We claimed in an earlier blog post that electron on the 2s orbital moves radially toward the proton or away from it. Then Coulomb focusing would have no effect.
But when we look at the radial probability of the electron on the 2s orbital, we see that the probability very close to the proton is much less than it would be if the electron would move exactly radially.
From the diagram at the link we see that the electron on the 2s orbital "typically" swings to a distance of 6 Bohr radii, and then dives back toward the proton again.
Let us calculate what would the total probability be for a sphere whose radius is the reduced Compton wavelength
r = 4 * 10⁻¹³ m = 1 / 137 Bohr radii,
if a classical electron would go almost directly toward the proton.
The electron moves slowly at 6 Bohr radii, and spends most of its time there. The potential at 6 radii is -4.5 eV, and the kinetic energy there is
4.5 eV - 13.6 eV / 4 = 1.1 eV.
The kinetic energy at r is
511 keV / 137 = 4 keV.
We see that the classical electron moves 60 times faster at r than at its typical distance.
The probability of the classical electron being in the r-sphere is
P' = 1 / (137 * 6 * 60) = 1 / 50,000.
However we calculated in the previous blog post that the quantum mechanical probability is
P = 1 / 1,500,000.
The quantum mechanical probability is 1 / 30 of the classical. We may interpret this that the quantum electron misses the r-sphere in most cases, and instead goes through a sphere whose radius is
R = 4 * 10⁻¹² m = 10 r.
The speed of an electron in that sphere is just 20 times faster than at the typical distance.
In quantum mechanics, everything is diffuse about the location of particles. It is not surprising that the electron does not swing exactly through the nucleus.
The far electric field of the electron lags behind and does not have time to take part in the sudden movement of the electron in the R-sphere. The effective mass of the electron is slightly reduced, and the electron will pass the proton slightly closer. It is as if the potential which is felt by the electron would be slightly lower. This is kind of extra Coulomb focusing, which is not taken into account in the Schrödinger equation and its solution.
Let us calculate the effect of the mass reduction, on an electron which flies past the proton at the distance R.
v
e- ---------->
R = impact parameter
● Z+
The kinetic energy of the electron is
511 keV / (10 * 137) = 400 eV.
The speed of the electron is
v = 0.04 c = 1.2 * 10⁷ m / s.
The fly-by lasts
t = 2 * R / v = 7 * 10⁻¹⁹ s.
The force is
F = k e² / r² = 2 * 10⁻⁵ N,
and the acceleration
a = F / m_e = 2 * 10²⁵ m / s².
The acceleration makes the electron to come closer to the proton the distance:
s = 1/2 a (t / 2)² = 10⁻¹² m.
We see that the electron turns substantially when close to the proton, tens of degrees.
The field which is farther than
D = c t / 2 = 10⁻¹⁰ m
does not have time to react. The effective mass of the electron is
m_e' = m_e (1 - 1 / 70,000).
Since the mass is reduced, the electron will go
s / 70,000
closer than it would otherwise. Thus, the effective potential is reduced
400 eV / 280,000 = 0.0014 eV.
The quantum mechanical probability of finding the electron in the R-sphere is
1,000
times the probability of finding it in the r-sphere, or
1 / 1,500.
We get an effective potential reduction of
0.0014 eV / 1,500 = 0.9 μeV.
The reduction is not of the same order of magnitude as the Uehling potential reduction 0.1 μeV.
The Coulomb focusing effect certainly exists in a classical model. Does it show up in a quantum model?
Dirac's sea of negative energy electrons
Mass reduction and Coulomb focusing cannot explain vacuum polarization effects.
Paul Dirac in his 1934 paper sketched a gas of negative energy electrons whose average location is slightly displaced by the electric field of the proton. This displacement is the vacuum polarization. E. A. Uehling and Robert Serber in 1935 calculated the correct, empirically tested, potential which is caused by vacuum polarization.
Let us have a cubic meter of empty space. If we put a ceiling on the momentum |p| that a negative energy electron can possess, we can fit just a finite number of those electrons in the cube. This is because the Pauli exclusion principle prohibits the electrons having the exact same momentum p and spin. It is like the Fermi sea of electrons in metal.
Vacuum polarization in Dirac's model is a collective phenomenon. In a Feynman diagram, vacuum polarization is a dynamic process with just a few particles present. One can derive the Uehling potential formula from a Feynman diagram (but one must use regularization).
Question. Why is the Feynman model equivalent to the Dirac sea model?
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