Positronium

Positronium, the Hydrogen-like bound state of an electron and a positron, has a ``hyperfine'' correction which is as large as the fine structure corrections since the magnetic moment of the positron is the same size as that of the electron. It is also an interesting laboratory for the study of Quantum Physics. The two particles bound together are symmetric in mass and all other properties. Positronium can decay by anihilation into two or more photons.

In analyzing positronium, we must take some care to correctly handle the relativistic correction in the case of a reduced mass much different from the electron mass and to correctly handle the large magnetic moment of the positron.

The zero order energy of positronium states is

where the reduced mass is given by

The relativistic correction must take account of both the motion of the electron and
the positron.
We use
and
.
Since the electron and positron are of equal mass, they are always exactly
oposite each other in the center of mass and so the momentum vector we use is
easily related to an individual momentum.

We will add the relativistic correction for both the electron
and the positron.

This is just half the correction we had in Hydrogen (with essentially replaced by ).

The spin-orbit correction should be checked also.
We had
as the interaction
between the spin and the B field producded by the orbital motion.
Since
, we have

for the electron. We just need to add the positron. A little thinking about signs shows that we just at the positron spin. Lets assume the Thomas precession is also the same. We have the same fomula as in the fine structure section except that we have in the denominator. The final formula then is

again just one-half of the Hydrogen result if we write everything in terms of for the electron spin, but, we add the interaction with the positron spin.

The calculation of the spin-spin (or hyperfine) term also needs some attention.
We had

where the masses in the deonominator of the first term come from the magnetic moments and thus are correctly the mass of the particle and the mas in the last term comes from the wavefunction and should be replaced by . For positronium, the result is

Jim Branson 2013-04-22