bound state and continuum state terms have the same fractionalchange, it is convenient to just use
p^2
2 mfor all the corrections.
p^2
2 mobs=
p^2
2 mbare
−Cp^2∆En(obs) = ∆En+C〈n|p^2 |n〉= ∆En+2 α
3 πm^2 c^2Ecut−off〈n|p^2 |n〉Because we are correcting for the mass used to calculate the baseenergy of the state|n〉, our
correction is written in terms of the electron’s momentum in that state.
35.1 The Lamb Shift
In 1947, Willis E. Lamb and R. C. Retherford used microwave techniques to determine thesplitting
between the 2 S 21 and 2 P 12 states in Hydrogento have a frequency of 1.06 GHz, (a wavelength
of about 30 cm). (The shift is now accurately measured to be 1057.864 MHz.) This is about the
same size as the hyperfine splitting of the ground state.
The technique used was quite interesting. They made abeam of Hydrogen atoms in the 2 S (^12)
state, which has a very long lifetime because of selection rules. Microwave radiationwith a
(fixed) frequency of 2395 MHz was used to cause transitions to the 2P 32 state and amagnetic field
was adjusted to shift the energy of the statesuntil the rate was largest. Thedecay of the
2 P 32 state to the ground state was observedto determine the transition rate. From this, they
were able to deduce the shift between the 2S 12 and 2P 12 states.
Hans Bethe used non-relativistic quantum mechanicsto calculate the self-energy correction
to account for this observation.