Another heuristic proof of the Pauli exclusion principle: particles can only enter the same state through a spin-z = 0 state

Our blog post on March 7, 2021 left open several questions.

Our argument was that we have to be able to sum wave functions linearly. If ψ represents one particle, then 2 ψ would represent four particles, since the probability density is the wave function squared. That prohibits two particles from going into the same state.

If the requirement of a linear sum can be relaxed temporarily, then two particles could go to form the wave function

        sqrt(2) ψ,

which does conserve the number of particles, and is thus allowed.

We also claimed that a wave function must, in some sense, rotate if we want to conserve the number of particles. Rotation means a spin value which is not zero.

That suggests that if the spin-z of the particle can be 0, then the number of particles in that state is fuzzy, and we cannot demand linear summation of the wave function of each particle because there is no clear number of particles. Then linearity can be broken.

Two particles can then end up in the same state by going through the spin-z 0 state.

That explains why particles with a spin

        n + 1/2

can never end up in the same state, but particles with a spin

        n

can. Particles with a spin n possess a spin-z state 0, which allows linear summation of wave functions to be relaxed.

Electron-positron annihilation

If we sum the angular momenta of a spin-z 1/2 and spin-z -1/2 electrons, then apparently, the angular momentum is not zero. The angular momenta are not opposite.

But if we sum a spin-z 1/2 electron and a spin-z -1/2 positron, then the angular momentum is zero, we think. We have to check the spinor arithmetic from some source.

The positron is the time reverse of the electron. It makes sense that the angular momenta of an electron and a positron can be truly opposite.

In annihilation, an electron and a positron form a combination whose spin is zero. Annihilation changes the particle number. We suggest that a change in the particle number is possible when the total angular momentum of the system is zero.

There is a paradox in the Dirac equation: if it conserves the probability current, how can a particle be created at all? The solution is to use an interaction which breaks the linearity of the system. It may be that breaking the linearity requires that the total angular momentum is zero.

In the February 11, 2021 blog posting we introduced the hypothesis that a photon is a combination of a rotating electron and a positron. We conjectured that in pair annihilation, a zero energy photon reacts with the annihilating pair and helps to build the two outgoing photons. The concept of a "zero energy photon" is dubious, however. It assumes a particle in empty space. Maybe it is better to assume that the photons can be created from nothing, which means that pairs can be created from nothing, too.

The fact that the positron is a time reverse of the electron suggests that in annihilation, the wave function somehow forms a bridge between the two particles, just like in the Feynman diagram. Richard Feynman wrote that an electron is "scattered backward in time".

If the Dirac wave function stubbornly wants to conserve the number of particles, then changing the number may require turning the wave function backward in time. The backward branch is the positron.

Pauli exclusion says that the wave functions of electrons have to be orthogonal - particle-in-a-box

In the particle-in-a-box model known from textbooks, different stationary states (energy levels) have orthogonal wave functions. The integral (the inner product) over the product of two wave functions is zero.

Putting n particles to different states conserves the number of particles. We can sum the wave functions and get a description of the system state, without a need to construct a direct product of the wave functions.

The following two conditions are equivalent to the Pauli exclusion principle in this simple case.

1. We can sum the n individual wave functions and get the system wave function.

2. The particle number must be conserved.

Since the particles are indistinguishable, a direct product of wave functions does not sound right. If we cannot keep track of which particle is which, why and how to construct a product? How does that product react to spontaneus switching of particles?

However, if there is interaction between particles, then a direct product sounds the right solution. We have problems defining the interaction if we do not have separate coordinates for each particle.

In our blog postings in 2018 we remarked that there is no proof that an interacting system of n > 1 particles has any definite "quantum state" for each individual particle. It is an empirical observation that electrons in atoms seem to occupy states which are similar to the states of the hydrogen atom, and that the electrons seem to obey the exclusion principle.

Suppose that there really is a meaningful "quantum state" for each electron in an atom. Then we can claim like in the particle-in-a-box case that putting two electrons into the same state would increase the number of particles by two. Pauli exclusion would follow from the requirement that the particle number must be conserved.

Conclusions

We have argued that Pauli exclusion follows from:

1. Wave functions can generally be summed to obtain a new wave function.

2. The number of particles must be conserved.

Assumed antisymmetry of a direct product of electron wave functions may have nothing to do with the Pauli exclusion principle.

We presented a hypothesis that bosons can enter the same state through the spin-z state 0, where the number of particles cannot be defined. For fermions, that route is not available.