1/2 h / (2π) = 5 * 10^-35 kg m/s.
Suppose that the electron is a classical mass of 0.9 * 10^-30 kg spinning around 1/2 its Compton wavelength at a speed b * c. Then its angular momentum is
L = m_e * 10^-12 m * c
= b * 0.9 *10^-30 * 1.2 * 10^-12
* 3 * 10^8 kg m/s
= b * 3.2 * 10^-34 kg m/s,
from which we get the speed b c = 1/6 c.
A classical model as the electron as a rotating ball of size the Compton wavelength is quite reasonable.
The spin kinetic energy in our model is
E = 1/2 m_e v^2
= 1/2 m_e c^2 / 36
= m_e c^2 / 72.
That is, 1.5 % of the electron total energy would be spin energy.
The electron charge circulating in a loop of radius 1.2 * 10^-12 m makes an electric current of
I = 1.6 * 10^-19 / (2π * 1.2 * 10^-12 m)
* (0.5 * 10^8) A
= 1 A.
The magnetic field strength inside the current loop I is huge:
B = μ_0 I / (2r)
= 1.3 * 10^-6 / (2.4 * 10^-12) T
= 5 * 10^5 T.
The magnetic moment of our model electron is
m = I S
= 1 * π * (1.2 * 10^-12)^2 A m^2
= 5 * 10^-24 J/T.
The measured magnetic moment of the electron is
m = 9 * 10^-24 J/T,
that is, it is roughly double to our classical model.
If we put the electron in a strong magnetic field B, flipping it will give an energy
E = 2 * |m| B.
For a field of 1 T, the energy is only 2 * 10^-23 J, while the total energy of the electron is 10^-13 J and the rotation energy of the electron is 1.5 * 10^-15 J. If we put the electron to a magnetic field of 10^8 T, then the electron might speed up its rotation to acquire a larger magnetic moment, and consequently, a larger spin. Neutron stars have magnetic fields up to 10^11 tesla.
The electron orbit radius is only 1/(4π) Compton wavelengths?
Our model of the electron circling at a radius 1/2 of its Compton wavelength would mean that the electron actually would complete 3.14 waves in its orbit. That does not sound reasonable.
In our model, the electron is only moving at 1/6 the speed of light. What if we let the electron move at the speed of light? The radius is then only 1/6, and the electron does roughly 1/2 wavelengths during its orbit, which corresponds to the spin quantum number 1/2.
The Compton wavelength is
λ = h / (m_e c).
Let us calculate again the angular momentum if the electron circles at a radius λ / (4π):
L = m_e c h / (m_e c 4π)
= 1/2 h / (2π).
Thus, the electron moving at the speed of light at a radius of 1/(4π) Compton wavelengths is a natural model for the angular momentum of the electron. The spin kinetic energy in this case is 100 % of the total energy of the electron.
The associated electric current would be roughly 40 amperes and the magnetic field inside the current loop some 10^8 T.
The magnetic moment stays the same, at 5 * 10^-24 J/T.
We conclude that a better classical-relativistic model for the electron is an object circling with a radius of
r = λ / (4π)
= h / (4π m_e c)
= 2 * 10^-13 m.
The spin 1/2 corresponds to the electron doing 1/2 Compton wavelengths in its orbit.
Note that zitterbewegung has the speed of light and the orbit radius of 1/2 the Compton wavelength.
In the hydrogen atom, in the ground state, the electron does one "spatial wavelength" in its orbit around the proton. The spatial wavelength is determined by the momentum p of the electron in the non-relativistic Schrödinger equation. The electron does one rotation around the proton in 1.5 * 10^-16 seconds. The Compton frequency of the electron is
f = 3 * 10^8 / (2.4 * 10^-12) Hz
= 1.2 * 10^20 Hz.
The wave function of the electron will rotate 1.8 * 10^4 (= α^2?) times in the time it completes one rotation around the proton.
The electron in the hydrogen atom can be considered a particle which makes a small circle at the speed of light, so that its wave function at a fixed point rotates 1/2 rounds per circle. It also makes a large slow circle around the proton, such that its wave function at a fixed point rotates 18,000 times during the circle, but its "spatial wave function" only rotates 1 round in the circle.
exp(-i (E t - E x)),
where we have set c = 1, and E is the rest mass of the electron. For a particle moving at the speed of light, E = p.
The electron must make two complete rounds around the small circle, for the wave function phase to return to its original value at a fixed point. This is probably the origin of the strange 720 degree rotation symmetry of the electron wave function.
Magnetic fields in particle collisions
The electric field in a head-on collision of an electron and a positron reaches at least
511 kV / (2 * 10^-12 m) = 2 * 10^17 V/m,
that is, it is close to the Schwinger limit.
A Lorentz transformation of the electric field to a speed close to the speed of light produces a magnetic field strength of the same order of magnitude, 10^17 tesla. The magnetic field in a collision might temporarily excite an electron to a higher spin state.
In our model of the previous section, all electron energy is spin energy. An electron in the spin state 3/2 would have an energy 1.5 MeV.
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