Physics 215A
Midterm
Winter, 2000

Problem 1 [10 points]

Some Hamiltonian H has eigenstates $\vert n\rangle$ and corresponding eigenvalues E_n. E₀ is the ground state energy. A new Hamiltonian is

$$H_T=H+V\, ;$$

V is not necessarily small. V is real and all its diagonal matrix elements are zero, i.e. $\langle n\vert V\vert n\rangle=0$ and $\langle n\vert V\vert m\rangle =\langle m\vert V\vert n\rangle$. Take as a trial state for the ground state of H_Tto be of the form

$$\vert S\rangle=\cos\theta \vert\rangle +\sin\theta \vert 1\rangle \, .$$

Find the best $\theta$ and the corresponding energy.

Problem 2 [20 points]

A particle of mass m moves on a circular hoop of radius R. There is also a small periodic potential acting on the particle.

The total Hamiltonian is:

$$H=-\frac{\hbar^2}{2mR^2}\frac{\partial^2}{\partial\phi^2}+\epsilon \cos(2\phi)\, ,$$

where $\phi$ denotes the angular position of the particle and $\epsilon$ is small.

Find to second order in $\epsilon$ the energy of the ground state and to first order the energy of the first excited states.