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3.2: Notes

Last updated

Oct 10, 2020
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- 3.1: Time-Independent Degenerate Perturbation Theory
- 3.3: Example of degenerate perturbation theory - Stark Effect in Hydrogen

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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.

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