Time dispersion in time-of-arrival measurements

http://timeandquantummechanics.comTime Dispersion in Time of Arrival/John Ashmead IARD-2020
Time Dispersion in
Time-of-Arrival
John Ashmead
International Association for Relativistic Dynamics - 2020 Virtual
Can we prove that the Heisenberg uncertainty
principle does not apply along the energy/time
axis in the same way it applies along the space/
momentum axis?

Heisenberg
Uncertainty
Principle (HUP)
• ΔΕΔτ≥1 taken for
granted
• Critical in early days
of QM
• But basically
abandoned since
• ı
∂
∂τ
ψτ
x( )= −
1
2m
∂2
∂x2
ψτ
x( )

Three ways of saying
the same thing
• Heisenberg uncertainty
principle applies in
energy/time
• Wave function extends in
time (or energy)
• Time is an observable
ΔEΔt ≥1
ˆE = ı
∂
∂t
ψτ
x( )→ψτ
t,x( )

How to get to
Falsifiability?
• Look at the class of theories, not just
one particular theory
• Use the simplest possible approaches
• Look for order of magnitude
estimates
• Use covariance to eliminate need for
free parameters

Why haven’t we seen
this before?
• Time for photon
to cross atom?
• .35 Attoseconds
• Explains why we
haven’t seen

 
Can we see it now?

Time of Arrival
• Most obvious test
• Also part of most experiments:
there is almost always a time
measurement, even if not recorded
• Intrinsic uncertainty in time
competes with extrinsic
uncertainty
• Curiously, time-of-arrival even in
standard quantum mechanics not
well understood

p > 0
p < 0
x > 0x < 0
paxis
x axis
Detector

Gaussian test functions
• Normalizable
• Satisfy free
Schrödinger
equation
• Back-of-
envelope
• General
ϕτ
x( )=
1
πσ x
2
4
1
fτ
x( ) e
ıp0x−
1
2σ x
2
fτ
x( )
x−xτ( )2
−ı
p0
2
2m
τ
fτ
x( )
≡ 1+ ı
τ
mσ x
2 xτ
≡ x0
+
p
m
τ
ρτ
x( )=
1
πσ x
2
1+
τ 2
m2
σ x
4
⎛
⎝
⎜
⎞
⎠
⎟
exp −
x − xτ( )
2
σ x
2
1+
τ 2
m2
σ x
4
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
Δx( )
2
≡ xτ
2
− xτ
2
=
σ x
2
2
1+
τ 2
m2
σ x
4

Bullet versus Wave
• Bullet works
well
• But slow,
wide wave-y
GTF
produces
• Nonsense
p0
p0
p 0
p
p0
p0
+ p
p > 0
p < 0
p0
p0
+ p
p0
p 0
p
p0
p > 0
p < 0

Life span of an
atom
• accelerated
electron radiates
• once its energy is
gone, it spirals
into the nucleus
• in about one
hundred trillionth
of a second!

Checkpoint
Copenhagen

Conservation of
probability
0

What makes QM QM?
• Quantum Zeno
effect
• Anti-Zeno
• Backflow
• Partial detection
• Delayed detection
• …
“Indeed the finite interaction
between object and measuring
agencies conditioned by the very
existence of the quantum of action
entails…the necessity of a final
renunciation of the classical ideal of
causality and a radical revision of
our attitude towards the problem of
physical reality” — Bohr

GTF meets detector
0-d
d
v0
Counts
• Detector
only open for
Δτ
• So doesn’t
interfere
with itself
• Crude but
reasonable

Results
• Detection rate
• Steepest
descents
• Dispersion in
time
• Slower = wider
τ = τ +δτ,τ =
d
v0

3D versus 4D pathsLab
Clock
Jagged paths in space only
A
Jagged paths in space
– and in time
B

Derivation OF
Schrödinger equation
Kτ
′′x ; ′x( )= lim
N→∞
Dπe
ı d ′τ L π , !π⎡⎣ ⎤⎦
0
τ
∫
∫
Dπ ≡ −ı
m2
4π 2
ε2
⎛
⎝⎜
⎞
⎠⎟
N
d4
n=1
N−1
∏ xn
L xµ
, !xµ
( )= −
1
2
m!xµ
!xµ
− q!xµ
Aµ
x( )−
m
2
ı
∂ψτ
∂τ
= −
1
2m
pµ
− qAµ( ) pµ
− qAµ
( )− m2
( )ψτ
ı
∂
∂τ
ψτ
= −
E2
− p2
− m2
2m
ψτ
ı
∂
∂τ
ψτ
t,
!
x( )=
1
2m
∂2
∂t2
− ∇2
− m2
⎛
⎝⎜
⎞
⎠⎟ψτ
t,
!
x( )

