If you are curious how the magnetic dipole strength of the electron can just pop out of the relativistic Dirac equation, this note gives a quick derivation.
First, a problem must be addressed. Dirac's equation, chapter
12.12, assumes that Einstein's energy square root falls
apart in a linear combination of terms:
If you believe that the Dirac linear combination is the way physics really works, and its description of spin leaves little doubt about that, then the answer is clear: you need to put in the linear combination, not in the square root.
So, what are now the energy levels? That would be hard to say
directly from the linear form, so square it down to , using
the properties of the matrices, as given in chapter
12.12 and its note. You get, in index notation,
By multiplying out the expressions for the of chapter
12.12, using the fundamental commutation relation for the
Pauli spin matrices that
,
In the nonrelativistic case, the rest mass energy is much
larger than the other terms, and in that case, if the change in square
energy is , the change in energy itself is
smaller by a factor , so the energy due to the magnetic
field is
(D.52) |