Homework Problems

1. An electron is bound in a harmonic oscillator potential $V_0={1\over 2}m\omega^2x^2$. Small electric fields in the $x$ direction are applied to the system. Find the lowest order nonzero shifts in the energies of the ground state and the first excited state if a constant field $E_1$ is applied. Find the same shifts if a field $E_1 x^3$ is applied.

2. A particle is in a box from $-a$ to $a$ in one dimension. A small additional potential $V_1=\lambda \cos({\pi x\over 2b})$ is applied. Calculate the energies of the first and second excited states in this new potential.

3. The proton in the hydrogen nucleus is not really a point particle like the electron is. It has a complicated structure, but, a good approximation to its charge distribution is a uniform charge density over a sphere of radius 0.5 fermis. Calculate the effect of this potential change for the energy of the ground state of hydrogen. Calculate the effect for the $n=2$ states.

4. Consider a two dimensional harmonic oscillator problem described by the Hamiltonian $H_0={p_x^2+p_y^2\over 2m}+{1\over 2}m\omega^2(x^2+y^2)$. Calculate the energy shifts of the ground state and the degenerate first excited states, to first order, if the additional potential $V=2\lambda xy$ is applied. Now solve the problem exactly. Compare the exact result for the ground state to that from second order perturbation theory.

5. Prove that $\sum\limits_n (E_n-E_a)\vert\langle n\vert x\vert a\rangle \vert^2={\hbar^2\over 2m}$ by starting from the expectation value of the commutator $[p,x]$ in the state $a$ and summing over all energy eigenstates. Assume $p=m{dx\over dt}$ and write ${dx\over dt}$ in terms of the commutator $[H,x]$ to get the result.

6. If the general form of the spin-orbit coupling for a particle of mass $m$ and spin $\vec{S}$ moving in a potential $V(r)$ is $H_{SO}={1\over 2m^2c^2}\vec{L}\cdot\vec{S}{1\over r}{dV(r)\over dr}$, what is the effect of that coupling on the spectrum of a three dimensional harmonic oscillator? Compute the relativistic correction for the ground state of the three dimensional harmonic oscillator.