference of polarized rays of light and found in 1816 that two rays polarized at right angles to each other never interface. SOLO Dominique François Jean Arago 1786-1853 Augustin Jean Fresnel 1788-1827 Arago relayed to Thomas Young in London the results of the experiment he had performed with Fresnel. This stimulate Young to propose in 1817 that the oscillations in the optical wave where transverse, or perpendicular to the direction of propagation, and not longitudinal as every proponent of wave theory believed. Thomas Young 1773-1829 1816-1817 longitudinal waves transversal waves Classical Theories Run This
  • 20. 20 SOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. 1818Diffraction - History Classical Theories
  • 21. 21 Diffraction SOLO Augustin Jean Fresnel 1788-1827 In 1818 Fresnel, by using Huygens’ concept of secondary wavelets and Young’s explanation of interface, developed the diffraction theory of scalar waves. P 0P Q 1x 0x 1y 0y η ξ Fr  Sr  ρ  r  O 'θ θ Screen Image plane Source plane 0O 1O Sn1 Σ Σ - Screen Aperture Sn1 - normal to Screen From a source P0 at a distance from a aperture a spherical wavelet propagates toward the aperture: ( ) ( )Srktj S source Q e r A tU − = ' ' ω According to Huygens Principle second wavelets will start at the aperture and will add at the image point P. ( ) ( ) ( )( ) ( ) ( )( ) ∫∫ Σ ++− Σ +−− == dre rr A Kdre r U KtU rrktj S sourcerkttjQ P S 2/2/' ',', πωπω θθθθ where: ( )',θθK obliquity or inclination factor ( ) ( )SSS nrnr 11cos&11cos' 11 ⋅=⋅= −− θθ ( ) ( )   === === 0',0 max0',0 πθθ θθ K K Obliquity factor and π/2 phase were introduced by Fresnel to explain experiences results. Fresnel Diffraction Formula Fresnel took in consideration the phase of each wavelet to obtain: Run This Return to Table of Content Classical Theories
  • 22. 22 MAXWELL’s EQUATIONS SOLO Magnetic Field IntensityH  [ ]1− ⋅mA Electric DisplacementD  [ ]2− ⋅⋅ msA Electric Field IntensityE  [ ]1− ⋅mV Magnetic InductionB  [ ]2− ⋅⋅ msV Electric Current DensityeJ  [ ]2− ⋅mA Free Electric Charge Distributioneρ [ ]3− ⋅⋅ msA 1. AMPÈRE’S CIRCUIT LAW (A) 1821 eJ t D H    + ∂ ∂ =×∇ 2. FARADAY’S INDUCTION LAW (F) 1831 t B E ∂ ∂ −=×∇   3. GAUSS’ LAW – ELECTRIC (GE) ~ 1830 eD ρ=⋅∇  4. GAUSS’ LAW – MAGNETIC (GM) 0=⋅∇ B  André-Marie Ampère 1775-1836 Michael Faraday 1791-1867 Karl Friederich Gauss 1777-1855 James Clerk Maxwell (1831-1879) 1865 Electromagnetism MAXWELL UNIFIED ELECTRICITY AND MAGNETISM Classical Theories
  • 23. 23 SOLO ELECTROMGNETIC WAVE EQUATIONS For Homogeneous, Linear and Isotropic Medium ED  ε= HB  µ= where are constant scalars, we haveµε, t E t D H t t H t B E ED HB ∂ ∂ = ∂ ∂ =×∇ ∂ ∂ ∂ ∂ −= ∂ ∂ −=×∇×∇ = =       εµ µ ε µ Since we have also tt ∂ ∂ ×∇=∇× ∂ ∂ ( ) ( ) ( )                   =⋅∇= ∇−⋅∇∇=×∇×∇ = ∂ ∂ +×∇×∇ 0& 0 2 2 2 DED EEE t E E     ε µε t D H ∂ ∂ =×∇   t B E ∂ ∂ −=×∇   For Source less Medium 02 2 2 = ∂ ∂ −∇ t E E   µε Define meme KK c KK v === ∆ 00 11 εµµε where ( ) smc /103 10 36 1 104 11 8 9700 ×=       ×× == −− ∆ π π εµ c is the velocity of light in free space. Electromagnetism Run This Return to Table of Content Classical Theories
  • 24. Completeness of a Theory SOLO At the end of the 19th century, physics had evolved to the point at which classical mechanics could cope with highly complex problems involving macroscopic situations; thermodynamics and kinetic theory were well established; geometrical and physical optics could be understood in terms of electromagnetic waves; and the conservation laws for energy and momentum (and mass) were widely accepted. So profound were these and other developments that it was generally accepted that all the important laws of physics had been discovered and that, henceforth, research would be concerned with clearing up minor problems and particularly with improvements of method and measurement. "There is nothing new to be discovered in physics now. All that remains is more and more precise measurement" - Lord Kelvin 1900: 1894: "The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote.... Our future discoveries must be looked for in the sixth place of decimals." - Albert. A. Michelson, speech at the dedication of Ryerson Physics Lab, U. of Chicago 1894 This was just before Relativity and Quantum Mechanics appeared on the scene and opened up new realms for exploration. 24 Classical Theories
  • 26. Many classical particles, both slits are open http://www.mathematik.uni- muenchen.de/~bohmmech/Poster/post/postE.htmlThe Double Slit Experiment A single particle, both slits are open Many particles, one slit is open. Many atomic particles, both slits are open http://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics#Schr.C3.B6ding er_wave_equation SOLO Run This 26 QUANTUM THEORIES https://www.youtube.com/watch?v=Q1YqgPAtzho&src_vid=4C5pq7W5yRM&feature=iv&annotation_id=an
  • 27. According to the results of the double slit experiment, if experimenters do something to learn which slit the photon goes through, they change the outcome of the experiment and the behavior of the photon. If the experimenters know which slit it goes through, the photon will behave as a particle. If they do not know which slit it goes through, the photon will behave as if it were a wave when it is given an opportunity to interfere with itself. The double-slit experiment is meant to observe phenomena that indicate whether light has a particle nature or a wave nature. Richard Feynman observed that if you wish to confront all of the mysteries of quantum mechanics, you have only to study quantum interference in the two-slit experiment The Double Slit Experiment SOLO Run This 27 QUANTUM THEORIES
  • 28. QUANTUM THEORIES Some trajectories of a harmonic oscillator (a ball attached to a spring) in classical mechanics (A–B) and quantum mechanics (C–H). In quantum mechanics (C–H), the ball has a wave function, which is shown with real part in blue and imaginary part in red. The trajectories C,D,E,F, (but not G or H) are examples of standing waves, (or "stationary states"). Each standing-wave frequency is proportional to a possible energy level of the oscillator. This "energy quantization" does not occur in classical physics, where the oscillator can have any energy 28
  • 31. 31 SOLO http://thespectroscopynet.com/educational/Kirchhoff.htm Spectroscopy 1868 A.J. Ångström published a compilation of all visible lines in the solar spectrum. 1869 A.J. Ångström made the first reflection grating. Anders Jonas Angström a physicist in Sweden, in 1853 had presented theories about gases having spectra in his work: Optiska Undersökningar to the Royal Academy of Sciences pointing out that the electric spark yields two superposed spectra. Angström also postulated that an incandescent gas emits luminous rays of the same refrangibility as those which it can absorb. This statement contains a fundamental principle of spectrum analysis. http://en.wikipedia.org/wiki/Spectrum_analysis
  • 32. 32 ParticlesSOLO 1874 George Johnstone Stoney 1826 - 1911 As early as 1874 George Stoney had calculated the magnitude of his electron from data obtained from the electrolysis of water and the kinetic theory of gases. The value obtained later became known as a coulomb. Stoney proposed the particle or atom of electricity to be one of three fundamental units on which a whole system of physical units could be established. The other two proposed were the constant universal gravitation and the maximum velocity of light and other electromagnetic radiations. No other scientist dared conceive such an idea using the available data. Stoney's work set the ball rolling for other great scientists such as Larmor and Thomas Preston who investigated the splitting of spectral lines in a magnetic field. Stoney partially anticipated Balmer's law on the hydrogen spectral series of lines and he discovered a relationship between three of the four lines in the visible spectrum of hydrogen. Balmer later found a formula to relate all four. George Johnstone Stoney was acknowledged for his contribution to developing the theory of electrons by H.A. Lorentz , in his Nobel Lecture in 1902. George Stoney estimates the charge of the then unknown electron to be about 10-20 coulomb, close to the modern value of 1.6021892 x 10-19 coulomb. (He used the Faraday constant (total electric charge per mole of univalent atoms) divided by Avogadro's Number. Return to Table of Content
  • 33. 33 Physical Laws of RadiometrySOLO Stefan-Boltzmann Law Stefan – 1879 Empirical - fourth power law Boltzmann – 1884 Theoretical - fourth power law For a blackbody: ( ) ( ) ( ) ( )       ⋅ ⋅==       = − == − ∞∞ ∫∫ 42 12 32 45 2 4 0 2 5 1 0 10670.5 15 2 : 1/exp 1 Kcm W hc k cm W Td Tc c dMM BBBB  π σ σλ λλ λλ LUDWIG BOLTZMANN (1844 - 1906) Stefan-Boltzmann Law JOSEF STEFAN (1835 – 1893) 1879 1884 1893 Wien’s Displacement Law 0= λ λ d Md BB Wien 1893 from which: The wavelength for which the spectral emittance of a blackbody reaches the maximum is given by: mλ KmTm  ⋅= µλ 2898 Wien’s Displacement Law WILHELM WIEN (1864 - 1928) Nobel Prize 1911
  • 34. 34 SOLO Johan Jakob Balmer presented an empirical formula describing the position of the emission lines in the visible part of the hydrogen spectrum. Spectroscopy 1885 Johan Jakob Balmer 1825 - 1898 Balmer Formula ( )222 / nmmB −=λ ,6,5,4,3,106.3654,2 8 =×== − mcmBn δH violet blue - green red 1=n 2=n 3=n 4=n 5=n ∞=n Lyman serie Balmer serie Paschen serie Brackett serie 0=E Energy         −= 2232 0 4 11 8 1 nnhc em f ελ 1=fn 2=f n 3=fn 4=f n Balmer was a mathematical teacher who, in his spare time, was obsessed with formulae for numbers. He once said that, given any four numbers, he could find a mathematical formula that connected them. Luckily for physics, someone gave him the wavelengths of the first four lines in the hydrogen spectrum.
