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\begin{document}

\csection{Ph 340 -- Problem Set 11 -- Spring 1997}

\begin{enumerate}
\item
Time independent perturbation theory: Assume that the Hamiltonian
splits into two parts, $H = H_0 + \epsilon J$, where
\begin{displaymath}
	H_0 = - \frac{1}{2} \frac{\partial^2}{\partial x^2} + U_0 (x) ,
\end{displaymath}
$U_0 = 0$ for $0 < x < 1$ and $U_0 = \infty$ otherwise.
\begin{enumerate}
\item
Find the eigenfunctions and energies for $H_0$.  (This should be easy
by now, as this is just the scaled infinite square well.)
\item
If $J = 1$ for $0 < x < 1/2$ and $J = 0$ otherwise, find the energy
shifts due to this perturbation potential.
\end{enumerate}
\item
Time dependent perturbation theory:
\begin{enumerate}
\item
Write out equation (21) in the section of the web notes on perturbation
theory for the case of a system with two states with amplitudes $A_1$
and $A_2$ and energies $E_1$ and $E_2$.
\item
Suppose that $\omega = E_2 - E_1$.  Drop the exponential terms which
oscillate rapidly with time since their net effect on the solution is
small.  Then solve the pair of equations for $A_1 (t)$ and $A_2 (t)$.
\item
A real-world example of this type of solution is nuclear magnetic
resonance.  Look up the magnetic moment of the proton and compute
$E_1$ and $E_2$ for up and down spin orientations for a magnetic field
of $0.1 \unit{Tesla}$.  Determine the frequency for which $\omega =
(E_2 - E_1)/\hbar$.  (Note that we set $\hbar = 1$ in the first two
parts of the problem.)
\end{enumerate}
\item
Show either numerically or analytically that $( \psi_m , x \psi_n )$
is only non-zero for $m = n \pm 1$ for the harmonic oscillator
eigenfunctions.  (Hint: First show that $A^+ + A^- = 2^{1/2} x$ to
do it analytically.)
\end{enumerate}

\end{document}
