Skip to main content
Physics LibreTexts

7.1: The Problem

  • Page ID
    6371
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    7.1 The harmonic Hamiltonian

    The Hamiltonian for lattice vibrations, in the harmonic approximation, is

    \[ \mathcal{H}=\frac{1}{2} \sum_{i=1}^{3 N} m_{i} \dot{x}_{i}^{2}+\frac{1}{2} \sum_{i=1}^{3 N} \sum_{j=1}^{3 N} x_{i} A_{i j} x_{j}.\]

    Notice that this Hamiltonian allows the possibility that atoms at different lattice sites might have different masses. Accept the fact that any real symmetric matrix S can be diagonalized through an orthogonal transformation, i.e. that for any such S there exists a matrix B whose inverse is its transpose and such that

    \[ \mathrm{BSB}^{-1}\]

    is diagonal. Show that the Hamiltonian can be cast into the form

    \[ \mathcal{H}=\frac{1}{2} \sum_{r=1}^{3 N}\left(\dot{q}_{r}^{2}+D_{r} q_{r}^{2}\right)\]

    by a linear change of variables. (Clue: As a first step, introduce the change of variable \(z_{i}=\sqrt{m_{i}} x_{i}\).)


    This page titled 7.1: The Problem is shared under a CC BY-SA license and was authored, remixed, and/or curated by Daniel F. Styer.

    • Was this article helpful?