# 8.P: Exercises

- Demonstrate that when , where is the momentum operator, and is a real function of the position operator, . Hence, show that the Hamiltonian (870) is Hermitian.
- Find the selection rules for the matrix elements , , and to be non-zero. Here, denotes an energy eigenket of a hydrogen-like atom corresponding to the conventional quantum numbers, , , and .
- Demonstrate that
- Demonstrate that the spontaneous decay rate (via an electric dipole transition) from any 2p state to a 1s state of a hydrogen atom is

### Contributors

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