It takes about 150 attoseconds for an electron to circle the nucleus of an atom. An attosecond is seconds long, or, expressed in another way, an attosecond is related to a second as a second is related to the age of the universe.
In standard quantum theory the Bohr condition for the allowed wave functions makes sense: the potential of the nucleus creates a kind of box for the atomic electron; the only self-consistent waves are those that close on themselves.
But in temporal quantization the paths are open ended in time; there is no confinement in the time direction. If we think in terms of paths, a path associated with a electron could wind around the nucleus an infinite number of times, having a slightly different phase at each turn. Here there is no boundary on the wave functions, at least not in time. What confines them? Or more accurately, why we do see well-defined orbitals?
We will argue, as above, that while the paths can twitter in time as much as they want, only those paths that are roughly stationary in time will interact constructively with outside particles, so those will dominate interaction with the outside world.
We will use the criteria of stationary states, those with laboratory energy zero, to pick out the bound states. It turns out that we get the same energy levels as with the Bohr condition.
There will be "temporal fuzz" around these stationary states; we can estimate the dispersion of the fuzz from the electron orbit times, see below.
As with the free case, Time/space representation, we use absolute quantum time to define the stationary states; but we make the most natural connection to standard quantum theory using relative time:
We will make our treatment independant of the specifics of the potential, but use the relativistic hydrogen atom as a model.
First, all usual bound states have a temporal quantization extrusion.
Second, no unusual bound states do.
Have to develop in a way which is independent of the specifics of the potential.
Look at generic potentials
Show for every bound state there is exactly one temporal quantization state, consisting of plane wave in time times the bound state (in space) that has a zero expectation of the Hamiltonian.
Show there is no cross-product term of the temporal quantization states unless there is already one for the standard quantum theory bound states.
Give formula for temporal quantization stationary states in terms of standard quantum theory bound states
Hamiltonian:
Eigenfunctions:
Stationary states are those with eigenvalue zero. We expect a countably infinite number of these.
Bound Hamiltonian:
Stationary condition:
Compare to standard quantum theory Hamiltonian:
With eigenfunctions:
The set of quantum numbers that describe the standard quantum theory wave function might be given, par example , by:
For a hydrogen atom the will be the usual binding energy:
With the the usual non-relativistic bound states:
More generally we will have relativistic corrections to both energy levels and wave functions. We are not limiting the analysis to the hydrogen atom of course; we will be careful to keep ourselves ignorant of any details of the potential or its solutions.
Starting point, base Hamiltonian:
An arbitrary solution of the temporal quantization Hamiltonian can be written as a sum over the product space of the time and standard quantum theory solutions:
Where the are the plane wave solutions:
And where the n on the left stands for the full set of quantum numbers, i.e.:
Not to be confused with the n on the right, which is the principle quantum number.
Start by looking at a single product state, of form:
Looking for the value of E that corresponds to a specific set , using the condition:
We can write the Hamiltonian as:
Using for the kinetic energy. Expanded:
The main complication is the presence of the term in H . As noted above, the time gauge likely to give singular results, so not a good choice.
Our goal is a Hamiltonian linear in both and .
Rewrite Hamiltonian as:
Using:
And:
The on the left cancels against the on the right, giving:
For a single , we have:
We pick the with:
So the time part, to lowest order is:
Most of the terms in the Hamiltonian cancel, leaving:
Diagonal expectation of energy is zero:
So we have that with the correct choice of quantum energy, the single product wave function has a zero expectation of the Hamiltonian.
Therefore the stationary condition is picking out the usual energy levels.
We therefore have a countable infinity of orthogonal states with laboratory energy zero.
Look at linear combinations of the form:
With:
For matrix element have:
Which is not zero unless the standard quantum theory binding energies are equal:
Therefore linear combinations not stationary unless the space parts have same binding energy.
If this were not the case, temporal quantization would imply the existent of hybrid states not found in nature.
As with the free case, we can rewrite plane waves time in terms of relative time:
So that the plane wave becomes a product of relative time and laboratory time parts.
To construct Gaussian test functions that describe the quantum time extensions of the temporal quantization wave functions, we will need to estimate the dispersion in of the wave functions.
The simplest estimate, as above, is to take the uncertainty in time as of order the orbital period, and the uncertainty in energy as one over that.
In the case of argon, cited above, the orbital period is measured at 150as.
To get a qualitative understanding of this, apply Kepler's law for orbital periods (atoms, planets, they all come out in the wash):
Bohr radius is:
For principal quantum number n :
So period for principal quantum number n of order:
For lowest state:
This is not quantitative by any means. The key is that period is of order one over fine structure constant squared by mass. Taking this as also estimate of uncertainty in time, we get an estimate of uncertainty in energy:
So that uncertainty in energy expected of order fine structure constant squared times mass.
Accurate calculation requires treatment of many particle problem; outside the scope of this work.
Given non-zero uncertainty in quantum time, the bound/stationary states will be accompanied by a cloud of off-axis states, a kind of temporal fuzz. How does this cloud evolve in time?
Taking the stationary states as a starting point, construct an arbitrary state as:
With:
And with the sum over both stationary and non-stationary states.
This is completely general as any laboratory time dependence not captured in the:
Will be tracked in the coefficients:
To estimate the laboratory energy associated with each plane wave, compute matrix element for it.
Take advantage of fact that laboratory energy of stationary states is zero:
Take matrix element of Hamiltonian between off-axis states. Difference between offshell and onshell energies is:
So:
Which is similar to the analogous construction for free waves, see Time/space representation. As noted above, uncertainty in energy of order times mass, so the first order correction is of order:
Plus two terms of order , twice as small.
If we shift to relative time we can rewrite the effective laboratory energy:
The first term is the standard quantum theory part, the rest is the temporal quantization part. This is similar to the development in the free case Time/space representation.
Therefore the original expansion may be written as:
Where the laboratory energy has been broken out into the mass, the usual binding energy:
And the time part. Therefore the on-axis part matches standard quantum theory exactly – key – and the off-axis parts differ by order .
We are particularly interested in clouds in the shape of Gaussian test functions.
Gaussian test function in energy:
Quantum energy breaks out into space and time parts:
And space part breaks out into mass and binding energy:
Laboratory energy for each component also breaks out into space and time part:
So Gaussian test function in energy is:
Gaussian test function in time:
Dispersion in time reciprocal of dispersion in energy:
We get the Bohr condition from the stationary state condition; only the stationary part will coherent interact with external photons or other wave forms.
This line of attack was suggested to us by Gondran and Gondran ( [Gondran-2005]) where they analyzed the Stern-Gerlach experiment in terms of coherent interference.
This is also similar to the "consistent histories" approach [Omnes-1999], [Omnes-1999b]; the bound states in temporal quantization are sums over the self-consistent paths associated with the stationary states.
With wave functions as with armies: a small group acting coherently will completely dominate a much larger number interfering destructively with each other.