  • 35. 35 SOLO Spectroscopy 1887 Johannes Robert Rydberg 1854 - 1919 Rydberg Formula for Hydrogen 2 2 1 1 1 H i f R n nλ   = − ÷ ÷   1=n 2=n 3=n 4=n 5=n ∞=n Lyman serie Balmer serie Paschen serie Brackett serie 0=E Energy         −= 2232 0 4 11 8 1 nnhc em f ελ 1=fn 2=fn 3=f n 4=fn 34 6.62606876 10h J s− = × gPlank constant 31 9.10938188 10em kg− = ×Electron mass 19 1.602176452 10e C− = ×Electron charge 12 0 8.854187817 10 /F mε − = ×Permittivity of vacuum Rydberg generalized Balmer’s hydrogen spectral lines formula. Theodore Lyman 1874 - 1954 2in = Balmer series (1885) Johan Jakob Balmer 1825 - 1898 Friedric Paschen 1865 - 1947 3in = Paschen series (1908) 4in = Brackett series (1922) Lyman series (1906)1in = Rydberg Constant for Hydrogen 17 x105395687310973.1 − = mRH 4 2 3 08 e H m e R h cε = Later in the Bohr Model was fund that Frederick Sumner Brackett 1896 - 1988
  • 36. 36 PhotoelectricitySOLO In 1887 Heinrich Hertz, accidentally discovered the photoelectric effect. Hertz conducted his experiments that produced radio waves. By chance he noted that a piece of zinc illuminated by ultraviolet light became electrically charged. Without knowing he discovered the Photoelectric Effect. 1887 Heinrich Rudolf Hertz 1857-1894 - - - - - - - - -- - - - - metallic surface ejected electrons incoming E.M. waves http://en.wikipedia/wiki/Photoelectric_effect http://en.wikipedia/wiki/Heinrich_Hertz Return to Table of Content
  • 37. 37 SpectroscopySOLO Zeeman Effect Pieter Zeeman observed that the spectral lines emitted by an atomic source splited when the source is placed in a magnetic field. In most atoms, there exists several electron configurations that have the same energy, so that transitions between different configuration correspond to a single line. 1896 Because the magnetic field interacts with the electrons, this degeneracy is broken giving rice to very close spectral lines. no magnetic field B = 0 cba ,, fed ,, a b c d e f magnetic field B 0≠ http://en.wikipedia.org/wiki/Zeeman_effect Pieter Zeeman 1865 - 1943 Nobel Prize 1902 Return to Table of Content
  • 38. 38 Physical Laws of RadiometrySOLO Wien Approximation to Black Body Radiation Wien's Approximation (also sometimes called Wien's Law or the Wien Distribution Law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in 1896. The equation does accurately describe the short wavelength (high frequency) spectrum of thermal emission from objects, but it fails to accurately fit the experimental data for long wavelengths (low frequency) emission. WILHELM WIEN (1864 - 1928) Comparison of Wien's Distribution law with the Rayleigh–Jeans Law and Planck's law, for a body of 8 mK temperature The Wien ‘s Law may be written as where • I(ν,T) is the amount of energy per unit surface area per unit time per unit solid angle per unit frequency emitted at a frequency ν. • T is the temperature of the black body. • h is Planck's constant. • c is the speed of light. • k is Boltzmann's constant 1896 Return to Table of Content
  • 39. 39 SOLO Particles J.J. Thomson showed in 1897 that the cathode rays are composed of electrons and he measured the ratio of charge to mass for the electron. Discovery of the Electron 1897 Joseph John Thomson 1856 – 1940 Nobel Prize 1922 The total charge on the collector (assuming all electrons are stick to the cathode collector and no secondary emissions is: e qnQ ⋅= The energy of the particles reaching the cathode is: 2/2 vmnE ⋅⋅= Uvm q E Q e 12 2 = ⋅ = U v m qe 2 2 = Thomson Atom Model Wavelike Behavior for Electrons Return to Table of Content
  • 40. 40 Physical Laws of RadiometrySOLO Rayleigh–Jeans Law Comparison of Rayleigh–Jeans law with Wien approximation and Planck's law, for a body of 8 mK temperature In 1900, the British physicist Lord Rayleigh derived the λ−4 dependence of the Rayleigh–Jeans law based on classical physical arguments.[3] A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir James Jeans in 1905. The Rayleigh–Jeans law revealed an important error in physics theory of the time. The law predicted an energy output that diverges towards infinity as wavelength approaches zero (as frequency tends to infinity) and measurements of energy output at short wavelengths disagreed with this prediction. John William Strutt, 3rd Baron Rayleigh 1842- 1919 James Hopwood Jeans 1877 - 1946 Rayleigh considered the radiation inside a cubic cavity of length L and temperature T whose walls are perfect reflectors as a series of standing electromagnetic waves. At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The three wavelengths λ1, λ2 and λ3, in the three directions orthogonal to the walls can be: ,2,1,, 2 === i i i nzyxi n Lλ 1900 1905
  • 41. 41 Physical Laws of RadiometrySOLO Rayleigh–Jeans Law (continue ) The Rayleigh–Jeans law agrees with experimental results at large wavelengths (or, equivalently, low frequencies) but strongly disagrees at short wavelengths (or high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe. Comparison of Rayleigh–Jeans law and Planck's law The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, although the concept goes back to 1900 with the first derivation of the λ − 4 dependence of the Rayleigh–Jeans law; Solution Max Planck solved the problem by postulating that electromagnetic energy did not follow the classical description, but could only oscillate or be emitted in discrete packets of energy proportional to the frequency, as given by Planck's law. This has the effect of reducing the number of possible modes with a given energy at high frequencies in the cavity described above, and thus the average energy at those frequencies by application of the equipartition theorem. The radiated power eventually goes to zero at infinite frequencies, and the total predicted power is finite. The formula for the radiated power for the idealized system (black body) was in line with known experiments, and came to be called Planck's law of black body radiation. Based on past experiments, Planck was also able to determine the value of its parameter, now called Planck's constant. The packets of energy later came to be called photons, and played a key role in the quantum description of electromagnetism. ( ) ( ) λ λ π λ λ λλ d Tk d V N Tkdu 4 8 == Rayleigh–Jeans Law Return to Table of Content
  • 42. 42 Physical Laws of RadiometrySOLO MAX PLANCK (1858 - 1947) 4242( ) ν ννπ νν ν d e h c du kT h 1 8 3 2 − = ( ) ν νπ νν ν de c h du kT h − = 3 3 8 WILHELM WIEN (1864 - 1928) Wien’s Law 1896 ( ) ν νπ νν dTk c du 3 2 8 = Rayleigh–Jeans Law 1900 - 1905 John William Strutt, 3rd Baron Rayleigh 1842- 1919 James Hopwood Jeans 1877 - 1946 Comparison of Rayleigh–Jeans law with Wien approximation and Planck's law, for a body of 8 mK temperature Tkh <<ν Tkh >>ν
  • 43. 43 Physical Laws of RadiometrySOLO MAX PLANCK (1858 - 1947) Planck’s Law 1900 ( ) ν ννπ νν ν d e h c du kT h 1 8 3 2 − = Planck derived empirically, by fitting the observed black body distribution to a high degree of accuracy, the relation By comparing this empirical correlation with the Rayleigh-Jeans formula Planck concluded that the error in classical theory must be in the identification of the average oscillator energy as kT and therefore in the assumption that the oscillator energy is distributed continuously. He then posed the following question: If the average energy is defined as how is the actual oscillator energies distributed? ( ) ν νπ νν dTk c du 3 2 8 = 1/ − = kTh e h E ν ν KT KWk Wh   ineTemperaturAbsolute- constantBoltzmannsec/103806.1 constantPlanksec106260.6 23 234 −⋅⋅= −⋅⋅= − −
  • 44. 44 Physical Laws of RadiometrySOLO MAX PLANCK (1858 - 1947) If the average energy is defined as how is the actual oscillator energies distributed? 1/ − = kTh e h E ν ν Planck deviated appreciable from the concepts of classical physics by assuming that the energy of the oscillators, instead of varying continuously, can assume only certain discrete values νε hnn = Let determine the average energy ( ) ( )  +++ ++ === −− −− ∞ = − ∞ = − ∞ = − ∞ = − ∑ ∑ ∑ ∑ kThkTh kThkTh n kTnh n kTnh n kTE n kTE n ee eeh e enh e eE E n n /2/ /2/ 0 / 0 / 0 / 0 / 1 2 νν νν ν ν ν ν From Statistical Mechanics we know that the probability of a system assuming energy between ε and ε+dε is proportional to exp (-ε/kT) dε x ee kTh ex kThkTh − =+++ − = −− 1 1 1 / /2/ ν νν  ( ) ( )2 0 /2/ 0 / 11 1 2 / x x h xxd d xhxn xd d xheehenh n n ex kThkTh n kTnh kTh − =      − ==++= ∑∑ ∞ = = −− ∞ = − − ννννν ν ννν  where n is an integer (n = 0, 1, 2, …), and h =6.6260. 10-14 W. sec2 is a constant introduced empirically by Planck , the Planck’s Constant.
