130_notes.dvi

infinite although it can be rendered finite if we accept the fact that out theories are not valid up to
infinite energies.

The quantum self energy correction has important, measurable effects. It causes observable energy
shifts in Hydrogen and it helps us solve the problem of infinities due to energy denominators from
intermediate states.

The coupled differential equations from first order perturbation theory for the state under studyφn
and intermediate statesψjmay be solved for the self energy correction.

∆En =

∑

~k,α

∑

j

|Hnj|^2

1 −ei(ωnj−ω)t

̄h(ωnj−ω)

The result is, in general, complex. The imaginary part of the self energy correction is directly related
to the width of the state.

−

2

̄h

ℑ(∆En) = Γn

Thetime dependence of the wavefunction for the stateψnis modified by the self energy
correction.
ψn(~x,t) =ψn(~x)e−i(En+ℜ(∆En))t/ ̄he

−Γnt

2

This gives us theexponential decay behaviorthat we expect,keeping resonant scattering
cross sections from going to infinity.

The real part of the correction should be studied to understand relative energy shifts of states. It is
thedifference between the bound electron’s self energy and thatfor a free electronin
which we are interested. The self energy correction for a free particle can be computed.

∆Efree = −

2 αEcut−off

3 πm^2 c^2

p^2

We automatically account for this correction by a change in the observed mass of the electron. For
the non-relativistic definition of the energy of a free electron, an increase in the mass decreases the
energy.

mobs = (1 +

4 αEcut−off

3 πmc^2

)mbare

If wecut off the integral atmec^2 , the correction to the mass is only about 0.3%,

Since the observed mass of the electron already accounts for most of the self energy correction for
a bound state, we must correct for this effect to avoid double counting of the correction. The self
energy correction for a bound state then is.

∆E(nobs) = ∆En+

2 αEcut−off

3 πm^2 c^2

〈n|p^2 |n〉

In 1947, Willis E. Lamb and R. C. Retherford used microwave techniques to determine thesplitting
between the 2 S 12 and 2 P 12 states in Hydrogen. The result can be well accounted for by the