Linear adiabatic invariants and coherent states

Abstract

The Born‐Fock adiabatic theorem is extended to all orders for some quadratic quantum systems with finitely or infinitely degenerate energy spectra. A prescription is given for obtaining adiabatic invariants to any order. For any quadratic quantum system with N degrees of freedom there are 2N linear adiabatic invariant series, which correspond to the 2N exact invariants. The exact quantum mechanical solution for any nonstationary quadratic quantum system is also constructed by making use of the coherent‐state representation: The Green's function, coherent states, transition amplitudes and probabilities and their generating functions are obtained explicitly. Two particular systems, the N‐dimensional time‐dependent general oscillator and charged particle motion in a varying and uniform electromagnetic field, are considered in greater detail as examples.