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The Lamb Shift

According to Dirac and Schrödinger theory, states with the same \(n\) and \(j\) quantum numbers, but different \(l\) quantum numbers ought to be degenerate. However, a famous experiment by Lamb and Retherford in 1947 showed that the

\[^2S_{1/2} (n=2, l=0, j=1/2)\] 


\[^2P_{1/2} (n=2, l=1, j=1/2)\]

states of the hydrogen atom were not degenerate, but that the s state had slightly higher energy by an amount now known to be tex2html_wrap_inline1710 . The effect is explained by the theory of quantum electrodynamics, in which the electromagnetic interaction itself is quantized. Some of the effects of this theory which cause the Lamb shift are shown in the Feynman diagrams of figure 5.

Figure 5: Feynman loop diagrams showing some effects that contribute to the Lamb shift.

Table 3 shows how much each of these contribute to the splitting of tex2html_wrap_inline1204 and tex2html_wrap_inline1206 .

Table 3: Contribution of different effects to the energy splitting of tex2html_wrap_inline1204 and tex2html_wrap_inline1206 in hydrogen. Numbers are given in units of frequency tex2html_wrap_inline1208 .

The most important effect is illustrated by the center diagram, which is a result of the fact that the ground state of the electromagnetic field is not zero, but rather the field undergoes ``vacuum fluctuations'' that interact with the electron. Any discussion of the calculation is beyond the scope of this paper, so the answers will merely be given. For l=0,


where k(n,0) is a numerical factor which varies slightly with n from 12.7 to 13.2. For tex2html_wrap_inline1602 ,


for tex2html_wrap_inline1730 , where k(n,l) is a small numerical factor <0.05 which varies slightly with n and l. Notice that the Lamb shift is very small except for l=0.