Free case preparation for everything else.
We assemble useful properties of free particle wave functions and kernels here. Look at:
Kernel:
Kernel may be written as a product of time and space parts:
Time part:
Space part:
We push the free wave function into the future using:
Initial plane wave:
Apply Schrödinger equation to it:
Use script for energy complementary to laboratory time. Laboratory energy is:
And therefore that the plane wave is given in laboratory time as:
The onshell states have zero, so are stationary.
As a check, we can use the free kernel to move forward the wave function forward in laboratory time:
Giving:
Initial wave function:
With the dispersion matrix diagonal and defined as:
To calculate the wave function in future apply the free kernel:
Expectation of coordinates evolve with laboratory time:
As does the dispersion matrix:
This is an exact solution of the free Schrödinger equation:
With probability density:
And uncertainties:
Beam stays narrow in -th dimension for short times, defined as:
But begins to widen at longer times:
The wave function has a natural decomposition into time and space parts.
Time part of wave function:
With oscillating and envelope parts separated:
Space/standard quantum theory part:
With definition:
Relative time is defined as the difference between the absolute quantum time and the laboratory time:
Note this is true for all particles; each particle has its own proper time, but shares laboratory time. Alice defines laboratory time; her negotiations with Bob – travelling at relativistic speeds – are discussed in Geodesic definition of laboratory time.
Only the time part of the kernel is affected by a switch from block time to relative time:
Or:
In the argument of the exponential, the term quadratic in time has same form as in the block time kernel; term linear in time is also present.
Full kernel using relative time:
Plane wave using relative time:
Recall that is zero for an onshell wave function, so that the coefficients of both and are quantum energy E .
Break out into time and space parts:
With obvious definitions of space part:
And time part:
Get for coefficient of laboratory time:
With the usual definition of the relativistic dilation factor :
Define this as the laboratory energy in relative time:
Plane wave in relative time:
In general expect both the quantum energy and the laboratory energy will be approximately equal to .
Gaussian test function given by:
With:
The most important case is when we are onshell:
Here laboratory energy is equal to quantum energy:
Full onshell wave function (time and space):
With:
For short distances, all of the dependence on laboratory time is contained in the laboratory energy laboratory time factor. Over longer distances, the wave packet widens and shows relativistic corrections to the expectation of quantum time.
In nonrelativistic case, relativistic dilation factor is approximately one, so can write time part as:
And space part as:
Where we have broken up the dependence on laboratory time to have the space part be just the usual standard quantum theory wave function.
Define the Fourier transform using opposite signs for energy and momentum parts:
For kernels, gates, potentials and other two sided objects we have:
Full kernel:
Kernel as product of energy and momentum parts:
Energy part of kernel:
Momentum part:
Free wave function in future given by:
At laboratory time zero:
At later laboratory time:
Recall that is zero for onshell energies, so onshell plane waves do not move in laboratory time.
Have:
With dispersion matrix:
With the chosen conventions, time and energy dispersions are reciprocal:
And similarly for space and momentum.
Wave function as a function of laboratory time is:
Since zero for onshell Gaussian test functions, onshell Gaussian test functions do not change shape or phase with respect to laboratory time.
This is simpler than the corresponding time space form in that the dispersions do not depend on the laboratory time. Simpler but not simple. This is a function, in principle, of 18 variables: the four of the offset in space, the four of the average energy-momenta, and the ten of the (symmetric) dispersion matrix.
Break out into energy and momentum parts:
Energy part:
Momentum part:
With probability density
Expectations:
Uncertainties:
In coordinate space, if sharply focused, more particle like. If spread out more wavelike.
In energy momentum space, the opposite and identical statement is true.
Can get the energy part of the kernel in relative time as a Fourier transform of the time kernel:
Or:
With all four dimensions:
If the original wave function is onshell (expected), kernel reduces to the three dimensional kernel plus a function in quantum energy:
The full four dimensional kernel is approximately the onshell part of the kernel times some energetic fuzz.
From Fourier transform:
Or:
So the shift to relative time means we shift from the laboratory energy to the relative time laboratory energy:
The relative time laboratory energy has a natural decomposition into space and time parts:
Define Gaussian test function in relative time as Fourier transform of relative time Gaussian test function in time and space:
Momentum part as above.
Easier however to take the Fourier transform at zero, then move forward using relative time kernel.
Energy part:
Onshell, the:
Cancels against the
From the momentum part and we are left with:
Plane wave and Gaussian test function have the same laboratory energy. Not surprising.
Work in relative time.
In analysis of slit experiments useful to work with a hybrid representation, quantum time and momentum. Here we take x and p as the single x and p dimensions, rather than four-vectors.
The momentum side serves as the standard quantum theory part, and will serve as carrier. The quantum time side as the temporal quantization part; and functions as signal. Clean division of labor.
Can get this by starting with the onshell relative time Gaussian test function as above, then transforming only the space parts:
To get the hybrid wave function and kernel:
In nonrelativistic case:
Using:
To combine the dependencies on the time and the momentum sides.
Hybrid wave function:
Or:
Or:
Expanded:
In nonrelativistic case can break up the laboratory time dependence into mass and usual kinetic energy:
Or, in a form which is particularly convenient for comparison to standard quantum theory:
Note the time part depends on the momentum indirectly, via
We would like to estimate the energy and time uncertainties of a free particle, assuming we do not have any information on its provenance.
We will estimate the uncertainty of a wave packet found in the wild, starting from an estimate of its uncertainty in momentum to get an estimate of its uncertainty in energy and therefore in time.
Schrödinger equation for free particle:
We assume that over longer laboratory times only the stationary part will contribute in a coherent way to interactions:
Define:
We would like to estimate the dispersion in energy for a Gaussian test function.
We will assume we already have an estimate of the distribution of the momentum:
For a Gaussian test function this is:
With averages and dispersions of the three momenta:
We would like to use these to make a reasonable estimate of the dispersion in the energy:
We estimate the average of energy for the Gaussian test function as:
Estimate the average of energy for a specific three momentum as:
Now we only need the expectation of the square of the energy. With the stationary assumption we expect:
So the expectation of energy squared is given by:
And we get:
The mass squared term cancels out and the rest gives:
Because we did not rely on the wave function being a Gaussian test function, this result applies to free wave packets in general.
The principal assumption is that over longer times only the onshell part will interact will coherently with the rest of the universe, and therefore – unless we are looking at very short times – we only need to look at the onshell part.
This is not therefore not a full description of the wave function in E ; is a description of the part that interacts coherently with rest of universe over "sufficiently long" time scales.