Skip to main content
Physics LibreTexts

3.2: Notes

  • Page ID
    28754
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    • The perturbed eigenstates of \(\hat{H}\) are linear combinations of degenerate eigenstates of \(\hat{H}_0\). This means that they too are eigenstates of \(\hat{H}_0\) from a different eigenbasis.
    • If \(\hat{H}_0\) is compatible with \(\hat{V}\), i.e. \([\hat{H}_0, \hat{V} ] = 0\), then there is no mixing with non-degenerate states and the analysis above is exact.
    • Notice how the mathematics mimics the quantum mechanics. Without the perturbation the eigenbasis of \(\hat{H}_0\) is not unique. When we try to determine its energy shift we find a matrix equation which can only be solved for specific values of \(\Delta E_j\). These \(\Delta E_j\) in turn correspond to specific choices for the coefficients \(c_{ji}\), i.e. particular linear combinations of the unperturbed states. Thus to solve the equations we are forced to collapse the wavefunction onto an eigenstate of \(\hat{V}\). \(V_{ki}\) is a Hermitian matrix, and consequently has real eigenvalues.

    This page titled 3.2: Notes is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.