1. Kyle Schmitt University of Tennessee December 4, 2008
The Lamb Shift
Kyle Schmitt
University of Tennessee
December 4, 2008

2. The Lamb Shift -- Kyle Schmitt -- 12/5/08
Outline
Experimental motivation
Calculation
Conclusion
The Lamb Shift -- Kyle Schmitt -- 12/5/08

3. Experimental Motivation
Descriminating detector (sensitive only to excited H atoms)
Microwave source
Excited H atoms
Electron beam H2
The Lamb Shift -- Kyle Schmitt -- 12/5/08

4. The Lamb Shift -- Kyle Schmitt -- 12/5/08
Calculation
To calculate this shift we treat the interaction of the electron with the fluctuating electromagnetic field as a perturbation, beginning with the Schrodinger equation:
Here  is a wavefunction describing both the electron and the radiation field Aq:
The Lamb Shift -- Kyle Schmitt -- 12/5/08

5. The Lamb Shift -- Kyle Schmitt -- 12/5/08
We must treat this interaction as a perturbation:
This results in the energy correction:
Noting the limits of integration K~mc2 for k and the Bohr radius 1/(mZ) for x, we have ei k·x ~ 1. Noting further that
The Lamb Shift -- Kyle Schmitt -- 12/5/08

6. The Lamb Shift -- Kyle Schmitt -- 12/5/08
We find the correction for a bound election:
Subtracting the correction for a free electron, which affects all states indiscriminately, we find:
Integrating with respect to k noting that Em – En << K,
From here we proceed with numeric calculations. Assuming the logarithm is nearly constant with respect to n, we use its numerically calculated average value and the sum given by Bethe:
The Lamb Shift -- Kyle Schmitt -- 12/5/08

7. The Lamb Shift -- Kyle Schmitt -- 12/5/08
The shift is then found to be
This quantity is zero for states with non-zero angular momentum, whose wavefunctions vanish at the origin. Thus this correction affects the 2S1/2 state, but not the 2P1/2 state. Using
for s-wave states and the numerically calculated result for Em – En, we find for the 2S1/2 state
The Lamb Shift -- Kyle Schmitt -- 12/5/08

8. The Lamb Shift -- Kyle Schmitt -- 12/5/08
Conclusion
This result of 1040 MHz is in good agreement with the experimental result 1057 MHz, with a small discrepancy due to the non-relativistic treatment.
References:
H. Bethe, Phys. Rev. 72, 339 (1947)
W. Lamb and R. Retherford, Phys. Rev. 72, 241 (1947)
C. Itzykson “Quantum Field Theory”, McGraw-Hill 1980 Section 7-3-2
The Lamb Shift -- Kyle Schmitt -- 12/5/08