  • 45. 45 Physical Laws of RadiometrySOLO MAX PLANCK (1858 - 1947) Planck’s Postulate: The energy of the oscillators, instead of varying continuously, can assume only certain discrete values νε hnn = where n is an integer (n = 0, 1, 2, …). We say that the oscillators energy is Quantized. ( ) 11 1 1 1 // / / 2/ / 0 / 0 / 0 / 0 / − = − = − − === − − − − − ∞ = − ∞ = − ∞ = − ∞ = − ∑ ∑ ∑ ∑ kThkTh kTh kTh kTh kTh n kTnh n kTnh n kTE n kTE n e h e e h e e e h e enh e eE E n n νν ν ν ν ν ν ν ν ν νν The average energy is
  • 46. 46 Physical Laws of RadiometrySOLO Plank’s Law ( ) 1/exp 1 2 5 1 − = Tc c M BB λλ λ Plank’s Law applies to blackbodies; i.e. perfect radiators. The spectral radial emittance of a blackbody is given by: ( ) KT KWk Wh kmc Kmkhcc mcmWchc    ineTemperaturAbsolute- constantBoltzmannsec/103806.1 constantPlanksec106260.6 lightofspeedsec/458.299792 10439.1/ 107418.32 23 234 4 2 4242 1 −⋅⋅= −⋅⋅= −= ⋅⋅== ⋅⋅⋅== − − − µ µπ Plank’s Law 1900 MAX PLANCK 1858 - 1947 Nobel Prize 1918 Return to Table of Content
  • 47. 47 SOLO Particles J.J. Thomson showed in 1897 that the cathode rays are composed of electrons and he measured the ratio of charge to mass for the electron. In 1904 he suggested a model of the atom as a sphere of positive matter in which electrons are positioned by electrostatic forces. Thomson Atom Model 1904 -- -- -- -- -- -- -- -- -- -- Joseph John Thomson 1856 – 1940 Nobel Prize 1922 Plum Pudding Model Return to Table of Content
  • 48. 48 PhotoelectricitySOLO Einstein and Photoelectricity Albert Einstein explained the photoelectric effect discovered by Hertz in 1887 by assuming that the light is quantized (using Plank results) in quantities that later become known as photons. 1905 - - - - - - - - -- - - - - metallic surface ejected electrons incoming E.M. waves k E 0 ν ν 0 2 2 1 νν hhvmE ek −== The kinetic energy Ek of the ejected electron is: where: functionworksec frequencylight constantPlanksec106260.6 0 234 −⋅ − −⋅⋅= − Wh Hz Wh ν ν Albert Einstein 1879 - 1955 Nobel Prize 1921 To eject an electron the frequency of the incoming EM wave v must be above a threshold v0 (depends on metallic surface). Increasing the Intensity of the EM Wave will increase the number of electrons ejected, but not their energy. Return to Table of Content
  • 49. 1905 EINSTEIN’S SPECIAL THEORY OF RELATIVITY Special Relativity Theory 49
  • 50. EINSTEIN’s SPECIAL THEORY OF RELATIVITY (Continue) First Postulate: It is impossible to measure or detect the Unaccelerated Translation Motion of a System through Free Space or through any Aether-like Medium. Second Postulate: Velocity of Light in Free Space, c, is the same for all Observers, independent of the Relative Velocity of the Source of Light and the Observers. Second Postulate (Advanced): Speed of Light represents the Maximum Speed of transmission of any Conventional Signal. Special Relativity Theory 50
  • 51. 51 SOLO x z y 'x 'z 'y v  'u 'OO 'u− A B Consequence of Special Theory of Relativity The relation between the mass m of a particle having a velocity u and its rest mass m0 is: 2 2 0 1 c u m m − = Special Relativity Theory EINSTEIN’s SPECIAL THEORY OF RELATIVITY (Continue) The Kinetic Energy of a free moving particle having a momentum p = m u, a velocity u and its rest mass m0 is: 42 0 222 cmcpT += The velocity of a photon is u = c, therefore, from the first equation, it has a rest mass 00 =photonm And has a Kinetic Energy and Total Energy of νhEcpT VTE V === += =0 Therefore if v is the photon Frequency and λ is photon Wavelength, we have cm h p hc cmph cp = = === ν ν λ
  • 52. Locality and NonlocalitySOLO Event inside Light Cone EVENT HERE AND NOW Simultaneous Event at different place A Light Cone is the path that a flash of light, emanating from a single Event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through space-time. The Light Cone Equation is ( ) 022222 =−++ tczyx Events Inside the Light Cone ( ) 022222 <−++ tczyx Events Outside the Light Cone ( ) 022222 >−++ tczyx Einstein’s Theory of Special Relativity Postulates that no Signal can travel with a speed higher than the Speed of Light c. Thousands of experiments performed with Particles (Photons, Electrons. Neutrons,…) complied to this Postulate. However no experiments could be performed with Sub-particles, so, in my opinion the confirmation of this Postulate is still an open issue. Light Cone 52
  • 53. Locality and NonlocalitySOLO Event inside Light Cone EVENT HERE AND NOW Simultaneous Event at different place According to Einstein only Events within Light Cone (shown in the Figure) can communicate with an event at the Origin, since only those Space-time points can be connected by a Signal traveling with the Speed of Light c or less. We call those Events “Local” although they may be separated in Space-time. Locality The Postulates of Relativity require that all frames of reference to be equivalent. So, if the Events are “Local” in any realizable frame of reference, they must be “Local” in all equivalent Frame of Reference. Two Space-time Points within Light Cone are called “timelike”. Nonlocality Two Space-time Points outside Light Cone are said to have “Spacelike Separation”. “Nonlocality” connected Points outside the Light Cone. They have Space-time separation. Simultaneously Events (Time = 0), in any given Reference Frame , cannot be causally connected unless the signal between them travels at superluminal speed. Some physicists use the term “Holistic” instead of “Nonlocal”. “Holistic” = “Nonlocal” 53 Return to Table of Content
  • 54. 54 SOLO 1908 Geiger-Marsden Experiment. Ernest Rutherford 1871 - 1937 Nobel Prize 1908 Chemistry Hans Wilhelm Geiger 1882 – 1945 Nazi Physicist Sir Ernest Marsden 1889 – 1970 Geiger-Marsden working with Ernest Rutherford performed in 1908 the alpha-particle scattering experiment. H. Geiger and E. Marsden (1909), “On a Diffuse Reflection of the α- particle”, Proceedings of the Royal Society Series A 82:495- 500 A small beam of α-particles was directed at a thin gold foil. According to J.J. Thomson atom-model it was anticipated that most of the α-particles would go straight through the gold foil, while the remainder would at most suffer only slight deflections. Geiger-Marsden were surprised to find out that, while most of the α-particles were not deviated, some were scattered through very large angles after passing the foil. QUANTUM THEORIES
  • 55. 55 ParticlesSOLO Electron Charge R.A. Millikan measured the charge of the electron by equalizing the weight m g of a charged oil drop with an electric field E. 1909 Robert Andrews Millikan 1868 – 1953 Nobel Prize 1923
  • 56. 56 SOLO Rutherford Atom Model 1911 Ernest Rutherford finds the first evidence of protons. To explain the Geiger-Marsden Experiment of 1908 he suggested in 1911 that the positively charged atomic nucleus contain protons. Ernest Rutherford 1871 - 1937 Nobel Prize 1908 Chemistry Hans Wilhelm Geiger 1882 – 1945 Nazi Physicist Sir Ernest Marsden 1889 – 1970 -- -- -- -- -- -- -- -- -- -- +2 +2 +2 Rutherford assumed that the atom model consists of a small nucleus, of positive charge, concentrated at the center, surrounded by a cloud of negative electrons. The positive α-particles that passed close to the positive nucleus were scattered because of the electrical repealing force between the positive charged α-particle and the nucleus . QUANTUM MECHANICS Return to Table of Content
  • 57. 57 1913 SOLO Niels Bohr presents his quantum model of the atom. Niels Bohr 1885 - 1962 Nobel Prize 1922 QUANTUM MECHANICS Bohr Quantum Model of the Atom.
  • 58. 58 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model In 1911, Bohr travelled to England. He met with J. J. Thomson of the Cavendish Laboratory and Trinity College, Cambridge, and New Zealand's Ernest Rutherford, whose 1911 Rutherford model of the atom had challenged Thomson's 1904 Plum Pudding Model.[ Bohr received an invitation from Rutherford to conduct post-doctoral work at Victoria University of Manchester. He adapted Rutherford's nuclear structure to Max Planck's quantum theory and so created his Bohr model of the atom.[ In 1885, Johan Balmer had come up with his Balmer series to describe the visible spectral lines of a hydrogen atoms: that was extended by Rydberg in 1887, to Additional series by Lyman (1906), Paschen (1908) ( )222 / nmmB −=λ 2 2 1 1 1 H i f R n nλ   = − ÷ ÷   Bohr Model of the Hydrogen Atom consists on a electron, of negative charge, orbiting a positive charge nucleus. The Forces acting on the orbiting electron are AttractionofForceticElectrosta r e F ForcelCentripeta r m F e c 2 0 2 2 4 v επ = = m – electron mass v – electron orbital velocity r – orbit radius e – electron charge ( )229 0 /109 4 1 coulombmN ⋅×= επ QUANTUM MECHANICS
  • 59. 59 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 1) The Conditions for Orbit Stability are 2 0 22 4 v r e r m FF ec επ = = rm e 04 v επ = The Total Energy E, of the Electron, is the sum of the Kinetic Energy T and the Potential Energy V r e r e r e r em VTE 0 2 0 2 0 2 0 22 84842 v επεπεπεπ −=−=−=+= To get some quantitative filing let use the fact that to separate the electron from the atom we need 13.6 eV (this is an experimental result), then E = -13.6 eV = 2.2x10-18 joule. Therefore ( ) ( ) ( ) m joule coulombmN coulomb E e r 11 18 229 219 0 2 103.5 102.2 /109 2 106.1 8 − − − ×= ×− ⋅× × −=−= επ QUANTUM MECHANICS
  • 60. 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 2) The problem with this Model is, since the electron accelerates with a =v2 /r, according to Electromagnetic Theory it will radiate energy given by Larmor Formula (1897) ( ) sec/109.2sec/106.4 43 2 43 2 109 4233 0 6 3 0 22 evjoule rmc e c ae P ×=×=== − επεπ As the electron loses energy the Total Energy becomes more negative and the radius decreases, and since P is proportional to 1/r4 , the electron radiates energy faster and faster as it spirals toward the nucleus. Bohr had to add something to explain the stability of the orbits. He knew the results of the discrete Hydrogen Spectrum lines and the quantization of energy that Planck introduced in 1900 to obtain the Black Body Radiation Equation. Sir Joseph Larmor FRS (1857 – 1942) QUANTUM MECHANICS 60
  • 61. 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 3) To understand Bohr novelty let look at an Elastic Wire that vibrates transversally. At Steady State the Wavelengths always fit an integral number of times into the Wire Length. This is true if we bend the Wire and even if we obtain a Closed Loop Wire. If the Wire is perfectly elastic the vibration will continue indefinitely. This is Resonance. Bohr noted that the Angular Momentum of the Orbiting electron in the Atom Hydrogen Model had the same dimensions as the Planck’s Constant. This led him to postulate that the Angular Momentum of the Orbiting Electrons must be multiple of Planck’s Constant divided by 2 π. ,3,2,1 24 v 0 === n h nr rm e mrm n n n πεπ ,3,2,12 0 22 == n em hn rn π ε therefore QUANTUM MECHANICS 61
