# 7.P: Exercises

- 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

### Contributors

- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)