Sample Test Problems

1. Assume an electron is bound to a heavy positive particle with a harmonic potential $V(x)={1\over 2}m\omega^2x^2$. Calculate the energy shifts to all the energy eigenstates in an electric field $E$ (in the $x$ direction).

2. Find the energies of the $n=2$ hydrogen states in a strong uniform electric field in the z-direction. (Note, since spin plays no role here there are just 4 degenerate states. Ignore the fine structure corrections to the energy since the E-field is strong. Remember to use the fact that $[L_z,z]=0$. If you are pressed for time, don't bother to evaluate the radial integrals.)

3. An electron is in a three dimensional harmonic oscillator potential $V(r)={1\over 2}m\omega^2r^2$. A small electric field, of strength $E_z$, is applied in the $z$ direction. Calculate the lowest order nonzero correction to the ground state energy.

4. Hydrogen atoms in the $n=2$ state are put in a strong Electric field. Assume that the 2s and 2p states of Hydrogen are degenerate and spin is not important. Under these assumptions, there are 4 states: the 2s and three 2p states. Calculate the shifts in energy due to the E-field and give the states that have those energies. Please work out the problem in principle before attempting any integrals.