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\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#5, Rev. B, due Friday, 2015-10-30 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~2 \#1, %p29, tent wave function
      \#2, %p30, E > minimum V
      \#3, %p38, E > 0 explicit
      \#4. %p38, HUP

{\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 and determine the constants $z_0$ and $\zeta_n$
in terms of $m$, $g$, $\hbar$, $E$ to massage the TISE into the
\href{https://en.wikipedia.org/wiki/Airy_function}{Airy equation},
$d^2\psi/d\zeta^2 - \zeta \psi = 0$.  This equation has two known independent
solutions, the Airy functions Ai$(\zeta)$ and Bi$(\zeta)$.  Quantize the
energy by applying boundary conditions $\psi|_{z=0}=0$ and
$\psi|_{z\to\infty}\to0$ to show that $\zeta_n$ is the $n^{\rm th}$ root 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 $\langle E \rangle$.  What frequency
should one vibrate the mirror in order to excite a neutron from the ground
state to the first excited state?

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