Physics 215A
Midterm
Fall, 1999

Problem 1
Let $\vert a_1\rangle$ and $\vert a_2\rangle$ be two states with $\langle a_i\vert a_j\rangle =\delta_{ij}$. A Hamiltonian is given by

$$H=\vert a_1\rangle\beta\langle a_2\vert+\vert a_2\rangle\beta\langle a_1\vert\, ;\$$

$\beta$ is a real number.

a.

Find the eigenvectors of H as combinations of the two states $\vert a_i\rangle$ and their corresponding eigenvalues.

b.

Suppose at t=0 the system is in the state $\vert a_1\rangle$. Write down the state vector for t>0.

c.

What is the probability of finding the system in state $\vert a_2\rangle$ at time t>0?

Problem 2
Sometimes it is convenient to approximate a situation by using a complex potential

$$V(x)=V_R(x)-iV_I(x)\, ,$$

with V_R and V_I real. We find that $\partial\rho(x,t)/\partial t + \partial j(x,t)/\partial x$ is no longer equal to zero. What is it equal to?

Problem 3

a.

If $\vert E\rangle$ is an eigenstate of the Hamiltonian H, show that

$$\frac{\partial}{\partial t} \langle E\vert{\cal O}\vert E\rangle =0\, ,$$

for any operator ${\cal O}$ (not depending explicitly on time).

b.

For

$$H=\frac{P^2}{2m}+V(X)$$

evaluate $\partial (XP)/\partial t$.

c.

Using the two results above we can show that

$$\langle E\vert\frac{P^2}{2m}\vert E\rangle = c\langle E\vert X\frac{\partial V}{\partial X}\vert E\rangle\, ,$$

where c is a constant. Derive the above and find c?