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. . For the atomic orbital and the Coulomb potential, since it is known that For p orbitals, the nonrelativistic wave function vanishes at the origin (at the nucleus), 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: 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. Lamb–Retherford experiment 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,)) a small number (approx. −0.05) making k(n,) close to unity. For a derivation of ΔELamb see for example: In