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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#7, Rev. B, due Monday, 2015-11-09 }
\end{center}

\vspace{.3 in}

{\bf 1.} {\bf Scattering potential}.  Let a potential be confined to the
region $a<x<b$, ie. $V(x)=0$ if $x<a$ or $x>b$.  Then positive energy
scattering states will be plane waves on either side of the potential:
$\psi_1(x)=Ae^{ikx}+Be^{-ikx}$ for $x<a$ and $\psi_3(x)=Fe^{ikx}+Ge^{-ikx}$
for $x>b$.

{\bf \quad a)} Identify the forward and backward travelling plane waves on
each side of the potential and calculate the probability current density
$j(x)=\frac{i\hbar}{2m}\left[\psi(x)\frac{d}{dx}\psi^*(x) -
  \psi^*(x)\frac{d}{dx}\psi(x)\right]$ for each.

{\bf \quad b)} The {\it scattering matrix} $\pmatrix{B\cr F}=\pmatrix{S_{11} &
  S_{12} \cr S_{21} & S_{22}}\pmatrix{A\cr G}$ is the ratio of incoming and
outgoing coefficients.  Show that each element forms a reflection or
transmission coefficient: $|S_{fi}|^2=|j_f(x)/j_i(x)|$, where $j_i(x)$ is an
incoming current density, and $j_f(x)$ is an outgoing current density.
 
{\bf \quad c)} The {\it transfer matrix} $\pmatrix{F\cr G}=\pmatrix{M_{11} &
  M_{12} \cr M_{21} & M_{22}}\pmatrix{A\cr B}$ is the ratio of left to right
coefficients.  Calculate the entries of the $M$ in terms of $S$ and vice
versa, and check that they are consistent.

{\bf \quad d)} Calculate the transfer/scattering matrices and forward/backward
reflection/transmission coefficients of the following potentials:

{\bf \quad\quad i)} The $\delta$-potential, $V(x)=-\alpha \delta(x-a)$.

{\bf \quad\quad ii)} The finite square well, $V(x)=-V_0\theta(a-|x|)$.

{\bf \quad\quad iii)} The step potential $V(x)=V_0\theta(x)$.


{\bf 2.} A {\bf Double $\delta$-potential} has two identical separated wells
$V(x)=-\alpha[\delta(x+a)+\delta(x-a)]$.

{\bf \quad a)} Calculate the bound state wave functions of this potential.

{\bf \quad b)} Show that the transfer matrix for two potential wells is the
product of transfer matrices for each well $M=M_1M_2$.  Multiply the transfer
matrices of the two single wells to calculate the transmission and reflection
coefficients probabilities of $V(x)$.

{\bf 3. [bonus]} Classical wave media are characterized by {\bf velocity} $v$
and {\bf characteristic impedance} $Z_0$.  For waves on a string of tension
$T$ and density $\mu$, $v=\sqrt{T/\mu}$ and $Z_0=\sqrt{T\mu}$.  For EM waves,
$v=1/\sqrt{\mu\epsilon}$ and $Z_0=\sqrt{\mu/\epsilon}$.  A transmission line
is characterized by the propagation constant $\gamma=\sqrt{ZY}$ so that
$V(z)=V_0e^{-\gamma z-i\omega t}$, and $Z_0=\sqrt{Z/Y}$, where $Z=R+i\omega L$
and $Y=G+i\omega C$ are the series impedance and shunt admittance of the line,
respectively.

{\bf \quad a)} Compare the velocity and reflection/transmission coefficients
of a QM plane waves on a step potential $V(x)=V_0\theta(x)$ to obtain the
``propagation constant'' and ``characteristic impedance''.

{\bf \quad b)} Compare a point mass on a string to the $\delta$-potential to
obtain the QM point impedance.

{\bf \quad c)} The physical interpretation of impedance is the ratio the two
fields involved in the boundary conditions: $f(x)$ and $f'(x)$ for a string;
$\bf E$ and $\bf H$ for an EM wave; $V$ and $I$ for a wave guide.  Note that
the product of these two fields is the transfer of energy $\bf S=\bf E \times
\bf H$.  Is there a corresponding interpretation for quantum mechanical waves?

\end{document}
