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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 521 \\
    Homework \#10, Rev. B, due Wednesday, 2016-02-10 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~2 \#17, %Hermite polynomials
Ch.~3 \#35, %p127 Coherent states of SHO
      \#39, %p129 generator of translations in space and time
Ch.~4 \#18, %p166 matrix elements of raising/lowering operators
      \#19, %p166 commutation relations of L,x,y,H
      \#22. %p170 Yll eigenstate

{\bf 1.} Consider a two-dimensional isotropic harmonic oscillator with
Hamiltonian $\mathcal{H}_{xy} = \frac12\left(-\nabla^2 + \rho^2 \right)$,
where $\rho^2=x^2+y^2$, and $\hbar=m=\omega=1$ for convenience (so that $\rho$
is similar to the normalized coordinate $\xi$ of Griffiths Eq. [2.71]).

{\bf \quad a)} Show that the Hamiltonian separates into two independent
oscillators $\mathcal{H}_{xy}=\mathcal{H}_x+\mathcal{H}_y$ in cartesian
coordinates, and thus the energy levels are $E_{n_x n_y}=n_x+n_y+1$.  Identify
the degeneracy of each energy level, and write the wave functions of the
lowest three levels, both in terms of Hermite polynomials
$H_{n_x}(x)H_{n_y}(y)$ and in terms of creation operators $a_x\ad$ and
$a_y\ad$ acting on the ground state
$\left|n_xn_y\right\rangle=\left|00\right\rangle$, where $a_x =
\frac1{\sqrt2}(x+ip_x)$ and $a_y = \frac1{\sqrt2}(y+ip_y)$ are the
annihilation operators for each independent direction.

{\bf \quad b)} To obtain eigenstates of definite $L_z$, define annihilation
operators for right and left circular quanta $a_r = \frac1{\sqrt2}(a_x-ia_y)$
and $a_\ell = \frac1{\sqrt2}(a_x+ia_y)$.  Show that the only nonzero
commutators between $a_r,a_r\ad,a_\ell,a_\ell\ad$ are
$[a_r,a_r\ad]=[a_\ell,a_\ell\ad]=1$.  Show that $\mathcal{H}_{xy} = N_x + N_y
+ 1 = N_r + N_\ell +1$ and $L_z = i(a_xa_y\ad - a_x\ad a_y) = N_r - N_\ell$,
where $N_i=a_i\ad a_i$ as usual for $i=x,y,r,\ell$.  Show also that
$[\mathcal{H}_{xy},a_i\ad]=a_i\ad$, so that $a_x,a_y$ and $a_r,a_\ell$ act as
ladder operators for two independent sets of quanta
$\left|n_xn_y\right\rangle$ and $\left|n_rn_\ell\right\rangle$.  Thus $E_{nm}
= n + 1$, and $L_z = m$, where $n = n_r + n_\ell$ and $m = n_r - n_\ell$.
Show that the allowed values of $n$ are $2k+|m|$ where $k=0,1,2\ldots$ and
$m=0,\pm1,\pm2 \ldots$, and plot the energy levels $E_{nm}$ versus $m$.  Note
the checker pattern with every other value of $n$ missing.

