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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 521 \\
    Homework \#12, Rev. A, due Monday, 2017-01-23 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
% Ch.~2 %\#17, %Hermite polynomials
Ch.~3 %\#35, %p127 Coherent states of SHO
      % \#6,   %p100 d^2/d phi^2 hermitian
      \#7,   %p102 linear combinations of eigenfunctions
      \#10,  %p106 groundstate infsqwell momentum eigenfunction?
      \#13,  %p112 commutators
      \#14,  %p113 x,H uncertainty principle
      \#15;~  %p113 commuting operators common eigenfunctions
      % \#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 $\hbar=m=\omega=1$ so that $\rho$
plays the role of 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.  Write the wave functions of the lowest
six levels in terms of both Hermite polynomials $H_{n_x}(x)H_{n_y}(y)$ and
creation operators $a_{x,y}\ad$ acting on the ground state
$\left|n_xn_y\right\rangle=\left|00\right\rangle$.  The two annihilation
operators $a_x = \frac1{\sqrt2}(x+ip_x)$ and $a_y = \frac1{\sqrt2}(y+ip_y)$
act in independent directions.

{\bf \quad b)} The eigenfunctions of part a) do not have definite angular
momentum.  To obtain eigenstates of definite $L_z$ using the operator method,
define annihilation operators for right and left circular quanta
$a_r = \frac1{\sqrt2}(a_x-ia_y)$ and $a_\ell = \frac1{\sqrt2}(a_x+ia_y)$,
respectively.  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 $a_r,a_\ell$ both act as ladder operators for two
independent sets of quanta $\left|n_xn_y\right\rangle$ and
$\left|n_rn_\ell\right\rangle$.  From this we infer that $E_{nm} = n + 1$, and
$L_z = m$, where $n = n_r + n_\ell$ and $m = n_r - n_\ell$.  Plot the energy
levels $E_{nm}$ versus $m$ and show that the allowed values of $n$ are
$2k+|m|$ where $k=0,1,2\ldots$ and $m=0,\pm1,\pm2 \ldots$.  Note the
checkerboard pattern with every other value of $n$ missing.

{\bf \quad c)} Use the coordinate representation of $a\ad_{r,\ell}$ acting on
the ground state $\left|n_rn_\ell\right\rangle = \left|00\right\rangle$ to
obtain wavefunctions for the six $n\leq 2$ states and compare your results
with H11 part e).  Expand the $a\ad_{r,l}$ in terms of $a\ad_{x,y}$ and
simplify to show that the $\left|n_{l},n_{r}\right\rangle$ states of part b)
are linear combinations of the $\left|n_{x},n_{y}\right\rangle$ states of part
a).

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