Time-dependent perturbation theory

Expanding a state in unperturbed energy eigenstates with their free phases converts the time-dependent Schrodinger equation into coupled equations for slowly varying amplitudes. To first order, the perturbation on the right-hand side acts on the initial unperturbed state.

In the long-wavelength approximation, a particle of charge $q$ in a spatially uniform electric field has perturbationIts transition amplitudes are controlled by matrix elements of the position operator.

Because the position operator has odd parity, an electric-dipole matrix element vanishes between states of the same parity. For orbital-angular-momentum eigenstates, the full rule is $\Delta \ell =\pm 1$ and $\Delta m=0,\pm 1$, with the field polarization selecting the allowed change in $m$.

For a harmonic perturbation and continuum detuning $\Omega (k)$, first-order amplitudes containThe distributional limitselects energy-conserving continuum states and produces a transition probability linear in time.