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Chapter 07

Approximation methods

Almost nothing real is solvable. Perturbation theory, variation, WKB, golden rule, Berry phase.

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Chapter 06
  1. 7.1Time-independent non-degenerate perturbation theorySplit an unsolvable Hamiltonian into "solvable + small", then let the solvable part approximate the true answer order by order.
  2. 7.2Time-independent degenerate perturbation theoryA zero denominator is not a disaster but a hint: diagonalise the perturbation inside the degenerate subspace first, and let the perturbation pick out the "good" zeroth-order states itself.
  3. 7.3The variational methodAn approximation that needs no small parameter: guess any wavefunction you like — the energy it gives can never break below the true ground state, so the better the guess, the tighter the squeeze.
  4. 7.4The WKB approximationWhen the potential varies slowly enough, the wavefunction is a local plane wave at every point — yielding a quantisation condition and the exponential formula for tunnelling probabilities.
  5. 7.5Time-dependent perturbation theoryWhen the Hamiltonian oscillates in time, stationary states are no longer eternal — first-order perturbation theory yields transition probabilities, and the resonance condition explains why atoms absorb only particular colours of light.
  6. 7.6Fermi's golden ruleWhen the final states form a continuum, the oscillating transition probability becomes a constant transition rate — one formula that rules every calculation of emission, absorption and scattering.
  7. 7.7The adiabatic theorem and the Berry phaseChange things slowly enough and the system tracks the instantaneous eigenstate the whole way — but after one closed loop, the wavefunction returns wearing a phase set purely by the geometry of the path.
  8. 7.8The sudden approximationWhen the change is too fast for the wavefunction to react, the old wavefunction is frozen untouched into the new Hamiltonian — and the fate of tritium's electron after nuclear decay comes down to one overlap integral.
Start with 7.1 Time-independent non-degenerate perturbation theory →