On March 13, 2021 we showed that we can explain the Lamb shift by the fact that the electron static electric field "lags behind" in sudden movements, and the effective mass of the electron is reduced a little bit, causing its wave function have a somewhat longer wavelength close to the proton.
The effect is to raise the energy of the 2s orbital 1,000 MHz relative to the 2p orbital.
But we did not calculate the effect of "Coulomb focusing" which is caused by the reduced mass of the electron. The reduced mass makes the electron to pass the proton a little bit closer, which lowers the energy level of the 2s orbital.
E. A. Uehling in 1935 wrote that vacuum polarization is expected to lower the energy level of 2s. The effect is roughly -2.7% on the Lamb shift.
Classically, Coulomb focusing certainly happens because of the reduced mass of the electron. We need to calculate how large the effect is, and if it can explain the Uehling potential in the hydrogen atom.
Elliptic orbits do not close => the potential appears to be not 1 / r
If we reduce the mass of the electron when it comes close (~ 10⁻¹² m) to the proton, then the electron will come somewhat closer to the proton than it would otherwise do. The path of the electron spirals relative to the laboratory frame.
A spiral in the orbit also happens if the potential is not exactly 1 / r. Thus, the lagging behind of the far field of the electron will make the 1 / r potential to appear somewhat steeper.
Calculation of the Uehling potential
We have to find a way to calculate the effect. Is it equal to the Uehling potential?
The Uehling potential decreases exponentially if we increase r. Coulomb focusing does not have an exponential law. It looks like Coulomb focusing cannot explain vacuum polarization.
The 2s orbital is radially symmetric. The electron seems to travel only radially toward the proton, and away from it. Coulomb focusing cannot have any effect on the energy of the 2s orbital.
For 2p, there should be Coulomb focusing. The Sommerfeld orbit is an ellipse whose major axis is 2 times the minor axis. We are are not sure what the effect of focusing is, because vibrations in the electric field cannot escape (the electron in the 2p orbit does not radiate), and consequently, the behavior of the system is not classical.
Paul Dirac and Werner Heisenberg about vacuum polarization
We have failed to find a classical model for vacuum polarization using the framework of Feynman diagrams.
We will next look at the work of Dirac and Heisenberg in 1934. What was their view of vacuum polarization?
Dirac's paper is from April 1934.
In the link is an English translation of Heisenberg's paper Bemerkungen zur Diracschen Theorie des Positrons, Zeit. Phys. 90 (1934).
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