Start with Gaussian test function, manufactured as in Slits in time or Estimates of uncertainty in time/energy or otherwise.
Look for changes in time, position, energy, and momentum in x direction.
The case of a magnetic field not changing in time has nothing to contribute; separation between time and space exact.
Usable cases:
Magnetic field changing in time.
Electric field not changing in time. Even with unchanging electric field, could still see effects from the term.
Electric field changing in time.
Change coordinates to foreground the differences between temporal quantization and standard quantum theory as a small perturbation on the latter.
Start with rule for time evolution of expectation of an operator:
Break Hamiltonian up into standard quantum theory part and temporal quantization part:
Look at effects of difference between the two:
Treat the incoming beam as a carrier; since we know how quickly it will get through the interaction area, we know the total first order impact of the change:
We can generally compute the expectation for a specific Gaussian test function in a straight-forward way.
Since the Gaussian test functions also work as "typical" experimental arrangements, we can get a reasonable idea of which experiments are likely to be productive.
Therefore the value:
Serves as a metric of the merit.
And since the decomposition into Gaussian test functions is, via Morlet wavelets, general and well-defined, we can compute the first order impact of an arbitrary experimental arrangement this way.
We are looking at the differences between standard quantum theory and temporal quantization. Consider effect of potential:
Expand in power series in relative time:
Standard quantum theory effects come from first term:
Effects of temporal quantization from terms after the first:
Note the coefficients of these terms are in general functions of the laboratory time:
And so on.
What we expect a wave function extended in time will sample potentials in past and future, creating what we might think of as forces of anticipation and regret.
Take magnetic field in z direction:
In relative time:
Or:
So the first two coefficients are functions of laboratory time, the third is not.
The effects of the first time are presumed managed by the standard quantum theory analysis, so the effects of temporal quantization from:
Get this magnetic field as curl of the vector potential:
Now have V as terms first and second order in the relative time:
Assume weak field; ignore vector potential squared term.
Rate of change of expectation of operator is:
Gauge invariant operator is , however for the gauge choice we have made have:
So can use :
Break out in terms of standard quantum theory and temporal quantization parts:
So that the temporal quantization particle anticipates (and remembers) changes in the magnetic field.
For one of our test Gaussian test functions:
So equations of motion now:
The standard quantum theory wave function has no past and no future; the temporal quantization wave function can see a bit to past and to future – and is impacted by what it sees.
Look first at case of electric potential not dependent on time:
Use the time gauge.
Look at impulse delivered to particle. Look particularly for effects on velocity.
Field:
And corresponding potential:
Schrödinger equation:
Provided we are in a source free area, i.e. within a capacitor.
Time part:
Space part:
Potential:
commutes with constant field so does not change in time:
For field changing with x :
If term linear in relative time zero, then only term quadratic in relative time contributes, giving:
Definitely see effects of electric field in the future and the past.
We expect in general linear terms average out, but quadratic may show effects of dispersion of wave function in time.
Use time gauge.
Example: take a capacitor with a hole in it. Change voltage on capacitor while a particle is going through it. No space dependence of electric field, but clear time dependence.
Electric field:
Perturbative potential:
With the time gauge, have for interaction potential:
Or:
Additional acceleration:
Time variation of full momentum:
Assume terms linear in relative time drop out:
Additional acceleration on particle given by:
Since no space dependence by assumption, does not reduce to previous case.
As before, the effect of temporal quantization is to create look ahead and behind, to sample past and future.