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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#6, Rev. A, due Monday, 2015-10-26 }
\end{center}

\vspace{.3 in}

% ~/papers/ngrav/readme
{\bf 1.} A {\bf bouncy neutron} is trapped in the vertical $z$-direction on a
perfectly reflecting horizontal neutron mirror ($V=\infty$ for $z<0$) and by
the earth's gravitational potential $V=mgz$ where $m$ is the mass of the
neutron and $g$ is the acceleration due to gravity.  Ignore the independent
uniform horizontal motion in the $x$- and $y$-directions (see
\href{http://www.nature.com/nature/journal/v415/n6869/full/415297a.html}{Nature
  {\bf 415} 299 (2002)}).

{\bf \quad a)} Write down the Hamiltonian for this system and solve for the
energy eigenstates.  {\it Hint:} Substitute the dimensionless parameter $\zeta
= z/z_0-\zeta_n$ into the TISE to obtain the
\href{https://en.wikipedia.org/wiki/Airy_function}{Airy equation},
$d^2\psi/d\zeta^2 - \zeta \psi = 0$, with Airy function solutions $Ai(\zeta)$
and $Bi(\zeta)$.  Quantize the energy by applying boundary conditions to show that
$\zeta_n$ is the $n^{\rm th}$ zero of $Ai(\zeta)$. 

{\bf \quad b)} Calculate the quantum gravitational height scale $z_0$
[$\mu$m].  Calculate the total energy $E_n$ [peV], frequency $\omega_n/2\pi$
[Hz], and the classical turning points $z_n$ [$\mu$m] for the three lowest
quantum states $n=1,2,3.$ Plot the energies $E_n$ and wavefunctions
$\psi_n(z)$ on the graph of $V(z)$ as usual.

{\bf \quad c)} Given the initial wave function $\psi_0(z)=1/\sqrt{z_0}$ if
$0<z<z_0$ and 0 elsewhere, calculate the initial amplitudes of the first three
energy states at $t=0$, and at any later time $t$.  Using these three states,
calculate the expectation value of energy $\langle E \rangle$, and compare
with the expectation value.  What frequency
should one vibrate the mirror in order to excite a neutron from the ground
state to the first excited state?

{\bf 2.}  Show that if operators $A$ and $B$ are linear, then $A+B$, $AB$, and
$[A,B]$ are linear.  Which of the following operators are linear?  {\bf
  a)}~$A_1 \psi(x) = x^3 \psi(x)$, {\bf b)}~$A_2 \psi(x) = x (d/dx) \psi(x)$,
{\bf c)}~$A_3 \psi(x)=\lambda \psi^*(x)$, {\bf d)}~$A_4 \psi(x)=e^{\psi(x)}$,
{\bf e)}~$A_5 \psi(x)=d\psi/dx + a$, {\bf f)}~$A_6 \psi(x)=\int_{-\infty}^x
dx' \psi(x') x'$.

{\bf 3.}  Verify using the matrices for $x$ and $p$ in the hamonic oscillator
energy eigenbasis $|n\rangle$, that $[x,p]=i\hbar$.  Construct the matrix for
$H=p^2/2m + \frac12 mw^2x^2$ from the matrices for $x$ and $p$.  Use matrices
to calculate the expectation value of $x^4$ in the ground state: $\langle 0 |
x^4 | 0\rangle$.

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