  • 62. 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 4) Energy Levels and Spectra We obtained ,3,2,1 1 88 222 0 4 0 2 =      −=−= n nh em r e E n n εεπ ,3,2,12 0 22 == n em hn rn π ε and Energy Levels: The Energy Levels are all negative signifying that the electron does not have enough energy to escape from the atom. The lowest energy level E1 is called the Ground State. The higher levels E2, E3, E4,…, are called Excited States. In the limit n →∞, E∞=0 and the electron is no longer bound to the nucleus to form an atom. QUANTUM MECHANICS 62
  • 63. 63 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 5) According to the Bohr Hydrogen Model when the electron is excited he drops to a lower state, and a single photon of light is emitted Initial Energy – Final Energy = Photon Energy vh nnh em nh em nh em EE iffi fi =         −=         +        −=− 2222 0 4 222 0 4 222 0 4 11 8 1 8 1 8 εεε where v is the photon frequency. If λ is the Wavelength of the photon we have         −=         −= − == 222232 0 4 1111 8 1 if H if fi nn R nnch em ch EE c v ελ 2in = Balmer series (1885) 3in = Paschen series (1908) 4in = Brackett series (1922) Lyman series (1906)1in = We recovered the Rydberg Formula (1887) ( ) ( ) ( ) 17 3348212 41931 32 0 4 10097.1 sec1063.6/103/1085.88 106.1101.9 8 − −− −− ×= −×××××× ××× = m joulesmmfarad coulombkg ch em ε QUANTUM MECHANICS
  • 64. 64 1913 SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Hydrogen Model (continue – 6) 2. The Bohr model treats the electron as if it were a miniature planet, with definite radius and momentum. This is in direct violation of the uncertainty principle (formulated by Werner Heisenberg in 1927) which dictates that position and momentum cannot be simultaneously determined. 1. It fails to provide any understanding of why certain spectral lines are brighter than others. There is no mechanism for the calculation of transition probabilities. While the Bohr model was a major step toward understanding the quantum theory of the atom, it is not in fact a correct description of the nature of electron orbits. Some of the shortcomings of the model are: The electrons in free atoms can will be found in only certain discrete energy states. These sharp energy states are associated with the orbits or shells of electrons in an atom, e.g., a hydrogen atom. One of the implications of these quantized energy states is that only certain photon energies are allowed when electrons jump down from higher levels to lower levels, producing the hydrogen spectrum. The electron must jump instantaneously because if he moves gradually it will radiate and lose energy in the process. The Bohr model successfully predicted the energies for the hydrogen atom, but had significant failures. Quantized Energy States QUANTUM MECHANICS Return to Table of Content
  • 65. 1915Einstein’s General Theory of Relativity The “General” Theory of Relativity takes in consideration the action of Gravity and does not assume Unaccelerated Observer like “Special” Theory of Relativity. Principle of Equivalence – The Inertial Mass and the Gravitational Mass of the same body are always equal. (checked by experiments first performed by Eötvos in 1890) Principle of Covariance -- The General Laws of Physics can be expressed in a form that is independent of the choise of the coordinate system. Principle of Mach -- The Gravitation Field and Metric (Space Curvature) depend on the distribution of Matter and Energy. SOLO GENERAL RELATIVITY Dissatisfied with the Nonlocality (Action at a Distance) of Newton’s Law of GravityEinstein developed the General Theory of Gravity. Albert Einstein 1879 - 1955 Nobel Prize 1921 65
  • 66. GENERAL RELATIVITY Einstein’s General Theory Equation  TENSOR MOMENTUMENERGY CURVATURETIMESPACE TG c RgR − − =− µνµνµν π 2 8 2 1  The Matter – Energy Distribution produces the Bending (Curvature) of the Space-Time. All Masses are moving on the Shortest Path (Geodesic) of the Curved Space-Time. In the limit (Weak Gravitation Fields) this Equation reduce to the Poisson’s Equation of Newton’s Gravitation Law SOLO 66
  • 68. GENERAL RELATIVITY Einstein’s General Theory of Relativity (Summary) • Gravity is Geometry • Mass Curves Space – Time • Free Mass moves on the Shortest Path in Curved Space – Time. SOLO Newton’s Gravity The Earth travels around the Sun because it is pulled by the Gravitational Force exerted by the Mass of the Sun. Mass (somehow) causes a Gravitational Force which propagates instantaneously (Action at a Distance) and causes True Acceleration. Einstein’s Gravity The Earth travels around the Sun because is the Shortest Path in the Curved Space – Time produced by the Mass of the Sun. Mass (somehow) causes a Warping, which propagates with the Speed of Light, and results in Apparent Acceleration. 68 Return to Table of Content
  • 69. 69 Photons EmissionSOLO Theory of Light Emission. Concept of Stimulated Emission 1916 Albert Einstein 1879 - 1955 Nobel Prize 1921 http://members.aol.com/WSRNet/tut/ut4.htm Spontaneous Emission & Absorption Stimulated Emission & Absorption “On the Quantum Mechanics of Radiation” Run This Einstein’s work laid the foundation of the Theory of LASER (Light Amplification by Stimulated Emission) Return to Table of Content
  • 70. E. RUTHERFORD OTTO STERN W. GERLACH A. COMPTON L. de BROGLIE W. PAULI QUANTUM MECHANICS 1919: ERNEST RUTHERFORD FINDS THE FIRST EVIDENCE OF PROTONS. HE SUGGESTED IN 1914 THAT THE POSITIVELY CHARGED ATOMIC NUCLEUS CONTAINS PROTONS. 1922: OTTO STERN AND WALTER GERLACH SHOW “SPACE QUANTIZATION” 1923: ARTHUR COMPTON DISCOVERS THE QUANTUM NATURE OF X RAYS, THUS CONFIRMS PHOTONS AS PARTICLES. 1924: LOUIS DE BROGLIE PROPOSES THAT MATTER HAS WAVE PROPERTIES. 1924: WOLFGANG PAULI STATES THE QUANTUM EXCLUSION PRINCIPLE. 70
  • 71. W. HEISENBERG MAX BORN P. JORDAN S. GOUDSMITH G. UHLENBECK E. SCHRODINGER QUANTUM MECHANICS 1925: WERNER HEISENBERG, MAX BORN, AND PASCAL JORDAN FORMULATE QUANTUM MATRIX MECHANICS. 1925: SAMUEL A. GOUDSMITH AND GEORGE E. UHLENBECK POSTULATE THE ELECTRON SPIN 1926: ERWIN SCHRODINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 71
  • 72. W. HEISENBERGMAX BORN PAUL DIRAC J. von NEUMANN QUANTUM MECHANICS 1926: MAX BORN GIVES A PROBABILISTIC INTERPRATATION OF THE WAVE FUNCTION. 1927: WERNER HEISENBERG STATES THE QUANTUM UNCERTAINTY PRINCIPLE. 1928: PAUL DIRAC STATES HIS RELATIVISTIC QUANTUM WAVE EQUATION. HE PREDICTS THE EXISTENCE OF THE POSITRON. 1932: JHON von NEUMANN WROTE “THE FOUNDATION OF QUANTUM MECHANICS” 72 Return to Table of Content
  • 73. 73 SOLO 1922 Otto Stern and Walter Gerlach show “Space Quantization” Walter Gerlach 1889 - 1979 They designed the Stern-Gerlach Experiment that determine if a particle has angular momentum. http://en.wikipedia.org/wiki/Stern-Gerlach_experiment Otto Stern 1888 – 1969 Nobel Prize 1943 They directed a beam of neutral silver atoms from an oven trough a set of collimating slits into an inhomogeneous magnetic field. A photographic plate recorded the configuration of the beam. They found that the beam split into two parts, corresponding to the two opposite spin orientations, that are permitted by space quantization. Run This QUANTUM MECHANICS
  • 74. 74 SOLO 1923 Arthur Compton discovers the quantum nature of X rays, thus confirms photons as particles. Arthur Holly Compton 1892 - 1962 Nobel Prize 1927 incident photon ( ) ( ) chp hE photon photon /ν ν =− =− ( ) ( ) 0 2 0 =− =− electron electron p cmE target electron Compton effect consists of a X ray (incident photons) colliding with rest electrons incident photon scatteredphoton ( ) ( ) chp hE photon photon /ν ν =− =− ( ) ( ) 0 2 0 =− =− electron electron p cmE ( ) ( ) chp hE photon photon /' ' ν ν =+ =+ ( ) ( ) ( ) ( )' 2 2 0 2 2242 0 νν −= +=+ −+=+ hT TcmTp cpcmE electron photonelectron ϕ θ ( )ϕ νν λλ cos1 ' ' 0 −=−=− cm hcc scatteredelectron target electron is scattered in the φ direction (detected by an X-ray spectrometer) and the electrons in the θ direction. Run This QUANTUM MECHANICS Return to Table of Content
  • 75. 75 SOLO 1924 Louis de Broglie proposes that matter has wave properties and using the relation between Wavelength and Photon mass: Louis de Broglie 1892 - 1987 Nobel Prize 1929 cm h p hc cmph cp = = === ν ν λ He postulate that any Particle of mass m and velocity v has an associate Wave with a Wavelength λ. QUANTUM MECHANICS
  • 76. SOLO Niels Bohr 1885 - 1962 Nobel Prize 1922 Explanation of Bohr Model using de Broglie Relation To understand Bohr novelty let look at an Elastic Wire that vibrates transversally. At Steady State the Wavelengths always fit an integral number of times into the Wire Length. This is true if we bend the Wire and even if we obtain a Closed Loop Wire. If the Wire is perfectly elastic the vibration will continue indefinitely. This is Resonance. ,3,2,12 4 0 === nr m r e h nn n n π επ λ ,3,2,12 0 22 == n em hn rn π ε We found the Electron Orbital Velocity Return to Bohr Hydrogen Model using de Broglie Relation Louis de Broglie 1892 - 1987 Nobel Prize 1929 rm e 04 v επ = Using de Broglie Relation m r e h m h 04 v επ λ == At Steady State the Wavelengths always fit an integral number of times into the Wire Length. We obtain the same relation as Bohr for the Orbit radius: QUANTUM MECHANICS 76 Return to Table of Content
  • 77. 77 SOLO 1924 Wolfgang Pauli states the “Quantum Exclusion Principle” Wolfgang Pauli 1900 - 1958 Nobel Prize 1945 QUANTUM MECHANICS Return to Table of Content
  • 78. QUANTUM THEORIES Werner Heisenberg, Max Born, and Pascal Jordan formulate Quantum Matrix Mechanics. QUANTUM MATRIX MECHANICS. Werner Karl Heisenberg (1901 – 1976) Nobel Price 1932 Max Born (1882–1970) Nobel Price 1954 Ernst Pascual Jordan (1902 – 1980) Nazi Physicist http://en.wikipedia.org/wiki/Matrix_mechanics 1925 Matrix mechanics was the first conceptually autonomous and logically consistent formulation of quantum mechanics. It extended the Bohr Model by describing how the quantum jumps occur. It did so by interpreting the physical properties of particles as matrices that evolve in time. It is equivalent to the Schrödinger wave formulation of quantum mechanics, and is the basis of Dirac's bra-ket notation for the wave function. SOLO In 1928, Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics, but Heisenberg alone won the 1932 Prize "for the creation of quantum mechanics, the application of which has led to the discovery of the allotropic forms of hydrogen",[47] while Schrödinger and Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory".[47] On 25 November 1933, Born received a letter from Heisenberg in which he said he had been delayed in writing due to a "bad conscience" that he alone had received the Prize "for work done in Gottingen in collaboration — you, Jordan and I."[48] Heisenberg went on to say that Born and Jordan's contribution to quantum mechanics cannot be changed by "a wrong decision from the outside." 78 Return to Table of Content
