Homework Assignment #2
Physics 8.421, Spring 2012, Prof. W. Ketterle
Due Wednesday, February 29th, 2012
1. Atomic Units
Let the atomic unit of E field be EA≡e/a2
0, the field of the ground state electron at the site of the proton
in hydrogen.
a) (2 points) On the scale of atomic units, the energy of the electrostatic potential balances the energy of
quantum confinement. Use this equality to derive the atomic size, a0. (Ignore numerical factors.)
b) (1 point) Find the magnetic field of the electron at the proton, BN. (Assume a classical orbit for the
electron. If factors of 2 arise, ignore them.)
c) (1 point) Find the magnetic field, BH, which has an interaction energy of one Hartree with a Bohr mag-
neton.
d) (1 point) Express these fields in terms of EA(Gaussian units).
e) (2 points) Are there strong reasons to prefer BNor BHas the atomic unit of magnetic field?
2. Determination of the fine structure constant, α
As was discussed in the class, the fine structure constant (α) determines the ratio of energy stored in the
electromagnetic field produced by a system of charged particles to the total mechanical energy of the system
including its rest mass assuming that particles cannot approach each other closer than a Compton wave-
length (that’s when pair creation would start).
The fine structure constant, together with the rest masses of the particles also determines the minimal ener-
gies of the system i.e. the bound states. Atoms consist of bound charged particles and so α, together with
the ratios of the masses of elementary particles, becomes responsible for the relative scale of all the observed
electromagnetic phenomena in nature. In the words of Richard P. Feynman:
“[α] has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical
physicists put this number up on their wall and worry about it. Immediately you would like to know where
this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms?
Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with
no understanding by man. You might say the ‘hand of God’ wrote that number, and ‘we don’t know how
He pushed his pencil.’” Richard P. Feynman, QED: The Strange Theory of Light and Matter, Princeton
University Press 1985, p. 129.
If the ratios of masses of the various charged particles in a system are known, αcan be measured by com-
paring the energies of the different bound states. The simplest such system, consisting of a proton and an
electron, is hydrogen. Unfortunately, methods based on hydrogen spectra have limited accuracy due to the
transition frequencies and linewidths. A better choice are the fine structure splittings of the 23Pstate in
helium. The lifetime of 100 ns lifetime and the 40 GHz splitting allow a higher precision compared to 1.6
ns and 10 GHz for hydrogen. However, relating the helium spectrum to αrequires precise calculations of
the energy levels of the helium atom and, at present unresolved theoretical inconsistencies remain (see Phys.
Rev. Lett. 95, 203001 (2005) ).
The most precise measurement of αis done by measuring g−2 (the difference between g and 2) for a single
electron trapped in a Penning trap ( G. Gabrielse et al., Phys. Rev. Lett 97, 030802 (2006) ). In this case
QED calculations relating g−2 to αcan be done to sufficiently high accuracy. However, an independent
measurement of αwould provide a stringent comparison of the physics behind that measurement with QED.
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