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\begin{document}

\csection{Ph 340 -- Problem Set 8 -- Spring 1997}

\begin{enumerate}
\item
Solve the 1-D infinite square well problem for a relativistic particle.
Hint: Use the Klein-Gordon equation:
\begin{displaymath}
	\frac{\partial^2 \psi}{\partial t^2} =
	c^2 \frac{\partial^2 \psi}{\partial x^2} -
	\frac{m^2 c^4}{\hbar^2} \psi .
\end{displaymath}
\item
Show that
\begin{displaymath}
	\frac{d}{dt} \int_{-\infty}^{\infty}
	\left(
		\psi^* \frac{\partial \psi}{\partial t} -
		\frac{\partial \psi^*}{\partial t} \psi
	\right) dx = 0
\end{displaymath}
for any solution $\psi$ to the 1-D Klein-Gordon equation which goes to
zero at $\pm \infty$.
\item
Assume that the 1-D time-independent Schr\"{o}dinger equation
\begin{displaymath}
	\frac{\partial^2 \psi}{\partial x^2} + k^2 \psi = 0
\end{displaymath}
has a solution of the form
\begin{displaymath}
	\psi (x) = A(x) \exp ( \pm i \int k dx ) ,
\end{displaymath}
where $k^2 = 2m [E - U(x)]/ \hbar^2$.
\begin{enumerate}
\item
Find a differential equation for $A(x)$.
\item
For slowly varying potentials compared to the wavelength, we can
ignore the term $d^2 A/dx^2$.  Find the resulting equation for $A$
with this simplification.  (This is called the WKB approximation.)
\item
Making the WKB approximation, seek solutions of the form $A = k^n$,
where $n$ is a constant which must be determined.
\item
Compute the probability flux for your solution.
\item
Qualitatively compare your WKB results with the numerical solutions
for positive and negative Gaussian potentials produced by {\tt
scatter1\_demo.r}.
\end{enumerate}
\item
In elementary particle physics the neutral kaon particle has the
peculiar property that it can turn into its own antiparticle under the
effects of the weak interactions.  The state of this
particle-antiparticle system can be written as
\begin{displaymath}
	\psi = \left( \begin{array}{c}
		A \\ B
	\end{array} \right)
\end{displaymath}
where $A$ is the amplitude for it to be a particle and $B$ is the
amplitude for it to be an antiparticle.  The strong interaction Hamiltonian
is
\begin{displaymath}
	H_S = \left( \begin{array}{cc}
		Mc^2 & 0 \\ 0 & Mc^2
	\end{array} \right)
\end{displaymath}
where $M$ is the kaon-antikaon mass and $c$ is the speed of light.
The weak interaction Hamiltonian is
\begin{displaymath}
	H_W = \left( \begin{array}{cc}
		0 & W \\ W & 0
	\end{array} \right)
\end{displaymath}
where $W$ is a constant.
\begin{enumerate}
\item
Show that the eigenvalues of the strong Hamiltonian by itself are
degenerate.
\item
Find the eigenvalues and eigenvectors of the combined strong-weak
Hamiltonian $H = H_S + H_W$.
\item
What are the masses of these two states?
\item
The \emph{charge conjugation} operator $C$ turns particles into
antiparticles and antiparticles into particles:
\begin{displaymath}
	C = \left( \begin{array}{cc}
		0 & 1 \\ 1 & 0
	\end{array} \right) .
\end{displaymath}
Show that the eigenstates of the combined Hamiltonian are also
eigenstates of the charge conjugation operator.  What are the
eigenvalues of $C$ for the eigenstates of the combined Hamiltonian?
\end{enumerate}
\end{enumerate}

\end{document}
