- Calculate the energy-shift in the ground state of the one-dimensional harmonic
oscillator when the perturbation
- Calculate the energy-shifts due to the first-order Stark effect in the
state of a hydrogen atom. You do not
need to perform all of the integrals, but you should construct the correct linear combinations of states.
- The Hamiltonian of the valence electron in a hydrogen-like atom can be written
- Consider an energy eigenstate of the hydrogen atom characterized by the standard quantum numbers
,
, and
.
Show that if the energy-shift due to spin-orbit coupling (see Section 7.7) is added to that due to the electron's relativistic mass increase (see previous exercise) then the
net fine structure energy-shift can be written

Here, is the Bohr radius. Assuming that the above formula for the energy shift is valid for (which it is), show that fine structure causes the energy of the states of a hydrogen atom to exceed those of the and states by .