Time-Dependent Hamiltonians

Canonical treatment: Time-Dependent Hamiltonians maintains the Dyson-series derivation, examples, exercises, and references.

This bridge is intentionally limited to the nonautonomous evolution concepts needed by the core postulates.

Basic rule

The Schrödinger equation remains

$$i\hbar\frac{d}{dt}\lvert\psi(t)\rangle=H(t)\lvert\psi(t)\rangle.$$

Its propagator obeys

$$i\hbar\,\partial_t U(t,t_0)=H(t)U(t,t_0), \qquad U(t_0,t_0)=I.$$

If $[H(t),H(s)]=0$ for all relevant times, then

$$U(t,t_0)=\exp\!\left[-\frac{i}{\hbar}\int_{t_0}^{t}H(s)\,ds\right].$$

Otherwise operator order matters and the solution requires a time-ordered exponential. For piecewise-constant driving, multiply the interval propagators in chronological action order, with the earliest factor on the right.

Continue to the canonical page for Dyson series, pulses, driven two-level systems, energy balance, and Floquet orientation.