  • 79. 1925 SAMUEL A. GOUDSMITH AND GEORGE E. UHLENBECK POSTULATE THE ELECTRON SPIN George Eugene Uhlenbeck (1900 – 1988) Samuel Abraham Goudsmit (1902 – 1978) Two types of experimental evidence which arose in the 1920s suggested an additional property of the electron. One was the closely spaced splitting of the hydrogen spectral lines, called fine structure. The other was the Stern-Gerlach experiment which showed in 1922 that a beam of silver atoms directed through an inhomogeneous magnetic field would be forced into two beams. Both of these experimental situations were consistent with the possession of an intrinsic angular momentum and a magnetic moment by individual electrons. Classically this could occur if the electron were a spinning ball of charge, and this property was called electron spin. In 1925, the Dutch Physicists S.A. Goudsmith and G.E. Uhlenbeck realized that the experiments can be explained if the electron has an magnetic property of Rotation or Spin. They work actually showed that the electron has a quantum-mechanical notion of spin that is similar to the mechanical rotation of particles. http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html no magnetic field B = 0 cba ,, fed ,, a b c d e f magnetic field B 0≠ Zeeman’s Effect QUANTUM MECHANICS 79
  • 80. Spin In quantum mechanics and particle physics, Spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. Spin is a solely quantum-mechanical phenomenon; it does not have a counterpart in classical mechanics (despite the term spin being reminiscent of classical phenomena such as a planet spinning on its axis).[ Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. Orbital angular momentum is the quantum- mechanical counterpart to the classical notion of angular momentum: it arises when a particle executes a rotating or twisting trajectory (such as when an electron orbits a nucleus).The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which particles are observed to possess angular momentum that cannot be accounted for by orbital angular momentum alone.[ http://en.wikipedia.org/wiki/Spin_(physics) In some ways, spin is like a vector quantity; it has a definite “magnitude”; and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number.[2] However, in a technical sense, spins are not strictly vectors, and they are instead described as a related quantity: a Spinor. In particular, unlike a Euclidean vector, a spin when rotated by 360 degrees can have its sign reversed QUANTUM MECHANICS 80
  • 81. ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 In January 1926, Schrödinger published in Annalen der Physik the paper "Quantisierung als Eigenwertproblem" [“Quantization as an Eigenvalue Problem”] on wave mechanics and presented what is now known as the Schrödinger equation. In this paper, he gave a "derivation" of the wave equation for time-independent systems and showed that it gave the correct energy eigenvalues for a hydrogen-like atom. This paper has been universally celebrated as one of the most important achievements of the twentieth century and created a revolution in quantum mechanics and indeed of all physics and chemistry. A second paper was submitted just four weeks later that solved the quantum harmonic oscillator, rigid rotor, and diatomic molecule problems and gave a new derivation of the Schrödinger equation. A third paper in May showed the equivalence of his approach to that of Heisenberg. http://en.wikipedia.org/wiki/Erwin_Schr%C3%B6dinger QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 SOLO 81
  • 82. MAX BORN GIVES A PROBABILISTIC INTERPRATATION OF THE WAVE FUNCTION. 1926 Max Born (1882–1970) Nobel Price 1954 Max Born wrote in 1926 a short paper on collisions between particles, about the same time as Schrödinger paper “Quantization as an Eigenvalue Problem”. Born rejected the Schrödinger Wave Field approach. He had been influenced by a suggestion made by Einstein that, for photons, the Wave Field acts as strange kind of ‘phantom’ Field ‘guiding’ the photon particles on paths which could therefore be determined by Wave Interference Effects. Max Born reasoned that the Square of the Amplitude of the Waveform in some specific region of configuration space is related to the Probability of finding the associated quantum particle in that region of configuration space. Since Probability is a real number, and the integral of all Probabilities over all regions of configuration space, the Wave Function must satisfy 1* =∫ +∞ ∞− dVψψ Condition of Normalization of the Wave Function Therefore the probability of finding the particle between a and b is given by [ ] ( ) ( )∫=≤≤ b a xdxxbXaP ψψ * Einstein rejected this interpretation. In a 1926 letter to Max Born, Einstein wrote: "I, at any rate, am convinced that He [God] does not throw dice."[ QUANTUM MECHANICS SOLO 82 Return to Table of Content
  • 83. QUANTUM MECHANICS In December 1926 Einstein wrote a letter to Bohr which contains a phrase that has since become symbolic of Einstein’s lasting dislike of the element of chance implied by the quantum theory: J. Baggott, “The Meaning of Quantum Theory”, Oxford University Press, 1992, pp. 28-29 SOLO 1926 http://en.wikipedia.org/wiki/Max_Born “Quantum mechanics is very impressive. But an inner voice tells me that it is not the real thing. The theory produce a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.” 83
  • 84. 84 SOLO Wavelike Behavior for Electrons In 1927, the wavelike behavior of the electrons was demonstrated by Davisson and Germer in USA and by G.P. Thomson in Scotland. Quantum 1927 Clinton Joseph Davisson 1881 – 1958 Nobel Prize 1937 Lester Halbert Germer 1896 - 1971
  • 85. 85 SOLO Wavelike Behavior for Electrons Quantum 1927 G.P. Thomson carried a series of experiments using an apparatus called an electron diffraction camera. With it he bombarded very thin metal and celluloid foils with a narrow electron beam. The beam then was scattered into a series of rings. George Paget Thomson 1892 – 1975 Nobel Prize 1937 Using these results G.P. Thomson proved mathematically that the electron particles acted like waves, for which he received the Nobel Prize in 1937. J.J. Thomson the father of G.P. proved that the electron is a particle in 1897, for which he received the Nobel Prize in 1906. Discovery of the Electron Results of a double-slit- experiment performed by Dr. Tonomura showing the build-up of an interference pattern of single electrons. Numbers of electrons are 11 (a), 200 (b), 6000 (c), 40000 (d), 140000(e).
  • 86. 86 SOLO Optics HistoryRaman Effect 1928 http://en.wikipedia.org/wiki/Raman_scattering http://en.wikipedia.org/wiki/Chandrasekhara_Venkata_Raman Nobel Prize 1930 Chandrasekhara Venkata Raman 1888 – 1970 Raman Effect was discovered in 1928 by C.V. Raman in collaboration with K.S. Krishnan and independently by Grigory Landsberg and Leonid Mandelstam. Monochromatic light is scattered when hitting molecules. The spectral analysis of the scattered light shows an intense spectral line matching the wavelength of the light source (Rayleigh or elastic scattering). Additional, weaker lines are observed at wavelength which are shifted compared to the wavelength of the light source. These are the Raman lines. Virtual Energy States IR Absorbance Excitation Energy Rayleigh Scattering Stokes - Raman Scattering Anti-Stokes - Raman Scattering
  • 87. 87 SOLO Stimulated Emission and Negative Absorption 1928 Rudolph W. Landenburg confirmed existence of stimulated emission and Negative Absorption Lasers History Rudolf Walter Ladenburg (June 6, 1882 – April 6, 1952) was a German atomic physicist. He emigrated from Germany as early as 1932 and became a Brackett Research Professor at Princeton University. When the wave of German emigration began in 1933, he was the principal coordinator for job placement of exiled physicist in the United States. Albert Einstein and Rudolf Ladenburg, Princeton Symposium, on the occasion of Ladenburg's retirement, May 28, 1950. Hedwig Kohn is in the background on the left. Photo courtesy of AIP Emilio Segrè Visual Archives. Return to Table of Content
  • 88. QUANTUM MECHANICS SOLO Wave Packet A wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere constructively only over a small region of space, and destructively elsewhere Depending on the evolution equation, the wave packet's envelope may remain constant (no dispersion, see figure) or it may change (dispersion) while propagating. As an example of propagation without dispersion, consider wave solutions to the following wave equation: ψ ψ 2 2 2 2 v 1 ∇= ∂ ∂ t where v is the speed of the wave's propagation in a given medium. The wave equation has plane-wave solutions ( ) ( )trki eAtr ω ψ −⋅ =   , ( ) v,/1111 2222 kkkkknPropagatioWaveofDirectionkzkykxkk zyxzyx =++=++= ω A wave packet without dispersion A wave packet with dispersion ( ) ( ) ( )tcxiktcx etx −+−− = 0 2 ,ψ 88 Run This
  • 89. QUANTUM MECHANICS SOLO Wave Packet The wave equation has plane-wave solutions ( ) ( )rkti eAtr   ⋅−− = ω ψ , ( ) v,/1111 2222 kkkkknPropagatioWaveofDirectionkzkykxkk zyxzyx =++=++= ω ( )rptE h r k k ktE h rkt hv    ⋅− / =⋅−=⋅− = = 122 /E π ω νπω ( ) p h p h v k hhphv / ===== =/== 122 v 2 v 2/://v πλλ π λ ππω ( ) ( ) ( ) ( )rptEhirkti eAeAtr   ⋅−/−⋅−− == / , ω ψ where v is the velocity , v is the frequency, λ is the Wavelength of the Wave Packet. The Energy E and Momentum p of the Particle are ( ) ( ) λ π λ νπν ππ hh phhE hhhh / ==/== =/=/ 2 &2 2/:2/: de Broglie RelationEinstein Relation The wave packet travels to the direction for ω = kv and to direction for ω = - kv.k1 k1− 89
  • 90. QUANTUM MECHANICS SOLO Wave Packet A wave packet is a localized disturbance that results from the sum of many different wave forms. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. From the basic solutions in one dimension, a general form of a wave packet can be expressed as ( ) ( ) ( ) ( ) ( ) ( ) ( ) tEhirptEhi erepApd h tr //− +∞ ∞− ⋅−//− = / = ∫ //3 3 2 1 ,   ψ π ψ ( ) ( ) ( ) ( ) ( )perrd h pA rphi   Φ= / = ∫ +∞ ∞− ⋅/− :0, 2 1 /3 3 ψ π The factor comes from Fourier Transform conventions. The amplitude contains the coefficients of the linear superposition of the plane-wave solutions. Using the Inverse Fourier Transform we obtain: ( )3 2/1 π ( )pA  ( ) ( ) ( ) ( ) ∫ +∞ ∞− ⋅// / = rphi epApd h r  /3 3 2 1 π ψwhere zyx pdpdpdpd =3 dzdydxrd =3 Define ( ) ( ) ( ) ( ) ∫ +∞ ∞− ⋅//− / =Φ rphi etrrd h tp  /3 3 , 2 1 :, ψ π Wave Function in Momentum Space 90 Return to Table of Content
