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\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#7, Rev. A, due Monday, 2017-10-23 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed] Ch.~2
\#15, %p57, SHO tunnel
\#16, %p57, recursion coeff
\#17, %p57, Hermite formulas
\#42. %p86, half SHO

{\bf 1.} {\bf Parity harmonics} --- stationary states of the harmonic
oscillator alternate between even and odd functions of $x$ with increasing
energy.  We will explicitly solve the even or odd states by restricting the
domain to $x>0$ and applying the boundary condition $\psi'(0)=0$ or
$\psi(0)=0$, respectively.  Doing so leads to solutions in terms of the
associated Laguerre polynomials instead of the standard Hermite polynomials.

{\bf \quad a)} Show that the wave function $\psi(\xi) = \xi^p
e^{-\xi^2/2}L(\xi^2)$ satisfies the boundary condition for either even or odd
functions of $\xi$ with $p=0$ or 1, respectively, for any function $L(u)$.
Note that the exponential captures the asymptotic dependence of $\psi(\xi)$.

{\bf \quad b)} Substitute $\psi(\xi)$ into Griffiths Eq.~2.72 [to save effort,
substitute $h=\xi^p L(\xi^2)$ in Eq.~2.78], and change variables from $\xi$ to
$u=\xi^2$ to obtain the associated Laguerre differential equation
$uL'' + (p-\frac12 + 1 - u)L' + kL=0$
[\href{https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials}
{wikipedia.org/Laguerre\_polynomials}].  {\it Hint: use the chain rule to
  calculate $h'(\xi)$.}  What is the relation between $E$ and $k$?

{\bf \quad c)} Solve the associated Laguerre equation using the Frobenius
method: substitute the Taylor series $L(u)=\sum_{j=0}^\infty a_j u^j$, and
solve the recurrence relation for $a_j$.

{\bf \quad d)} Calculate the energy levels for both $p=0$ and $p=1$ by setting
$k=0,1,2, \ldots$ to truncate the series so that $\psi(\pm\infty)\to 0$.
Compare with the levels $E_n=\hbar\omega(n+\frac12)$, where $n=2k+p$.

{\bf \quad e)} Use the recursion relation to calculate the first three
solutions $L_k(u)$, where $k=0,1,2$, for both $p=0$ and $p=1$, using the
normalization $L_k(0)={k+\alpha\choose
  k}=\frac{(1+\alpha)(2+\alpha)\cdots(k+\alpha)}{~~1~~\cdot ~~2~~\cdot \cdots
  ~~k~~}$, where $\alpha=p-\frac12$.  Compare your solutions with the
associated Laguerre polynomials $L_k^{p-\frac12}(u)$.  Note that the first
three are $L_0^{(\alpha)}(u)=1$,\quad $L_1^{(\alpha)}(u)=(1+\alpha-u)$, \quad
and $L_2^{(\alpha)}(u)=\frac{(1+\alpha)(2+\alpha)}{~~1~~\cdot~~2~~} -
(2+\alpha)u + \frac12 u^2$.

{\bf \quad f) [bonus]} Normalize the wave functions $\psi_k^p(\xi)$.  Compare
these solutions to the standard wavefunctions, Griffiths Eq.~2.85, to obtain
the following relations between Hermite and associated Laguerre polynomials,
Eqs. 22.5.40 and 22.5.41 of
[\href{http://people.math.sfu.ca/~cbm/aands/page_779.htm}{Abramowitz and
  Stegun, p. 779}]
\begin{eqnarray}
  H_{2k}(x) &=& (-1)^k 2^{2k} k! L_k^{-\frac12}(x^2) \\ H_{2k+1}(x) &=&
  (-1)^k 2^{2k+1} k! x L_k^{\frac12}(x^2) .
\end{eqnarray}
We will encounter associated Laguerre polynomials in the solution of other
potentials such as the 2d and 3d harmonic oscillators and the hydrogen atom.

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