Kτ
′′x ; ′x( )= lim
N→∞
Dπe
ı d ′τ L π , !π⎡⎣ ⎤⎦
0
τ
∫
∫
L xµ
, !xµ
( )= −
1
2
m!xµ
!xµ
− q!xµ
Aµ
x( )−
m
2
τ = t
Δt( )
2
≡ t2
−τ 2
∼10−18
sec
ı
∂
∂τ
ψτ
t,
!
x( )=
1
2m
∂2
∂t2
− ∇2
− m2
⎛
⎝⎜
⎞
⎠⎟ψτ
t,
!
x( ) 0 =
∂2
∂τ 2
− ∇2
− m2
⎛
⎝⎜
⎞
⎠⎟ψτ
t,
!
x( )
Dπ ≡ −ı
m2
4π 2
ε2
⎛
⎝⎜
⎞
⎠⎟
N
d4
n=1
N−1
∏ xn
Dπ ≡
ım
2πε
3
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
N
d3
n=1
N−1
∏ xn
4D Paths 3D Paths

Rules for detection
• Rules for detection
must be the same in
time as in space
• We are mapping from
the coordinate in the
path to the coordinate
in detector
• Paths are tied at ends
but free elsewherex
x
t
t
Detector
Source

Initial Wave
Function?
E( )
E
1 = 1
E E = m2
+ p
2
E2
= m2
+ p2
= m2
+ p2
• We have
constraints on
norm, average, &
dispersion
• Get maximum
entropy solution
using Lagrange
multipliers

Clock time as average over
the rest of the universe
space
coordinatetime
Detector &
post-detector
analyzed
classically
Source & pre-source
analyzed classically
Paths analyzed
quantum
mechanically
Alice
A
B
τ ≡ ψ rest−of −universe
t ψ rest−of −universe

Quantum Pointillism

Convolutions
over clock
time
Source
Detector
t
1
2
3
4 5
6
7
8
9
10
11
Different paths can take a wildly
different number of steps to go
between the same starting
and ending points but
what they have in
common is that they
all hit the plane of
the detector only once.
We
number
the steps
for only
one path,
otherwise
the figure
would be
unreadable.

Definite but small
effects
• Small deﬁnite effects
• Present everywhere
• For non-relativistic
objects, additional
dispersion small
σt
2
∼ σ x
2
1
v0
≫1⇒στ
2
≫ "στ
2
στ
2
= στ
2
+ !στ
2
=
τ 2
m2
v0
2
σ x
2
+
τ 2
m2
σt
2

Ultra fast camera
Source
Detector
t
Ultra Fast
Camera Shutter
Clicks
If the HUP does not apply in
time/energy
then paths that reach the slit
will be clipped by it:
total dispersion in time
at the detector
will be reduced …
… but if the HUP does apply
in time/energy
then paths that reach
the slit will be diffracted by it:
total dispersion in time
at the detector
will be increased.

Technically challenging
experiments
• Have to calculate
time on a per particle
basis (strobed?)
• Have to model the
detector (& in time as
well as space)
• Do not look like easy
experiments

• More likely than not
(from symmetry)
• Present everywhere, but
small
• Present large, but in only
specialized experiments
• Falsiﬁable — just not easy
ΔEΔt ≥1?
0-d
d
v0
Counts
Clipped
Diffracted
Diffracted
Clipped
W
Source Ultra Fast
Camera Shutter
Detector

• Quantum gravity without costs & risks of
black holes
• New communication channels, new
circuit elements, as memristors
• Attosecond chemistry; attoseconds are
the time scale for electron/molecule
interactions
• Attosecond biology; better understanding
of DNA, protein formation, …
• Let’s take a ﬂyer on the future; you never
know where the lightning will strike!
Δt( )
2
≡ t2
−τ 2
∼ attoseconds?

Time dispersion in time-of-arrival measurements

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