How to define a standing wave which describes the movement and has a sensible probability density and a conserved probability current?
____ ____
/ \___/ \ t = 0
________________ t = 1
linearly polarized standing
wave is identically
zero at certain moments
If we try a linearly polarized wave, that is not going to succeed. A linearly polarized standing wave is at certain moments identically zero. That would spoil the probability density.
_______
____/ \____
jump rope rotates
around the axis
A better idea is to make a rotating standing wave, like girls' jump rope. The value of the wave function stays non-zero at most places. It is a circularly polarized wave.
Where would that wave rotate? In some abstract space?
Hypothesis. Waves in nature must live in the familiar 3 + 1 -dimensional physical space and must hold energy and momentum. They have to be classical waves in that sense.
Electromagnetic waves satisfy the hypothesis.
We suggest that the rotating electron wave lives in the familiar 3 + 1 -dimensional space. It contains angular momentum. That angular momentum is the spin of the electron.
The Dirac equation solves the following problems in the massive Klein-Gordon equation:
1. There is too much freedom in specifying the initial values for the massive Klein-Gordon equation because it is a second order differential equation. We have to specify both the wave function value at a time t, and its temporal derivative.
2. One cannot define a probability density and a conserved probability current. This is probably associated with item 1.
The Dirac equation is a reasonably simple way to implement a probability density and a conserved probability current. Classically, the simple way is to use girls' jump rope, which inevitably includes a spin. This may be the underlying reason why there is a spin in the Dirac equation.
According to this reasoning, the existence of the spin follows from:
1. The wave equation has to be relativistic. It cannot be the Schrödinger equation.
2. There has to be a probability density and a conserved probability current. A spin is a simple way to implement that.
Why is the electron spin 1/2?
Why is the electron spin 1/2? Why not 1/4 or 1?
To have a conserved probability current, we have to avoid linearly polarized standing waves.
If the spin would be 1, then we could model the electron as a classical particle which moves at the speed of light along a circle whose length is one Compton wavelength. What if we would put another electron to circle to the opposite direction? The result might look like a linearly polarized standing wave. Particles with spin 1 (like the photon) can posses spin-z values 1, 0, -1. The value 0 is linearly polarized.
The electron wave function in the Pauli equation is a two-component spinor wave function.
Spin-z states 1/2 and -1/2 are orthogonal in the spinor space of the Pauli equation. Suppose that we have two electrons with spin-z 1/2 and -1/2 confined into an "almost same" state. That is, the wave functions of the two electrons are of the form
(0, ψ) and (ψ, 0),
where we have used spin-z as the basis.
The sum of the two wave functions is
(ψ, ψ),
which is just right, so that the square
ψ^2 + ψ^2
of the wave function corresponds to exactly two particles.
This did not yet explain why the electron spin has to be 1/2. The Dirac equation is kind of a "square root" of the massive Klein-Gordon equation. That might be associated with the value 1/2.
An electron spin 1 would be ok if we could ban the spin-z state 0.
A new proof of the Pauli exclusion principle, and a Pauli exclusion principle for photons
Note that if the spin-z for the two electrons were the same, then the sum of the two wave functions would be, e.g.,
(2 ψ, 0).
The square of the wave function
4 ψ^2
would correspond to four particles. That would break conservation of the probability current.
We get a new proof for the Pauli exclusion principle. If two electrons would end up in the exact same state, then the probability density of the summed wave function would be 4-fold, which would break conservation of the probability current.
The proof assumes:
1. We can sum two wave functions linearly.
2. The probability density of the summed wave function is obtained by squaring it.
In nature, many waves allow linear summation. The square of the wave function is often the energy of the wave. It makes sense to associate the energy with the number of particles.
More precisely, conservation of the probability current requires that electrons in a similar state must have their spinor vectors orthogonal:
Strong Pauli exclusion principle. If two electrons are in states
(s, s') ψ and (r, r') ψ,
then the vectors (s, s') and (r, r') have to be orthogonal.
We get a "Pauli exclusion principle" for photons, too:
Pauli exclusion for photons. If two photons were created separately, they can never end up in the same state. The electric and the magnetic fields would be double in such a configuration, which would make the energy 4-fold. Conservation of energy prohibits such a state.
Photons can be created in the same state. A radio transmitter creates a huge number of coherent photons. Energy is conserved because the radio transmitter supplies the required energy.
There is a classical version of Pauli exclusion:
Classical Pauli exclusion principle. If two classical waves are created separately, they can never end up (without adding energy) in a configuration where constructive interference would dominate over destructive interference. That would break energy conservation.
It turns out that the Pauli exclusion principle is an energy conservation principle.
We cannot create electrons in the same state. If we could, we might be able to break the Pauli exclusion principle.
Question. Is the electric repulsion between electrons a result from the Pauli exclusion principle? Suppose that we have a small box and put more and more electrons there. They must occupy higher and higher energy states because of Pauli exclusion. Classically, we must win the electric repulsion. How do the energies compare?
Pauli exclusion causes electron degeneracy pressure.
At short distances, degeneracy pressure is stronger than the Coulomb repulsion. In the hydrogen atom, degeneracy pressure balances the Coulomb attraction.
We cannot derive the Coulomb force from degeneracy pressure.
Could it be that some kind of degeneracy anti-pressure is the mysterious attractive 1 / r^2 potential in pair annihilation?
How does the traditional antisymmetric wave function proof differ from our proof for Pauli exclusion?
In the traditional proof, the two electrons are distinct individual particles, and their combined wave function is some kind of a product of individual wave functions.
In our proof, the combined wave function of two electrons is the sum of wave functions. The electrons are kind of "bulk material" for the wave function. This is analogous to photons, for which we believe summing the wave functions is the correct procedure.
Which approach is right?
1. Electrons are indistinguishable particles. The traditional proof seems to assume that we have somehow marked them. That is dubious.
2. We do not know how to model the Coulomb potential between electrons if they are not particles. Simply summing the wave functions ignores this aspect.
Another question is how an electron-positron pair is created. If they are particles, where exactly these particles pop up in spacetime?
Photon wave functions can be summed, and must be summed to explain, for example, the interference pattern of two laser beams.
Let us check if coherent beams of electrons have been made.
D. Ehberger et al. (2015) report about a highly coherent electron beam from a tungsten needle tip. It looks like electrons can behave like "bulk matter" just like photons. Then summing the wave functions is the right procedure.
Let us check the literature. Has anyone invented our simple proof for Pauli exclusion before us?
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