Lamb Shift - Derivation

Derivation

This heuristic derivation of the electrodynamic level shift following Welton is from Quantum Optics.

The fluctuation in the electric and magnetic fields associated with the QED vacuum perturbs the Coulomb potential due to the atomic nucleus. This perturbation causes a fluctuation in the position of the electron, which explains the energy shift. The difference of potential energy is given by

Since the fluctuations are isotropic,

.

So we can obtain

.

The classical equation of motion for the electron displacement induced by a single mode of the field of wave vector and frequency is

,

and this is valid only when the frequency is greater than in the Bohr orbit, .

For the field oscillating at ,

,

therefore

.

By the summation over all ,

,

where

.

The summation is changed into the integral because of the continuity of, so

.

This result diverges when there is no limit about the integral. But this method is valid only when, or equivalently . It is also valid only for wavelengths longer than the Compton wavelength, or equivalently . Therefore we can choose the upper and lower limit of the integral and these limits make the result converge.

.

For the atomic orbital and the Coulomb potential,

,

since we know that

.

For p orbitals, the nonrelativistic wave function vanishes at the origin, so there is no energy shift. But for s orbitals there is some finite value at the origin,

,

where the Bohr radius is

.

Therefore

.

Finally, the difference of the potential energy becomes

.

This shift is about 1 GHz, very similar with the observed energy shift.