Physics · Advanced university
Perturbation theory and approximation checks · revision notes
Write H=H0+λV and expand observables or states about a solvable reference problem. Small λ is useful only relative to relevant spectral scales.
Key ideas and conditions
Setup
Write H=H0+λV and expand observables or states about a solvable reference problem. Small λ is useful only relative to relevant spectral scales.
First order
For a normalized nondegenerate state, the first-order energy coefficient is ⟨n∣V∣n⟩. Include λ when calculating the first-order energy shift.
Degeneracy
Degenerate levels require diagonalizing the perturbation in the relevant subspace rather than applying a single arbitrary diagonal element.
Validation
Check dimensions, symmetry, limiting behaviour and the size of corrections. Near-degeneracy can spoil a naive small-coupling expansion.
Explain without looking
- How can symmetry make a first-order correction vanish without making all higher corrections vanish?
- State two checks on a claimed perturbative energy correction.
Further reading and review references
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