  • 91. ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 Following Max Planck's quantization of light (see black body radiation), Albert Einstein interpreted Planck's quanta to be photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, one of the first signs of wave–particle duality. Since energy and momentum are related in the same way as frequency and wavenumber in special relativity, it followed that the momentum p of a photon is proportional to its wavenumber k. c k h hwherekh h p c πν λ π πλ νλ 22 :, 2 : /= ===//== Louis de Broglie hypothesized that this is true for all particles, even particles such as electrons. He showed that, assuming that the matter waves propagate along with their particle counterparts, electrons form standing waves, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.[7] These quantized orbits correspond to discrete energy levels, and de Broglie reproduced the Bohr model formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum: hn h nL /== π2 According to de Broglie the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit: n λ = 2 π r http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Historical Background and Development QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 SOLO 91
  • 92. ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 ( ) ( ) ( )λνπ νπω νλ ω ψ /2 2 /v v/ , xtixti eAeAtx −− = = −− == http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Historical Background and Development (continue – 1) Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by William R. Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system — the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action. For a general form of a Progressive Wave Function in + x direction with velocity v and frequency v: The Energy E and Momentum p of the Particle are λ π λ νπν hh phhE / ==/== 2 2 ( ) ( ) ( )xptEhi eAtx −/− = / ,ψTherefore QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 SOLO 92
  • 93. ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Historical Background and Development (continue – 2) We want to find the Differential Equation yielding the Wave Function . We have Wave Function: ( ) ( ) ψ ψ 2 2 / 2 2 2 2 h p eA h p x xptEhi / −= / −= ∂ ∂ −/− At particle speeds small compared to speed of light c, the Total Energy E is the sum of the Kinetic Energy p2 /2m and the Potential Energy V (function of position and time): ψψψ ψ V m p EV m p E +=⇒+= × 22 22 2 2 22 x hp ∂ ∂ /−= ψ ψ ti h E ∂ ∂/ −= ψ ψ cV xm h ti h <<− ∂ ∂/ = ∂ ∂/ v 2 2 22 ψ ψψ QUANTUM MECHANICS SOLO ( ) ( ) ( )xptEhi eAtx −/− = / ,ψ 93
  • 94. ERWIN SCHRÖDINGER STAES THE NONRELATIVISTIC QUANTUM WAVE EQUATION AND FORMULATES THE QUANTUM MECHANICS. HE PROVES THAT WAVE AND MATRIX FORMULATIONS ARE EQUIVALENT 1926 Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Historical Background and Development (continue – 3) cV xm h ti h <<− ∂ ∂/ = ∂ ∂/ v 2 2 22 ψ ψψ Non-Relativistic One-Dimensional Time Dependent Schrödinger Equation In the same way cV m h ti h <<−∇ / = ∂ ∂/ v 2 2 2 ψψ ψ Non-Relativistic Three-Dimensional Time Dependent Schrödinger Equation QUANTUM MECHANICS SOLO This is a Linear Partial Differential Equation. It is also a Diffusion Equation (with an Imaginary Diffusion Coefficient), but unlike the Heat Equation, this one is also a Wave Equation given the imaginary unit present in the transient term. 94
  • 95. 1926 Schrödinger Equation Time-dependent Schrödinger equation (single non-relativistic particle) A wave function that satisfies the non-relativistic Schrödinger equation with V=0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary". QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 SOLO 95
  • 96. Schrödinger Equation: Steady State Form Using ti h E ∂ ∂/ −= ψ ψ and the Time-dependent Schrödinger equations cV xm h ti h <<− ∂ ∂/ = ∂ ∂/ v 2 2 22 ψ ψψ Non-Relativistic One-Dimensional Time Dependent Schrödinger Equation cV m h ti h <<−∇ / = ∂ ∂/ v 2 2 2 ψψ ψ Non-Relativistic Three-Dimensional Time Dependent Schrödinger Equationwe can write ( ) cVE h m x <<=− / + ∂ ∂ v0 2 22 2 ψ ψ Non-Relativistic One-Dimensional Steady-State Schrödinger Equation ( ) cVE h m <<− / +∇ v 2 2 2 ψψ Non-Relativistic Three-Dimensional Steady-State Schrödinger Equation QUANTUM MECHANICS Erwin Rudolf Josef Alexander Schrödinger (1887 – 1961) Nobel Prize 1933 1926 SOLO 96 Return to Table of Content
  • 97. Operators in Quantum Mechanics Since, according to Born, ψ*ψ represents Probability of finding the associated quantum particle in a region we can compute the Expectation (Mean) Value of the Total Energy E and of the Momentum p in that region using ( ) ( ) ( ) ( )∫ +∞ ∞− = xdtxtxEtxtE ,,,* ψψ ( ) ( ) ( ) ( )∫ +∞ ∞− = dxtxtxptxtp ,,,* ψψ But those integrals can not compute exactly, since p (x,t) is unknown if x is know, according to Uncertainty Principle. A way to find is by differentiating the Free-Particle Wave Function pandE ( ) ( )xptEhi eA −/− = / ψ ( ) ( ) ( ) ( ) ψ ψ ψ ψ E h i eAE h i t p h i eAp h i x xptEhi xptEhi / −= / −= ∂ ∂ / = / = ∂ ∂ −/− −/− / / Rearranging we obtain ψψ ψψ t hiE xi h p ∂ ∂ /= ∂ ∂/ = t hiE xi h p ∂ ∂ /= ∂ ∂/ = :ˆ :ˆ QUANTUM MECHANICS SOLO We can look at p and E as Operators on ψ (the symbol means “Operator”)∧ Note: One other way to arrive to this result by manipulating the integrals will be given in the following presentations. 97
  • 98. Operators in Quantum Mechanics (continue – 1) We obtained Moment Operatorxi h p ∂ ∂/ =:ˆ t hiE ∂ ∂ /=:ˆ Total Energy Operator Although we derived those operators for free particles, they are entire general results, equivalent to Schrödinger Equation. To see this let write the Operator Equation    Operator Energy Potential Operator Energy Kinetic Operator Energy Total VTE ˆˆˆ +=  2 2222 22 1 2 ˆ xm h xi h mm p T Operator Energy Kinetic ∂ ∂/ −=      ∂ ∂/ ==We have  V xm h t hiE Operator Energy Total + ∂ ∂/ −= ∂ ∂ /= 2 22 2 ˆ Applying this Operator on Wave Function ψ we recover the Schrödinger Equation ψ ψψ V xm h t hi + ∂ ∂/ −= ∂ ∂ / 2 22 2 The two descriptions (Operator and Schrödinger’s) are equivalent. QUANTUM MECHANICS SOLO 98
  • 99. QUANTUM MECHANICS Operators in Quantum Mechanics (continue – 3) We obtained Moment Operatorxi h p ∂ ∂/ =:ˆ t hiE ∂ ∂ /=:ˆ Total Energy Operator Because p and E can be replaced by their Operators in an equation, we can use those Operators to obtain Expectation Values for p and E. ∫∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− ∂ ∂/ =      ∂ ∂/ == dx xi h dx xi h dxpp ψ ψψψψψ *** ˆ ∫∫∫ ∞+ ∞− ∞+ ∞− ∞+ ∞− ∂ ∂ /=      ∂ ∂ /== xd x hixd x hixdEE ψ ψψψψψ *** ˆ Let define the Hamiltonian Operator V xm h H ˆ 2 :ˆ 2 22 + ∂ ∂/ −= Schrödinger Equation in Operator form is ψψ EH ˆˆ = This Equation has a form of an Eigenvalue Equation of the Operator with Eigenvalue Ê and Eigenfunction as the Wavefunction ψ. Hˆ SOLO 99
  • 100. Dirac bracket notation Paul Adrien Maurice Dirac ( 1902 –1984) A elegant shorthand notation for the integrals used to define Operators was introduced by Dirac in 1939 onWavefunctiket nn ψψ ⇔"" Instead of dealing with Wavefunctions ψn, we defined a related Quantum “State”, denoted |ψ› which is called a “ket”, “ket vector”, “state” or “state vector”. The complex conjugate of |ψ› is called the “bra” and is denoted by ‹ψ|. onWavefunctibra nn * "" ψψ ⇔   ket m bra n ψψ When a “bra” is combined with a “ket” we obtain a “bracket”. The following integrals are represented by “bra” and “ket” mnmn AdA ψψτψψ |ˆ|ˆ* ≡∫ mnmn d ψψτψψ |* ≡∫ nnnnnn aAaA ψψψψ =⇔= ˆˆ Operators in Quantum Mechanics (continue – 5) ( ) ( ) ( ) ( ) mnmnmnmnmn AAdAAdA ψψψψτψψψψτψψ |ˆ|ˆ|ˆ|ˆˆ *** ==== ∫∫ nnnnnn aAaA ψψψψ ****** ˆˆ =⇔= QUANTUM MECHANICS SOLO 100 Return to Table of Content
  • 101. QUANTUM THEORIES HILBERT SPACE AND QUANTUM MECHANICS. Ernst Pascual Jordan (1902 – 1980) Nazi Physicist http://en.wikipedia.org/wiki/Matrix_mechanics Born had also learned Hilbert’s theory of integral equations and quadratic forms for an infinite number of variables as was apparent from a citation by Born of Hilbert’s work “Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen” published in 1912. Jordan, too was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert’s book Methoden der mathematischen Physik I, which was published in 1924. This book, fortuitously, contained a great many of the mathematical tools necessary for the continued development of quantum mechanics. In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert Space to describe the algebra and analysis which were used in the development of quantum mechanics Max Born (1882–1970) Nobel Price 1954 John von Neumann (1903 –1957) David Hilbert (1862 –1943) Richard Courant (1888 –1972) SOLO 101
  • 102. 102 Functional AnalysisSOLO Vector (Linear) Space: E is a Linear Space if Addition and Scalar Multiplication are Defined Addition Scalar Multiplication From those equations follows: The null element 0 ∈ E is unique. The addition inverse |η› of |ψ›, (|ψ›+|η›= 0) is unique. E∈∀=⋅ ψψ 00 |η› = (-1) |ψ› is the multiplication inverse of |ψ›. αβ −= E∈∀+=+ χψψχχψ ,1 Commutativity ψψ +=+∈∃ 00..0 tsE3 Identity 0.. =+∈∃∈∀ χψχψ tsEE4 Inverse E∈∀=⋅ ψψψ15 Normalization ( ) ( ) βαψψβαψβα ,& ∀∈∀= E6 Associativity 8 ( ) αηψηαψαηψα ∀∈∀+=+ &, E Distributivity 7 ( ) βαψψβψαψβα ,& ∀∈∀+=+ E Distributivity 2 Associativity( ) ( ) E∈∀++=++ ηχψηψχηχψ ,, The same apply for “bra” ‹ψ| the “conjugate” of the “ket” |ψ›. See also “Functional Analysis ” Presentation for a detailed description
  • 103. 103 Functional AnalysisSOLO Vector (Linear) Space: E is a Linear Space if Addition and Scalar Multiplication are Defined Linear Independence, Dimensionality and Bases    ∈≠ = ⇒=∑= CsomefortrueifDependentLinear allifonlytrueiftIndependenLinear i in i ii 0 0 01 α α ψα A set of vectors |ψi› (i=1,…,n) that satisfy the relation Dimension of a Vector Space E , is the maximum number N of Linear Independent Vectors in this space. Thus, between any set of more that N Vectors |ψi› (i=1,2,…,n>N), there exist a relation of Linear Dependency. Any set of N Linearly Independent Vectors |ψi› (i=1,2,…,N), form a Basis of the Vector Space E ,of Dimension N, meaning that any vector |η› ∈ E can be written as a Linear Combination of those Vectors. Ci N i ii ∈≠= ∑= αψαη 01 In the case of an Infinite Dimensional Space (N→∞), the space will be defined by a “Complete Set” of Basis Vectors. This is a Set of Linearly Independent Vectors of the Space, such that if any other Vector of the Space is added to the set, there will exist a relation of Linear Dependency to the Basis Vectors.
