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3.2: Notes

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    • 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.