Time-Dependent Perturbation Theory

Abstract

In previous chapters we have learned how to solve either exactly or approximately the time-independent Schrödinger equation. This is however not enough in general, for there are many instances where one is required to understand the time evolution of a quantum system. If the Hamiltonian in the Schrödinger picture is H(t), the equation we want to integrate is

$$ i\hbar \frac{{d\left| {\Psi (t)} \right\rangle }}{{dt}} = H(t)\left| {\Psi (t)} \right\rangle $$

(11.1)

assuming |Ψ(t₀)〉 is known. This is equivalent to the computation of the operator U(t, t₀) introduced in Chap. 2. In most cases the mathematical structure of H(t) is so complicated that no practical methods of computation are available.