  • 104. SOLO Functional Analysis Use of bra-ket notation of Dirac for Vectors. ketbra TransposeConjugateComplexHfefeefef HH − =⋅==⋅= ,| operatorkete operatorbraf | | Paul Adrien Maurice Dirac (1902 – 1984) Assume the are a basis and the a reciprocal basis for the Hilbert space. The relation between the basis and the reciprocal basis is described, in part, by: je| |if ketbra ji ji efef jij H iji −    = ≠ === 1 0 | ,δ 104 The Inner Product of the Vectors f and e is defined as Inner Product Using Dirac Notation ( ) ( )** & ψψψψ == To every “ket” corresponds a “bra”.
  • 105. 105 Functional AnalysisSOLO Inner Product Using Dirac Notation If E is a complex Linear Space, for the Inner Product (bracket) < | > between the elements (a complex number) is defined by: E∈∀ 321 ,, ψψψ * 1221 || >>=<< ψψψψ1 Commutative Law Using to we can show that:1 4 If E is an Inner Product Space, than we can induce the Norm: [ ] 2/1 111 , ><= ψψψ 2 Distributive Law><+>>=<+< 3121321 ||| ψψψψψψψ 3 C∈><>=< αψψαψψα 2121 || 4 00|&0| 11111 =⇔>=<≥>< ψψψψψ ( ) ( ) ( ) ><+><=><+><=>+<=>+< 1312 1 * 31 * 21 2 * 321 1 132 |||||| ψψψψψψψψψψψψψψ ( ) ( ) ( ) ><=><=><=>< 21 * 1 * 12 * 3 * 12 1 21 |||| ψψαψψαψψαψαψ ( ) ( ) * 1 1 111 2 11 |000|0|0|00|0| ><=>=<⇒><+><=>+>=<< ψψψψψψ
  • 106. 106 Functional AnalysisSOLO Inner Product ηψηψ ≤>< | Cauchy, Bunyakovsky, Schwarz Inequality known as Schwarz Inequality Let |ψ›, |η› be the elements of an Inner Product space E, than : x y ><= >< y y x y yx , , y y y y xxy y yx x ><−= >< − , , 2 0||||| 2* ≥><+><+><+>>=<++< ηηλψηληψλψψηλψηλψ Assuming that , we choose:0| 2/1 ≠= ηηη >< >< −= ηη ηψ λ | | we have: 0| | | | || | || | 2 2* ≥>< >< >< + >< ><>< − >< ><>< −>< ηη ηη ηψ ηη ψηηψ ηη ηψηψ ψψ which reduce to: 0 | | | | | | | 222 ≥ >< >< + >< >< − >< >< −>< ηη ηψ ηη ηψ ηη ηψ ψψ or: ><≥⇔≥><−><>< ηψηψηψηηψψ |0||| 2 q.e.d. Augustin Louis Cauchy )1789-1857( Viktor Yakovlevich Bunyakovsky 1804 - 1889 Hermann Amandus Schwarz 1843 - 1921 Proof:
  • 107. 107 Functional Analysis SOLO Hilbert Space A Complete Space E is a Metric Space (in our case ) in which every Cauchy Sequence converge to a limit inside E. ( ) 2121, ψψψψρ −= David Hilbert 1862 - 1943 A Linear Space E is called a Hilbert Space if E is an Inner Product Space that is complete with respect to the Norm induced by the Inner Product ||ψ1||=[< ψ1, ψ1>]1/2 . Equivalently, a Hilbert Space is a Banach Space (Complete Metric Space) whose Norm is induced by the Inner Product ||ψ1||=[< ψ1, ψ1>]1/2 . Orthogonal Vectors in a Hilbert Space: Two Vectors |η› and |ψ› are Orthogonal if 0|| == ηψψη Theorem: Given a Set of Linearly Independent Vectors in a Hilbert Space |ψi› (i=1,…,n) and any Vector |ψm› Orthogonal to all |ψi›, than it is also Linearly Independent. Proof: Suppose that the Vector |ψm› is Linearly Dependent on |ψi› (i=1,…,n) ∑= =≠ n i iim 1 0 ψαψ But ∑= ==≠ n i imimm 1 0 00  ψψαψψ We obtain a inconsistency, therefore |ψm› is Linearly Independent on |ψi› (i=1,…,n) Therefore in a Hilbert Space, of Finite or Infinite Dimension, by finding the Maximum Set of Orthogonal Vectors we find a Basis that “Complete” covers the Space. q.e.d.
  • 108. 108 Functional Analysis SOLO Hilbert Space Orthonormal Sets Let |ψ1›, |ψ2›, ,…, |ψn›, denote a set of elements in the Hilbert Space H. ( )             ><><>< ><><>< ><><>< = nnnn n n nG ψψψψψψ ψψψψψψ ψψψψψψ ψψψ ,,, ,,, ,,, :,,, 21 22212 12111 21      Jorgen Gram 1850 - 1916 Define the Gram Matrix of the set: Theorem: A set of functions |ψ1›, |ψ2›, ,…, |ψn› of the Hilbert Space H is linearly dependent if and only if the Gram determinant of the set is zero. zeroequalallnot inn αψαψαψα 02211 =+++ Proof: Linearly Dependent Set: Multiplying (inner product) this equation consecutively by |ψ1›, |ψ2›, ,…, |ψn›, we obtain: ( ) 0,,,det 0 0 0 ,,, ,,, ,,, 21 2 1 21 22212 12111 =⇔             =                         ><><>< ><><>< ><><>< n Solution nontrivial nnnnn n n G ψψψ α α α ψψψψψψ ψψψψψψ ψψψψψψ       q.e.d.
  • 109. 109 Functional Analysis SOLO Hilbert Space Orthonormal Sets (continue – 2) Theorem: A set of functions |ψ1›, |ψ2›, ,…, |ψn›, of the Hilbert space H is linearly dependent if and only if the Gram Determinant of the Set is zero. Proof: The Gram Matrix of an Orthogonal Set has only nonzero diagonal; therefore Determinant G (|ψ1›, |ψ2›, ,…, |ψn› ).=║|ψ1 ›║2 ║ |ψ2 › ║2 … ║ |ψn › ║2 ≠ 0, and the Set is Linearly Independent. q.e.d. Corollary: The rank of the Gram Matrix equals the dimension of the Linear Manifold L (|ψ1›, |ψ2›, ,…, |ψn› ). If Determinant G (ψ1›, |ψ2›, ,…, |ψn›) is nonzero, the Gram Determinant of any other Subset is also nonzero. Definition 1: Two elements |ψ›,|η› of a Hilbert Space H are said to be orthogonal if <ψ|η>=0. Definition 2: Let S be a nonempty Subset of a Hilbert Space H. S is called an Orthogonal Set if |ψ›┴|η› for every pair |ψ›,|η› є S and |ψ› ≠ |η›. If in addition ║ |ψ›║=1 for every |ψ› є S, then S is called an Orthonormal Set. Lemma: Every Orthogonal Set is Linearly Independent. If |η› is Orthogonal to every element of the Set (|ψ1›, |ψ2›, ,…, |ψn› ), then |η› is Orthogonal to Manifold L (|ψ1›, |ψ2›, ,…, |ψn› ). If then for every we have:nii ,,2,10, =∀=>< ψη ( )n n i ii L ψψψαχ ,,1 1 ∈= ∑= 0,, 1 =><>=< ∑= n i ii  ψηαχη
  • 110. 110 Functional AnalysisSOLO Hilbert Space Orthonormal Sets (continue – 3) Gram-Schmidt Orthogonalization Process Jorgen Gram 1850 - 1916 Erhard Schmidt 1876 - 1959 Let Ψ=(|ψ1›, |ψ2›, ,…, |ψn› ) any finite Set of Linearly Independent Vectors and L (|ψ1›, |ψ2›, ,…, |ψn› ) the Manifold spanned by the Set Ψ. The Gram-Schmidt Orthogonalization Process derive a Set (|e1›, |e2›, ,…, |en› ) of Orthonormal Elements from the Set Ψ. 11 : ψη = 1 11 21 22 11 21 21 1121212112122 , , , , ,,,0: η ηη ψη ψη ηη ψη α ηηαψηηψαψη >< >< −=⇒ >< >< =⇒ ><−>>=<=<⇒−= y ∑ ∑∑ − = − = − = >< >< −=⇒ >< >< =⇒ ><−>>=<=<⇒−= 1 1 1 1 1 1 , , , , ,,,0: i j j ji ji ii kk ki ik i j jkkjikki i j jijii kj η ηη ηψ ψη ηη ηψ α ηηαψηηηηαψη δ  
  • 111. 111 Functional AnalysisSOLO Hilbert Space Orthonormal Sets (continue – 4) Gram-Schmidt Orthogonalization Process (continue) Jorgen Gram 1850 - 1916 Erhard Schmidt 1876 - 1959 11 : ψη = 1 11 21 22 , , : η ηη ψη ψη >< >< −= ∑ − = >< >< −= 1 1 , , : i j j ji ji ii η ηη ηψ ψη   2/1 11 1 1 , : >< = ηη η e   Orthogonalization Normalization ∑ − = >< >< −= 1 1 , , : n j j ji jn nn η ηη ηψ ψη 2/1 22 2 2 , : >< = ηη η e 2/1 , : >< = ii i ie ηη η 2/1 , : >< = nn n ne ηη η
  • 112. 112 Functional AnalysisSOLO Hilbert Space Discrete |ei› and Continuous |wα› Orthonormal Bases From those equations we obtain ijji ee δ=| The Orthonormalization Relation ( )'| ' ααδαα −=ww A Vector |ψ› will be represented by ( ) ψψψψ ∑∑∑∑ ==== ==== n i ii n i ii n i ii n i ii eeeeeeec 1111 || ( )ψαψαψααψ αααααααα ∫∫∫∫ ==== wwdwwdwwdwcd i n j jiji n j jj ceeceec ij ==⇒= ∑∑ == 11  δ ψψ ( ) α ααδ αααααα αψαψ cwwcdwwcd ==⇒= ∫∫ −  ' ''''' Therefore Iee n i ii =∑=1 Iwwd =∫ ααα The Closure Relations I – the Identity Operator (its action on any state leaves it unchanged). α- a real number or vector, not complex-valued The Vectors in Hilbert Space can be Countable (Discrete) or Uncountable (Continuous).
