b) In the spin–orbit interaction we take the derivative and find
The matrix element is a constant, which simplifies the calculation. We
evaluate the factor by defining the total angular momentumJ as
For the ground state of the harmonic oscillator, and
The above expectation value of is zero. The ground state is unaf-
fected by the spin–orbit interaction, although it is affected by relativistic
corrections (see Problem 5.44) as well as by other states (see Problem 5.45).
The first excited states have so that For
we find that
5.47 Interacting Electrons (MIT)
a) The wave function for a single electron bound to a proton is that of the
hydrogen atom, which is
where is the Bohr radius. When one can neglect the Coulomb repulsion
between the two electrons, the ground state energy and eigenfunctions are
For we find that
298 SOLUTIONS