Schrödinger equation with time broken out:
Schrödinger equation in relative time:
Schrödinger equation in energy momentum space:
Look at wave functions stationary with respect to laboratory time:
Therefore condition for a stationary relative time wave function has the same form as condition for a stationary block time wave function:
In energy momentum space:
This is just the Klein-Gordon equation, with the minimal substitution.
Requirement of stationarity picks out the standard quantum theory solutions, something we will see again in analysis of bound states.
Solution of free Schrödinger equation in block time:
With laboratory energy:
Free Schrödinger equation in relative time:
Solution of free Schrödinger equation in relative time:
With laboratory energy:
Stationary states in block time are defined by:
These are the only states that will interfere coherently with external fields, and will therefore dominate the physics except at very high energies/short frequencies. Stationary condition on energy is:
Makes sense to define the space or standard quantum theory part of the laboratory energy as:
Break out the quantum energy:
The space part of the quantum energy is chosen equal to the standard quantum theory energy:
For nearly stationary states we will have:
Have the stationary relative time plane waves:
With the standard quantum theory part being the familiar:
General relative time plane wave:
With obvious definition of the time part of the laboratory energy:
And the relativistic dilation factor being the usual:
Properties of free wave functions summarized in appendix Free particles.
Will see essentially the same breakout of relative time and standard quantum theory parts for bound states below.
For static magnetic field, temporal quantization has nothing to say: Schrödinger equation factors into free quantum time part and usual standard quantum theory part.
However, for time varying magnetic fields, expect effects.
Schrödinger equation:
Natural to develop in terms of relative time.
Vector potential, to second order in :
As a simplification, assume vector potential is in Coulomb gauge:
Letting us interchange position of and freely.
We have both and terms.
Linear corrections:
Quadratic corrections:
Because of the term in the Lagrangian, there is coupling between time and even a static electric potential. We look at this.
In the Schrödinger equation, this coupling leads to a term:
This is not at all like form in standard quantum theory, where enters linearly:
We can gauge away the that accompanies the . This creates, as a side effect, the term linear in , at the expense of creating some cross terms. The cross terms are all of order to the first or higher, so that a wave function very narrow in time will show no effect.
We would like to express the Hamiltonian in a form like:
With time part:
And space part as above. is defined as whatever is left.
Gauged wave function:
Try gauge:
On left, get an Alice's potential:
On right, in the time part is cancelled out.
However we have to look at the result of applying the momentum squared term to the gauge itself. Using:
Get for the cross term:
The full Schrödinger equation is now:
The divergence of the electric field is the charge density:
So in source-free regions only the second of the two relative time terms is non-zero. Therefore, to lowest order expect effects proportional to dispersion in quantum time:
The field squared term implies that the particle senses – and dislikes – volumes of large field strength. For a simple Gaussian test function, the effective potential is (to first order):
For a 1/r potential, the Laplacian acting on the gauge terms gives a function:
So we have:
The first term on the right is reasonable enough: non-zero only for s states in the case of hydrogen atom. However the electric field squared gives:
Which is singular at the origin. This will blow up for s states. This makes the use of the time gauge suspect for 1/r (and more singular) potentials. (Of course the fact the gauge transformation is singular in such cases might have warned us off in the first place.) We will look at how to manage 1/r potentials in the next section.
Now consider time dependent potential:
Again we need:
Giving:
If the potential is constant, we recover the previous result.
Substituting back into the Schrödinger equation, we lose the term next to the term, and we pick up the desired term on the left:
The cross potential is similar to above, except that the time-smoothed electric field replaces the electric field:
Cross potential:
With the time-smoothed electric field:
This result is exact. The Schrödinger equation is:
If the expectation of the relative time is small, i.e. if the wave function is centered on , then we can use this as the starting point of a perturbative expansion in :
To second order in :
The three terms involving derivatives with respect to relative time (or equivalently laboratory time) are new; the other two we have seen above.
Again, corrections will be small if dispersion in relative time is small.
Time gauge here is closely related to the gauge used in Quantum Paradoxes, [Aharonov-2005], p49. The difference is that we integrate over relative time; they integrate (necessarily) over laboratory time.
We may identify the relative time wave function with the standard quantum theory wave function, in the limit as the effects of quantum time become negligible:
Note even for time independent potentials, we cannot completely separate the time and space parts. The extension of the wave function in time causes it to sample potential earlier and later than the present. However these corrections will be small if the dispersion in time is small.
Now consider time dependent potential:
Again we need:
Giving:
If the potential is constant, we recover the previous result.
Substituting back into the Schrödinger equation, we lose the term next to the term, and we pick up the desired term on the left:
The cross potential is similar to above, except that the time-smoothed electric field replaces the electric field:
Cross potential:
With the time-smoothed electric field:
This result is exact. The Schrödinger equation is:
If the expectation of the relative time is small, i.e. if the wave function is centered on , then we can use this as the starting point of a perturbative expansion in :
To second order in :
The three terms involving derivatives with respect to relative time (or equivalently laboratory time) are new; the other two we have seen above.
Again, corrections will be small if dispersion in relative time is small.
Time gauge here is closely related to the gauge used in Quantum Paradoxes, [Aharonov-2005], p49. The difference is that we integrate over relative time; they integrate (necessarily) over laboratory time.
We may identify the relative time wave function with the standard quantum theory wave function, in the limit as the effects of quantum time become negligible:
Note even for time independent potentials, we cannot completely separate the time and space parts. The extension of the wave function in time causes it to sample potential earlier and later than the present. However these corrections will be small if the dispersion in time is small.