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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#8, Rev. A, due Friday, 2016-10-28 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed] Ch.~2
\#52, %p90, scattering matrix
\#53. %p91, transfer matrix

{\bf 1.} The {\bf step potential} $V(x)=V_0\theta(x) = \{V_0$ if $x>0$, or 0
if $x<0\}$ is the quantum mechanical analog of a wave on a string crossing
over the junction of two strings of characteristic impedance $Z_1$ and $Z_2$
connected at $x=0$.

{\bf \quad a)} Calculate the transfer matrix for this potential.

{\bf \quad b)} Calculate the scattering matrix from the transfer matrix of
part a).

{\bf \quad c)} Calculate the forward (coming in from the left) and backward
(coming in from the right) probabilities of reflection and transmission for a
particle of energy $E>0$ using the elements of the scattering matrix of part
b).  {\it Hint: the forward transmission probability must take into account
  the difference of incoming and outgoing velocities:}
\[ T_\ell=\left.\frac{j_F}{j_A}\right|_{G=0} =
\left.\frac{k_2|F|^2}{k_1|A|^2}\right|_{G=0} = \frac{k_2}{k_1} |S_{21}|^2 .\]

{\bf \quad d)} Show that the forward reflection coefficient is $R=1$ if
$E<V_0$, similar to total internal reflection of a light wave in a fibre optic
cable.

{\bf \quad e) [bonus]} What are the quantum mechanical analogs of the velocity
$v$ and impedance $Z$ of a classical wave medium?

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