Homework Problems

1. Show that the Hamiltonian $H={1\over 2\mu}[\vec{p}+{e\over c}\vec{A}(\vec{r},t)]^2-e\phi(\vec{r},t)$ yields the Lorentz force law for an electron. Note that the fields must be evaluated at the position of the electron. This means that the total time derivative of $\vec{A}$ must also account for the motion of the electron.

2. Calculate the wavelengths of the three Zeeman lines in the $3d\rightarrow 2p$ transition in Hydrogen atoms in a $10^4$ gauss field.

3. Show that the probability flux for system described by the Hamiltonian
   
   $$H={1\over 2\mu}[\vec{p}+{e\over c}\vec{A}]^2$$
   
   
   is given by
   
   $$\vec{j}={\hbar\over 2i\mu}[\psi^*\vec{\nabla}\psi- (\vec{\nabla}\psi^*)\psi+{2ie\over \hbar c}\vec{A}\psi^*\psi].$$
   
   
   Remember the flux satisfies the equations ${\partial (\psi^*\psi)\over\partial t}+\vec{\nabla}\vec{j}=0$.

4. Consider the problem of a charged particle in an external magnetic field $\vec{B}=(0,0,B)$ with the gauge chosen so that $\vec{A}=(-yB,0,0)$. What are the constants of the motion? Go as far as you can in solving the equations of motion and obtain the energy spectrum. Can you explain why the same problem in the gauges $\vec{A}=(-yB/2,xB/2,0)$ and $\vec{A}=(0,xB,0)$ can represent the same physical situation? Why do the solutions look so different?

5. Calculate the top left $4\times 4$ corner of the matrix representation of $x^4$ for the harmonic oscillator. Use the energy eigenstates as the basis states.

6. The Hamiltonian for an electron in a electromagnetic field can be written as $H={1\over 2m}[\vec{p}+{e\over c}\vec{A}(\vec{r},t)]^2-e\phi(\vec{r},t) +{e\hbar\over 2mc}\vec{\sigma}\cdot\vec{B(\vec{r},t)}$. Show that this can be written as the Pauli Hamiltonian
   
   $$H={1\over 2m}\left(\vec{\sigma}\cdot[\vec{p}+{e\over c}\vec{A}(\vec{r},t)]\right)^2-e\phi(\vec{r},t).$$