Physics 215A
Final
Fall, 1999

Open book, open notes.
Relevant Clebsch-Gordan coefficients and spherical harmonics are attached.

Problem 1 [15 points]

A wave function of some particle of mass m moving in one dimension is

$$\psi(x,t)=f(x,t)e^{ikx}\, ,$$

with f(x,t) a real, positive function of position and time.

a.

Using the continuity of current, relate $\partial f(x,t)/ \partial x$ to $\partial f(x,t)/\partial t$.

b.

Using the result above, find the most general form for f(x,t).

Problem 2 [10 points]

Let $\Psi (t)$ be a two component wave function,

$$\Psi(t)=\left(\begin{array}{c} \psi_1(t) \\ \psi_2(t) \end{array}\right)\, ,$$

satisfying the wave function

$$i\hbar \frac{\partial\Psi}{\partial t}=H\Psi\, ,$$

with H a hermitian, 2x2 matrix. Show that if

$$\Psi^{\dag }(0)\Psi(0)=\vert\psi_1(0)\vert^2+\vert\psi_2(0)\vert^2=1\, ,$$

then

$$\Psi^{\dag }(t)\Psi(t)=\vert\psi_1(t)\vert^2+\vert\psi_2(t)\vert^2=1\, ,$$

for any time t; namely, normalization is maintained. Problem 3 [15 points]

The Hamiltonian for a magnetized rotator in an external magnetic field is

$$H=\frac{ L^2}{2I}-\frac{\mu}{\hbar} BL_z\, .$$

I is the moment of inertia, $\mu$ is the magnetic moment and B is the magnetic field. L_i are the angular momentum operators

a.

What are the eigenvalues of this Hamiltonian?

b.

If the wave function of this system at t=0 is

$$\psi(\theta,\, \phi;\, t=0)=\sqrt{\frac{3}{4\pi}}\sin\theta\sin\phi\, ,$$

what is $\psi(\theta,\, \phi;\, t)$ for $t\ge 0\,$?

c.

At t=0, find the expectation value of L_z and L_z² in the state of part b above.

Problem 4 [10 points]

An electron is in an l=1 state. The total angular momentum is j=1/2 and the z component is $-{\hbar}/2$.

a.

Write out the wave function as a sum of the states $\vert l=1,m_l;s=1/2,m_s\rangle$.

b.

What is the expectation value of ${\hbar}\sigma_z/2$ in this state?