A heuristic derivation of the Pauli exclusion principle from the spin magnetic interaction

As we showed in the previous blog post, it is doubtful if the current proofs of the Pauli exclusion principle are correct.

The magnetic potential of the spin up-down electrons is

       -10^-53 / r^3   meter^3 J,

when the electrons are in the x,y plane, and r is their separation, and r is large enough so that we can consider the electron as a pointlike magnetic dipole.

The Coulomb potential is

       3 * 10^-28 / r   meter J.

For a "typical" separation in a helium atom of 2 * 10^-11 m, the magnetic potential is 10^-21 J and the Coulomb potential is 1.5 * 10^-17 J. Thus, the Coulomb potential has a 7,000 X larger absolute value.

When the electrons are not close, we can ignore the magnetic potential. When the separation of the electrons is one Compton wavelength 2 * 10^-12 m, the ratio of potentials is much lower, only 70 X.

We calculated in a previous blog post that the electron spin can be modeled as a classical particle moving in a circle of a radius 2 * 10^-13 m at the speed of light. In that model, when the distance is that, the magnetic attractive potential dominates greatly. Its potential is of the order -511 keV, while the Coulomb potential is 10 keV.

If spins are parallel, then an electron cannot tunnel through the potential of another electron

Let us assume that the electron is a pointlike magnetic dipole, an assumption which may not be true.

Let us assume that electron B stays at a fixed position. Electron B forms a repulsive potential barrier for electron A. If the spins are parallel, the barrier height grows as 1 / r^3 when r decreases. But if the spins are antiparallel, the barrier levels off at about 3 * 10^-13 m and becomes attractive at distances less than 2 * 10^-13.

Let us calculate the potential when r = 10^-15 m and we have electrons B and A with parallel spins. The potential of the magnetic repulsion is 10^-8 J, that is, some 70 GeV. How far can an electron tunnel through such a potential barrier? Our new energy-momentum relation

       E^2 = (p c)^2 + (m c^2 + 10^-8 J)^2

has E roughly zero. Then

       p c = i 10^-8  J
      =>
       p = i * 3 * 10^-17   kg m/s,

where i is the imaginary unit. For a relativistic particle,

       p c = h c / λ
      =>
       λ = h / p = i * 2 * 10^-17 m.

The imaginary wavelength means that the wave function decays by a factor e = 2.718 during that distance. Since the barrier width in our case is 2 * 10^-15 m, we see that tunneling through the barrier is negligible.

What about the case where the spins are antiparallel? The potential levels off at about r = 3 * 10^-13 m, where the potential is of the order 10^-15 J. The momentum p is i * 3 * 10^-24 kg m/s. We have then

       λ = i * 2 * 10^-10 m.

Electron A can easily tunnel through the potential of B if the spins are antiparallel.

Modeling two electrons with a single particle in 6 dimensions

We can model electrons A and B as a single particle which moves in 6 spatial dimensions. If the spins of A and B are parallel, that means that the wave function

      Ψ(t, x, y, z, x', y', z') = 0,

      if x = x', y = y', and z = z',

because A cannot tunnel into the potential barrier of B. The probability of finding A and B very close to each other is zero.

The equation above restricts the set of possible wave functions. The restriction may force the lowest energy state much higher than in the antiparallel spins case.

The setup has 6 spatial dimensions and a 3-dimensional plane where Ψ is restricted to be zero. It is hard to visualize what that means for the wave. But if we consider a wave in a 3-dimensional spherical cavity, then restricting the wave function to be zero at a 0-dimensional object, that is, at a point, will obviously raise the lowest resonant frequency greatly.

An analogue is a drum skin.
 
             drum skin
           ____________
          |             /\        |
                     fixed
                   support

If we attach one point of the skin to a fixed support, so that the displacement of the skin is always zero at that point, then the lowest resonance frequency of the skin will be much higher. The fixed point forces a node to be in the oscillation at that point.

Suppose that we are adding the 2n +1'th electron as spin up. There are already n spin up electrons. We will add 3 new spatial dimensions and introduce n new constraints for the new electron.
For 2n electrons, there are

        2 + 4 +... + 2(n - 1)
        = (n - 1) * n

constraints. For 2n + 1 electrons, there are n^2 constraints

The number of constraints increases rapidly with n. Nature seems to cope with them by placing electrons in different shells.

If the two lowest energy electrons are close to the nucleus, then all other electrons must have the wave function close to zero there. The 2s orbital of lithium would not be like the 2s orbital of hydrogen. In hydrogen, all s orbitals are non-zero at the nucleus.

Solving the helium and lithium atoms hierarchically

Let us try to solve the helium atom hierarchically, so that we let electron B to be in the lowest energy state. B will fly around on the 1s orbital. B will mostly stay rather close to the nucleus. If the spins of A and B are parallel, then the wave function of A will have a zero amplitude at B, and a small amplitude at the nucleus. The energy level of A is then a lot higher than in the orbital 1s.

If the spins of A and B are antiparallel, then A can easily tunnel through the potential of B, and A can choose its wave function relatively freely. This is the electron structure of helium.

In lithium, let us add a third electron C with spin up, for instance. If B has spin up, then the wave function of C must have a zero amplitude at B, and a small amplitude at the nucleus. The energy level of C will be substantially higher than the energy level of A and B.

The Pauli exclusion principle

Electrons tend to fill orbitals in pairs of antiparallel spins. Each electron has to avoid other electrons which have a parallel spin. That gives rise to the shell structure of electron orbitals.

Such organization of electrons will appear to fulfill the Pauli exclusion principle. Our reasoning did not assume anything about the spins being 1/2 or integer values. What is needed is a very strong magnetic repulsion between parallel spins. We did not assume anything about the wave function being antisymmetric under "interchanges" either.

Are there examples of bosons having strong magnetic repulsion, but still occupying the same quantum states? An internet search does not reveal anything.