The technique used was quite interesting.
They made a **beam of Hydrogen atoms in the state**, which has a very long lifetime because of selection rules.
**Microwave radiation** with a (fixed) frequency of 2395 MHz was used to cause transitions to the
state
and a **magnetic field was adjusted to shift the energy of the states** until the rate was largest.
The **decay of the state to the ground state was observed** to determine the transition rate.
From this, they were able to deduce the shift between the
and
states.

**Hans Bethe used non-relativistic quantum mechanics** to calculate the self-energy correction to account for this observation.

It is now necessary to discuss **approximations needed** to complete this calculation.
In particular, the electric dipole approximation will be of great help, however, it is certainly not warranted for
large photon energies.
For a good E1 approximation we need
eV.
On the other hand, we want the cut-off for the calculation to be of order
.
We will use the E1 approximation and the high cut-off, as Bethe did, to get the right answer.
At the end, the result from a relativistic calculation can be tacked on to show why it turns out to be the right answer.
(We aren't aiming for the worlds best calculation anyway.)

The log term varies more slowly than does the rest of the terms in the sum. We can

This sum can now be reduced further to a simple expression proportional to the

This

Only the

This

The

So the

The

The **Lamb shift splits the and states which are otherwise degenerate**.
Its origin is **purely from field theory**.
The experimental **measurement of the Lamb shift stimulated theorists to develop Quantum ElectroDynamics**.
The correction increases the energy of s states.
One may think of the physical origin as the electron becoming less pointlike as virtual photons are emitted and reabsorbed.
Spreading the electron out a bit decreases the effect of being in the deepest part of the potential,
right at the origin.
Based on the energy shift,
I estimate that the **electron in the 2s state is spread out by about 0.005 Angstroms**,
much more than the size of the nucleus.

The **anomalous magnetic moment of the electron**,
, which can also be calculated in field theory,
makes a small contribution to the Lamb shift.

Jim Branson 2013-04-22