{\bf \quad c)} Show that the Laplacian in cylindrical coordinates is $\nabla^2
= \frac1{\sqrt\rho}\frac{\partial^2}{\partial\rho^2}{\scriptstyle\sqrt{\rho}}
- \frac{(m-\frac12)(m+\frac12)}{\rho^2}$ using the eigenvalue $L_z
\Phi_m(\phi) = m \Phi_m(\phi)$, where $\Phi_m(\phi) = e^{i m\phi}$.  Factor
out the asymptotic dependence of $\psi$ as $\rho\to0$ and $\infty$ by making
the substitution $\psi(\rho,\phi) = \rho^{|m|} e^{-\rho^2/2} F(\rho)
\Phi_m(\phi)$ and use the energy eigenvalues $E_n = n+{2\over2}$ to put the
Schr\"odinger equation in the form
\begin{equation} {d^2F \over d\rho^2} - \left(2\rho - {2|m| + 1 \over\rho}
  \right) {dF\over d\rho} + 2\left( n-|m| \right)F = 0.
\end{equation}
Change variables to $f(u=\rho^2)=F(\rho)$ to obtain Laguerre's associated
differential equation,
\begin{equation}
  u {d^2f\over du^2} + \left( |m| + 1 - u \right) {df \over du} + k f = 0,
\end{equation} 
with the solution $f(u) = L^{|m|}_{k} (u)$, where $k = \frac12
(n-|m|)=0,1,2\ldots$.  Write out the lowest three energy levels and show that
the ground state is equal to that in part a).  Write the next two degenerate
levels as linear combinations of $\psi_{10}$ and $\psi_{01}$ from part a).

\newpage

{\bf \quad d) [bonus]} Show that the three-dimensional Laplacian has a similar
form after factoring out angular momentum: $\nabla^2 = \frac1{r}
\frac{\partial^2}{\partial r^2} r - \frac{(l)(l+1)}{r ^2}$, and by analogy
with part c) solve for the normalized energy eigenfunctions
\begin{equation}
  \psi_{nlm} = (-1)^{k} \sqrt{ 2 k ! \over \Gamma  
    (n\!-\!k\!+\!{3\over2})} r^{l} e^{-\frac12 r^2} L^{l+\frac12}_{k}
  (r^2) Y_{lm}(\theta,\phi)
\end{equation}
and eigenvalues $E_{nlm}= n+\frac32$, where $n=2k+l$ with $k=0,1,2\ldots$.
Determine the degeneracy of each energy level and make a correspondence of
states with linear combinations of the cartesian solution $E_{n_x n_y
  n_z}=n_x+n_y+n_z+\frac32$.

{\bf \quad e) [bonus]} Show that the one-dimensional Laplacian can also be put
into the same form, $\nabla^2 = {\partial^2 \over \partial \xi^2} - {(p-1)p
  \over \xi^2}$, using the `radial' coordinate $\xi=|x|$ and discrete `angle'
$\sigma=\mbox{sgn}(x)=x/|x|$, and factoring out the {\it parity harmonics}
$\Pi_p(\sigma)=e^{i \pi p \sigma}=(-1)^{\sigma p}$.  (Note the second term,
the centrifugal barrier, is always zero!)  Show that $p=0,1$ are the
eigenvalues for even, and odd functions, respectively.  By analogy with parts
c) and d), show that $E_{np}=n+\frac12$, where $n=2k + p$, and write the 1-d
eigenfunctions in terms of Laguerre polynomials.  Equate these with the normal
Hermite solutions to obtain the two relations [Abramowitz \& Stegun,
Eqs. 22.5.40-41].
\begin{eqnarray}
  H_{2m}(x) &=& (-1)^m 2^{2m} m! L_m^{-\frac12}(x^2) \\ 
  H_{2m+1}(x) &=& (-1)^m 2^{2m+1} m! x L_m^{\frac12}(x^2) .
\end{eqnarray}

{\bf \quad f) [bonus]} Show that the same dimensional symmetry of solutions
occurs for the free particle wave equation:
\begin{eqnarray}
  \Psi(\xi,\sigma) & = & \sqrt{\xi} J_{p-\frac12}\left(k \xi\right)
  \Pi_p(\sigma) \\ \Psi(\rho,\phi) & =
  & J_m(k \rho) \Phi_m(\phi) \\ \Psi(r,\theta,\phi) & = &
  {1\over\sqrt r} J_{l+\frac12} (k r) Y_{lm}(\theta,\phi)
\end{eqnarray}
The order of the Bessel function is raised by a half as you go to a
higher dimension.  The one- and three-dimensional solutions are
usually written using circular functions $e^{ikx}$ and spherical
Bessel functions $j_l(k r)$.

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