Part VI: Approximation Methods
Most quantum systems cannot be solved exactly. We develop systematic approximation techniques: perturbation theory (time-independent and time-dependent), variational methods, and adiabatic approximations.
60+ pages | 7 chapters
1. Time-Independent Perturbation (10 pages)
Non-degenerate perturbation theory, energy corrections
$\hat{H} = \hat{H}_0 + \lambda\hat{V}$, $E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots$
2. Degenerate Perturbation Theory (8 pages)
Degenerate states, good quantum numbers, matrix diagonalization
Secular equation: $\det(V_{ij} - E^{(1)}\delta_{ij}) = 0$, good basis selection
3. Fine Structure of Hydrogen (9 pages)
Relativistic corrections, spin-orbit coupling, Lamb shift
$\Delta E_{fs} \propto \alpha^2 E_n$, $\hat{H}_{SO} = \frac{1}{2m^2c^2}\frac{1}{r}\frac{dV}{dr}\hat{\vec{L}}\cdot\hat{\vec{S}}$
4. Time-Dependent Perturbation (12 pages)
Interaction picture, Dyson series, transition probabilities
$c_n(t) = -\frac{i}{\hbar}\int_0^t V_{ni}(t')e^{i\omega_{ni}t'}dt'$, $P_{i\to n} = |c_n(t)|^2$
5. Fermi's Golden Rule (7 pages)
Transition rates, density of states, selection rules
$\Gamma_{i\to f} = \frac{2\pi}{\hbar}|V_{fi}|^2\rho(E_f)$, continuum approximation
6. Variational Method (7 pages)
Ground state energy bounds, trial wavefunctions
$E_{gs} \leq \langle\psi_{trial}|\hat{H}|\psi_{trial}\rangle$, Rayleigh-Ritz principle
7. Adiabatic Approximation (7 pages)
Slowly varying Hamiltonians, Berry phase
$|\psi(t)\rangle = e^{i\gamma_n(t)}e^{-\frac{i}{\hbar}\int_0^t E_n(t')dt'}|n(t)\rangle$, geometric phase $\gamma_n$