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2.5: Notes

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    • The results in 2.2 2.3 and 2.4 are worth memorising: physicists use them without proof.
    • Energy shifts are real numbers, but matrix elements may be complex.
    • If the perturbation operator commutes with the Hamiltonian, “Off-diagonal” matrix elements \((V_{ij}, i \neq j)\) are zero. Such perturbations change the energy, but not the wavefunction.
    • If the perturbation is turned on and off again, the off-diagonal matrix elements determine whether the quantum state is changed.
    • To help with notation, we have derived results for perturbation to a state labelled by 0. This is not necessarily the ground state - the above derivation is general.
    • For the first-order changes to the eigenfunction to be small we must have:

    \[ \langle n_k|\hat{V} |n_0 \rangle \equiv V_{k0} \ll |(E_0 − E_k)| \quad \text{ for all } k \neq n \nonumber\]

    • Similarly, we require that the level shift be small compared to the level spacing in the unperturbed system:

    \[|\Delta E_0| \ll \text{ min } |(E_0 − E_k| \nonumber\]

    • These conditions may break down if there are degeneracies in the unperturbed system. However, we need only assume that the particular energy level whose shift we are calculating is non-degenerate for the preceding analysis to be correct.
    • The first order corrected wavefunctions are not fully normalised.
    • The second order term always lowers the energy of the ground state.

    This page titled 2.5: 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.

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