vacuum energy fluctuations and the hydrogen electron in these different orbitals is the cause of the Lamb shift, as was shown subsequent to its discovery. The Lamb shift has since played a significant role through vacuum energy fluctuations in theoretical prediction of
Hawking radiation from
In 1978, on Lamb's 65th birthday,
Freeman Dyson addressed him as follows: "Those years, when the Lamb shift was the central theme of physics, were golden years for all the physicists of my generation. You were the first to see that this tiny shift, so elusive and hard to measure, would clarify our thinking about particles and fields."
and this is valid only when the
frequencyν is greater than ν0 in the Bohr orbit, . The electron is unable to respond to the fluctuating field if the fluctuations are smaller than the natural orbital frequency in the atom.
For the field oscillating at ν,
where is some large normalization volume (the volume of the hypothetical "box" containing the hydrogen atom). By the summation over all
This result diverges when no limits about the integral (at both large and small frequencies). As mentioned above, this method is expected to be valid only when , or equivalently . It is also valid only for wavelengths longer than the
Compton wavelength, or equivalently . Therefore, one can choose the upper and lower limit of the integral and these limits make the result converge.
Finally, the difference of the potential energy becomes:
where is the
fine-structure constant. This shift is about 500 MHz, within an order of magnitude of the observed shift of 1057 MHz.
Welton's heuristic derivation of the Lamb shift is similar to, but distinct from, the calculation of the
Darwin term using
Zitterbewegung, a contribution to the
fine structure that is of lower order in than the Lamb shift.: 80–81
In 1947 Willis Lamb and
Robert Retherford carried out an experiment using
microwave techniques to stimulate radio-frequency transitions between
2S1/2 and 2P1/2 levels of hydrogen. By using lower frequencies than for optical transitions the
Doppler broadening could be neglected (Doppler broadening is proportional to the frequency). The energy difference Lamb and Retherford found was a rise of about 1000 MHz (0.03 cm−1) of the 2S1/2 level above the 2P1/2 level.
This particular difference is a
one-loop effect of
quantum electrodynamics, and can be interpreted as the influence of virtual
photons that have been emitted and re-absorbed by the atom. In quantum electrodynamics the electromagnetic field is quantized and, like the
harmonic oscillator in
quantum mechanics, its lowest state is not zero. Thus, there exist small
zero-point oscillations that cause the
electron to execute rapid oscillatory motions. The electron is "smeared out" and each radius value is changed from r to r + δr (a small but finite perturbation).
The Coulomb potential is therefore perturbed by a small amount and the degeneracy of the two energy levels is removed. The new potential can be approximated (using
atomic units) as follows:
The Lamb shift itself is given by
with k(n, 0) around 13 varying slightly with n, and
with log(k(n,ℓ)) a small number (approx. −0.05) making k(n,ℓ) close to unity.
Hans Bethe was the first to explain the Lamb shift in the
hydrogen spectrum, and he thus laid the foundation for the modern development of
quantum electrodynamics. Bethe was able to derive the Lamb shift by implementing the idea of mass renormalization, which allowed him to calculate the observed energy shift as the difference between the shift of a bound electron and the shift of a free electron.
The Lamb shift currently provides a measurement of the
fine-structure constant α to better than one part in a million, allowing a
precision test of quantum electrodynamics.