Units#

Review & Summary#

Perturbation series#

\(E_n = E_n^{(0)} + \lambda E_n^{(1)} + \lambda^2 E_n^{(2)} + \cdots\) for \(H = H_0 + \lambda V\).

First-order energy#

\(E_n^{(1)} = \langle n^{(0)}|V|n^{(0)}\rangle\) (Hellmann-Feynman).

Second-order energy#

\(E_n^{(2)} = \sum_{m\neq n} \frac{|\langle m^{(0)}|V|n^{(0)}\rangle|^2}{E_n^{(0)} - E_m^{(0)}}\). Always lowers the ground state energy.

Degenerate perturbation#

Diagonalize \(V\) within the degenerate subspace via the effective Hamiltonian.

Interaction picture#

\(V_I(t) = e^{iH_0 t/\hbar}V(t)e^{-iH_0 t/\hbar}\). Dyson series: \(U(t) = \mathcal{T}\exp\!\left(-\frac{i}{\hbar}\int_0^t V_I(t')dt'\right)\).

Fermi’s golden rule#

\(\Gamma_{i\to f} = \frac{2\pi}{\hbar}|\langle f|V|i\rangle|^2 \rho(E_f)\).

Adiabatic theorem#

Slow evolution keeps the system in the instantaneous eigenstate, acquiring a Berry phase.