Extensions of the Adiabatic Theorem in Quantum Mechanics

Abstract

In a quantum-mechanical problem with a slowly varying Hamiltonian, such as appears in a semiclassical treatment of inelastic atomic collisions, it is common to restrict attention to a small number of states among which transitions are anticipated. This procedure is justified by an extension of the conventional adiabatic theorem to nearly degenerate states. The correct mixing within the restricted basis is determined by a time-dependent variational principle. The contribution of states outside the basis vanishes as the time scale for variation of the Hamiltonian becomes large. A further generalization deals with pseudoeigenfunctions, such as appear in the theory of the Stark effect.