  • 113. 113 Functional AnalysisSOLO Hilbert Space Series Expansions of Arbitrary Functions Definitions: Approximation in the Mean by an Orthonormal Series |ψ1›, |ψ2›, ,…, |ψn› : Let |η› be any function. The numbers: are called the Expansions Coefficients or Components of |η› with respect to the given Orthonormal System nn ψηα ,:= From the relation we obtain or 2 1 2 , ηηηα =≤∑= n i i 0| 11 2 1 ≥      −      −=      − ∑∑∑ === n i ii n i ii n i ii ψαηψαηψαη 0|2| ||| 1 * 1 * 1 * 1 * 11 * ≥−=+−= +−− ∑∑∑ ∑∑∑ === === n i ii n i ii n i ii n i ii n i ii n i ii ααηηααααηη ααηψαψηαηη
  • 114. 114 Functional AnalysisSOLO Hilbert Space Series Expansions of Arbitrary Functions Approximation in the Mean by an Orthonormal Series |ψ1›, |ψ2›, |ψ3›,… : Let |η› be any function. The numbers: are called the Expansions Coefficients or Components of |η› with respect to the given Orthonormal System nnc ψη,:= 2 1 2 ηα ≤∑= n i i Since the sum on the right is independent on n, is true also for n →∞, we have 2 1 2 ηα ≤∑ ∞ =i i Bessel’s Inequality Bessel’s Inequality is true for every Orthonormal System. It proves that the sum of the square of the Expansion Coefficients always converges. Friedrich Wilhelm Bessel 1784 - 1846
  • 115. 115 Functional AnalysisSOLO Hilbert Space Series Expansions of Arbitrary Functions Approximation in the Mean by an Orthonormal Series |ψ1›, |ψ2›, |ψ3›,… : If for a given Orthonormal System |ψ1›, |ψ2›, |ψ3›,… any piecewise continuous function |η› can be approximated in the mean to any desired degree of accuracy ε by choosing a n large enough ( n>N (ε) ), i.e.: ( )εεψαη Nnfor n i ii >≤− ∑=1 then the Orthonormal System |ψ1›, |ψ2›, |ψ3›,… is said to be Complete. For a Complete Orthonormal System |ψ1›, |ψ2›, |ψ3›,… the Bessel’s Inequality becomes an Equality: 2 1 2 ηα =∑ ∞ =i i Parseval’s Equality applies for Complete Orthonormal Systems This relation is known as the “Completeness Relation”. ( )( ) ∑∑∑∑∑ ∞ = ∞ = ∞ = ∞ = ∞ = +++=++=+ ++=++=+ 1 * 1 * 1 * 1 * 1 ,2, i ii i ii i ii i ii i iiii dcdc βββαβαααχη χχηηχηχηχη ∑∑ ∞ = ∞ = += 1 * 1 * ,2 i ii i ii βαβαχη A more general form, for , can be derived as follows:∑∑ ∞ = ∞ = == 1 * 1 * & i ii i ii ββχααη Marc-Antoine Parseval des Chênes 1755 - 1836
  • 116. Functional AnalysisSOLO Hilbert Space Linear Operators in Hilbert Space An Operator L in Hilbert Space acting on a Vector |ψ›, produces a Vector |η›. ψη L  = L  ψη = The arrow over L means that the Operator is acting on the Vector on the Left, ‹ψ|. An Operator L is Linear if it Satisfies ( ) CLLL ∈+=+ βαηβψαηβψα ,  Consider the quantities . They are in general not equal.( ) ( ) ψηψη || LandL  Eigenvalues and Eigenfunction of a Linear Operator are defined by CL ∈= λψλψ  The Eigenfunction |ψ› is transformed by the Operator L into multiple of itself, by the Eigenvalue λ. The conjugate equation is ( ) CLL ∈== λψλψψ **  The corresponding Operator which transforms the “bra” ‹ψ| , called the Adjoint Operator, is L  The arrow over means that the Operator is acting on the Vector on the Right, |ψ›. L  116
  • 117. Functional AnalysisSOLO Hilbert Space Adjoint or Hermitian Conjugates Operators An Operator L1 in Hilbert Space acting on a Vector |ψ›, produces a Vector |η›. Let have another Operator in Hilbert Space acting on the Vector |η›, and produce a Vector |χ›. ( ) 1111 LLorLL  =⇔ 1L  Operator ψη 1L  = 1L  Adjoint Operator 1L  ψη = 22 & LL  ηχηχ == Therefore 2112 & LLLL  ψχψχ == ( ) 21122112 LLLLorLLLL  =⇔ The Adjoint of a Product of Operators is obtained by Reversing the order of the Product of Adjoint of Operators. 117
  • 118. Functional AnalysisSOLO Hilbert Space ILLLLILLLL ==== −−−− 1111 &  Inverse Operator Given ψη L  = L  ψη = The Inverse Operator on is the Operator that will return .ψL  ψ1− L  ψψη == −− LLL  11 Therefore ηψη ==− LLL  1 The Inverse Operator on is the Operator that will return .L  ψ ψ1− L  ψψη == −− 11 LLL  In the same way ηψη ==− LLL  1 Not all Operators have an Inverse. 118
  • 119. Functional AnalysisSOLO Hilbert Space ( ) ( ) ( ) ηψηψψηψη ,||| * ∀== LLL  Hermitian Operator In Quantum Mechanics the Operators for which are equal present a great importance. They are called Hermitian or Self-Adjoint Operators. ( ) ( ) ψηψη || LandL  Properties of Hermitian Operators From the definition we can see that the direction of the arrow is not important and we can write ( ) ( ) ηψηψψηψηψη ,||||:|| * ∀=== LLLL  1 2 All the Eigenvalues of a Hermitian Operator are Real ( ) ( ) ( ) ( ) ( ) ψηψηψηηψψη |||||| ******* LLLLL  ==== ( ) ψψλψψλψλψ || =⇒∈= LCL  ( ) ψψλψψλψλψ || ** =⇒∈= LCL  Hermitian Operator : ( ) ( ) ( ) * 0 * 0||| λλψψλλψψψψ =⇒=−⇒= >   LL An Operator is Hermitian if it is equal to its Adjoint: Hermitian or Self-Adjoint Operators ( ) LLL  == 119
  • 120. Functional AnalysisSOLO Hilbert Space ( ) ( ) ( ) ηψηψψηψη ,||| * ∀== LLL  Hermitian Operator In Quantum Mechanics the Operators for which are equal present a great importance. They are called Hermitian Operators. ( ) ( ) ψηψη || LandL  Properties of Hermitian Operators If all the Eigenvalues of an Operator are Real the Operator is Hermitian3 iiii i iiiiiiii iiiiiiii iii iii LL L L i L L ii ψψψψ ψψλψψλψψ ψψλψλψψψ ψλψ ψλψ λλ || ||| ||| * * *      =⇒     == == ⇒∀     = = = ∀ Hermitian Operator 120
  • 121. Functional AnalysisSOLO Hilbert Space ( ) ( ) ( ) ηψηψψηψη ,||| * ∀== LLL  Hermitian Operator In Quantum Mechanics the Operators for which are equal present a great importance. They are called Hermitian Operators. ( ) ( ) ψηψη || LandL  Properties of Hermitian Operators 4 All the Eigenfunctions of a Hermitian Operator corresponding to different Eigenvalues are Orthogonal, the others can be Orthogonalized using the Gram-Schmidt Procedure. Therefore for a Hermitian Operator we can obtain a “complete Set” of Orthogonal (and Linearly Independent) Eigenfunctions     == = ⇒     == == ** * ||| || nmmmnmmn nmnnm mmmmm nnnnn L L L L ψψλψψλψψ ψψλψψ λλψλψ λλψλψ If |ψn› and |ψm› are two Eigenfunctions of the Hermitian Operator L, with eigenvalues λn and λm, respectively Hermitian Operator: nmmnmnmnnm LL ψψλψψλψψψψ |||| =⇒= If λm ≠ λn this equality is possible only if ψn and ψm are Orthogonal 0| =nm ψψ If λm = λn we can use the Gram-Schmidt Procedure to obtain a new Eigenfunction Orthogonal to |ψn›. n nn mn mm ψ ψψ ψψ ψψ         −= | | :~ 0| | | |~| =        −= nn nn mn mnmn ψψ ψψ ψψ ψψψψ The Hermitian Operators have Real Eigenvalues and Orthogonal Eigenfunctions. λm ≠ λn 121
  • 122. Functional AnalysisSOLO Hilbert Space 1− =UU  Unitary Operator Properties of Unitary Operators A Unitary Operator U is defined by the relation where I is the Identity Operator.IUUUU ==  A Unitary Matrix is such that it’s Adjoint is equal to it’s Inverse. All Eigenvalues of a Unitary Matrix have absolute values equal to 1. Suppose |ψi› is an Eigenfunction and λi is the corresponding Eigenvalue of a Unitary Operator.  iUU U U iiiiiii I i iii iii ∀=⇒=⇒     = = 1| ** * λλψψλλψψ ψλψ ψλψ    1 2  ψηψηψη ,| ∀= I UU  For all <η| and |ψ› the Inner Product of equals‹η|ψ›ψη UandU  3 ψψψ ∀=U   ψψψψψψψψ ∀=== 2/1 2/1 2/1 || I UUUUU  122

Editor's Notes

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