% begin preamble \documentclass[11pt,titlepage]{article} \usepackage[margin=1in,noheadfoot]{geometry} \usepackage[parfill]{parskip} \usepackage{graphicx} \usepackage[colorlinks,urlcolor=blue]{hyperref} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{wrapfig} \newcommand{\e}[1]{\times 10^{#1}} \newcommand{\q}{\mathrm} \newcommand{\where}[1][where]{\qquad\mbox{#1}\qquad} \newcommand{\bs}[1]{\boldsymbol{#1}} \newcommand{\bh}[1]{\hat{\boldsymbol{#1}}} \newcommand{\bv}[1]{\vec{\boldsymbol{#1}}} \newcommand{\bt}[1]{\tilde{\boldsymbol{#1}}} \newcommand{\ad}{^{\dagger}} \newcommand{\flow}[1]{\mathcal{E}_{#1}} \newcommand{\flux}[1]{\Phi_{#1}} \newcommand{\dslash}{d \hspace{-0.5ex}\rule[1.2ex]{0.6ex}{.1ex}} \def\rcurs{{\mbox{$\resizebox{.16in}{.08in}{\includegraphics{ScriptR}}$}\!\!}} \def\brcurs{{\mbox{$\resizebox{.16in}{.08in}{\includegraphics{BoldR}}$}\!\!}} \def\hrcurs{{\mbox{$\hat \brcurs$}\!}} %end preamble \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