Lamb shift

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In physics, the Lamb shift, named after Willis Lamb, is a small difference in energy between two energy levels $2 s 1 / 2$ and $2 p 1 / 2$ of the hydrogen atom in quantum mechanics. According to Dirac and Schrödinger theory, hydrogen states with the same $n$ and $j$ quantum numbers but different $l$ quantum numbers ought to be degenerate.

[edit] Experimental work

In 1947 Lamb and Robert Retherford carried out an experiment using microwave techniques to stimulate radio-frequency transitions between $2 s 1 / 2$ and $2 p 1 / 2$ 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 1060MHz of the $2 s 1 / 2$ level above the $2 p 1 / 2$ 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 (QED) the electromagnetic field is quantized and, like the harmonic oscillator in quantum mechanics, its lowest state is not zero. So there exist little zero-point oscillations that cause the electron to execute rapid oscillatory motions. The electron is kind of "smeared out" and the radius is changed by $r + δ r$ .

The Coulomb potential is therefore perturbed by a small amount and the degeneration of the two energy levels is removed. The new potential can be approximated (using Atomic units) as follows:

$\langle E_\mathrm{pot} \rangle=-\frac{Ze^2}{4\pi\epsilon_0}\left\langle\frac{1}{r+\delta r}\right\rangle.$

The Lamb shift itself is given by

$\Delta E_\mathrm{Lamb}=\alpha^5 m_e c^2 \frac{k(n,0)}{4n^3}\ \mathrm{for}\ \ell=0\,$

with $k (n,0)$ around 13 varying slightly with $n$ , and

$\Delta E_\mathrm{Lamb}=\alpha^5 m_e c^2 \frac{1}{4n^3}\left[k(n,\ell)\pm \frac{1}{\pi(j+\frac{1}{2})(\ell+\frac{1}{2})}\right]\ \mathrm{for}\ \ell\ne 0\ \mathrm{and}\ j=\ell\pm\frac{1}{2},$

with $k(n,\ell)$ a small number (< 0.05).

[edit] Lamb Shift in the Hydrogen Spectrum

In 1947, 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. The Lamb shift currently provides a measurement of the fine structure constant α to better than a part in a million, allowing a precision test of quantum electrodynamics.