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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#10, Rev. A, due Friday, 2015-11-11 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed] App.~A
\#19, %p454 Jordan matrix
\#22, %p454 commuting matrices
\#25. %p457 diagonalize T

{\bf 1.}  The 2nd order linear \mbox{\bf Sturm-Liouville} differential
operator
\begin{equation}
  L[y(x)] \equiv \frac{1}{w(x)}\left[\frac{d}{dx} p(x) \frac{d}{dx}
    - q(x)\right] y(x)
\end{equation}
is self-adjoint, $L\ad=L$, with respect to the inner product
\begin{equation}
  \langle y_1|y_2\rangle \equiv \int_a^b w(x) dx \; y_1(x)^*\, y_2(x)
\end{equation}
if we impose the boundary conditions $y(a)w(a)=y(b)w(b)=0$.  Thus $L$ has real
eigenvalues $\lambda_i$ and a complete set of orthogonal eigenfunctions
$u_i(x)=\langle x|u_i\rangle$.  In particular, $L|u_i\rangle=\lambda_i
|u_i\rangle$, with $\langle u_i|u_j\rangle=\delta_{ij}$ [orthonormality], and
$\sum_i|u_i\rangle\langle u_i|=I$ [completeness].  This mean that any smooth
function $|f\rangle$ can be expanded in the basis $|u_i\rangle$ as $|f\rangle
= \sum_i |u_i\rangle f_i$ or $f(x) = \sum_i u_i(x) f_i$, where $f_i=\langle
u_i|f\rangle = \int_a^b w(x)dx\; u_i(x)^* f(x)$.

{\bf \quad a)} Show that $L$ is self-adjoint or Hermitian.  {\it Hint:} use
the definition $\langle f|H\ad g\rangle \equiv \langle Hf|g\rangle$ to show
that the derivative operator $\frac{d}{dx}$ is antiHermitian and apply it to
the composition \mbox{of operators in $L$.}

{\bf \quad b)} Given eigenfunctions $L|u_i\rangle=\lambda_i |u_i\rangle$, show
that $\lambda_i\in \mathbb{R}$ and that $\langle u_i|u_j\rangle = 0$ if
$\lambda_i\neq\lambda_j$.  Note that it is much harder to prove completeness.
Operate $L$ on the expansion of $|f\rangle$ in the basis $|u_i\rangle$ to show
that its {\it spectral decomposition} is $L=\sum_i \lambda_i |u_i\rangle
\langle u_i|$.  What is the decomposition of the identity operator
$I|f\rangle=|f\rangle$ in the same orthogonal basis $|u_i\rangle$?

{\bf 2.~a)} The {\bf Legendre polynomials} $P_n(\cos\theta)$, used for
spherically symmetric potentials, are eigenfunctions of the operator
$L=\frac{d^2}{d\theta^2}+\cot\theta\frac{d}{d\theta}$.  Show that this is a
Sturm-Liouville system on the domain $0<\theta<\pi$, with
$w(\theta)=\sin\theta$, $p(\theta)=\sin\theta$, and $q(\theta)=0$.  Change
variables to $x=\cos\theta$ and calculate the new functions $w(x)$, $p(x)$,
$q(x)$ and domain $a<x<b$.  Note the sign change!

{\bf \quad b)} Show that $\langle x^m|x^n\rangle = \frac{2}{m+n+1}$ if $m+n$
is even and, 0 if $m+n$ is odd.  Apply the Gram-Schmidt procedure to the basis
functions $1$, $x$, and $x^2$ to obtain the first three Legendre polynomials
$P_\ell(x)$, and find their eigenvalues $\lambda_\ell$.

{\bf 3.} Compile a chart of $w,p,q,\lambda_i$ for each of the following {\bf
  orthogonal functions} $\phi_i(x)$:

\quad\begin{tabular}{llll@{\quad}r@{}c@{}l}
i) & Cylindrical harmonics & $e^{im\phi}$ & on & $0<$&$\phi$&$<2\pi$.\\
ii) & Associated Legendre functions & $P_l^{|m|}(x)$ & on & $-1<$&$x$&$<1$.\\
iii) & Fourier series & $\sin(k_nx)$ & on & $0<$&$x$&$<b$.\\
iv) & Bessel functions & $J_m(k_nx)$ & on & $0<$&$x$&$<b$.\\
v) & Spherical Bessel functions & $j_l(k_nx)$ & on & $0<$&$x$&$<b$.\\
vi) & Hermite polynomials & $H_n(x)$ & on & $-\infty<$&$x$&$<\infty$.\\
vii) & Associated Laguere polynomials & $L^{(\alpha)}_n(x)$ & on & $0<$&$x$&$<\infty$.
\end{tabular}

{\bf 4. Simultaneously diagonalize} the matrices %
$A=\scriptsize{\pmatrix{ %
  \phantom{-}9 & -2 & -6 \cr %
  -2 & \phantom{-}9 & -6 \cr %
  -6 & -6 & -7 }}$ %
and $B=\scriptsize{\pmatrix{ %
  54 & \phantom{-}10 & -3 \cr %
  10 & -45 & 30 \cr %
  -3 & \phantom{-}30 & 46 }}$.\\ %
Are the eigenvectors orthogonal? {\it Hint:}
\href{https://www.gnu.org/software/octave}{Octave} or
\href{http://www.wolfram.com/solutions/education/students}{Mathematica} is
your friend!

\end{document}
