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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#8, Rev. A, due Wednesday, 2017-11-01 }
\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 traveling from a string
of characteristic impedance $Z_1$ to another of impedance $Z_2$, joined 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
$E<V_0$.  This is the analog of total internal reflection of light wave inside
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?

{\bf 2. a)} From H03 1d), the mechanical impedance of an oscillating mass is
$Z=i\omega m$.  We can also apply impedance to {\bf elastic collisions},
interpreting the boundary conditions as conservation of momentum $p=mv$ and
kinetic energy $E = \frac12 p v$, and defining impedance as $m=p/v$.  For two
particles of mass $m_1$ and $m_2$, initial velocities $v_1$ and $v_2$, and
final velocities $v_1'$ and $v_2'$ confined to the $x$-axis, factorize the
energy equation $\frac12 m_1(v_1^2-v_1'^2)=\frac12 m_2(v_2'^2-v_2^2)$ and use
conservation of momentum to show that $v_1+v_1'=v_2+v_2'$, and therefore %
${\scriptsize \pmatrix{v_1'\cr v_2'} = \frac{1}{m_1+m_2} %
  \pmatrix{ m_1{-}m_2 & 2m_2 \cr 2m_1 & m_2{-}m_1 } \pmatrix{v_1 \cr v_2}}$.
In particular, if two particles' impedances match, their velocities are
exchanged during the collision [see
\href{http://en.wikipedia.org/wiki/Newton's_cradle}{Newton's cradle}].  For
$v_2=0$, show the that the {\it reflection coefficient} is
$R \equiv E_2/E_0 = (m_1-m_2)^2/(m_1+m_2)^2$ and the {\it transmission
  coefficient} is $T \equiv E_2/E_0 = 4m_1m_2/(m_1+m_2)^2$.

{\bf \quad b) [bonus]} Apply the concept of impedance (moment of inertia) to
conservation of angular momentum and energy: given an incoming baseball of
mass $m$ and velocity $v$, and a bat with moment of inertia $I$ swinging at
angular velocity $\omega$, calculate the optimum radius $r$ for transfer of
energy from the bat to the ball.


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