Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators

A class of exact invariants for oscillator systems whose Hamiltonians areH=(1/2ε)p2+Ω2(t)q2is given in closed form in terms of a function ρ(t) which satisfies ε2d2ρ/dt2+Ω2(t)ρ−ρ−3=0. Each particular solution of the equation for ρ determines an invariant. The invariants are derived by applying an asymptotic theory due to Kruskal to the oscillator system in closed form. As a consequence, the results are more general than the asymptotic treatment, and are even applicable with complex Ω(t) and quantum systems. A generating function is given for a classical canonical transformation to a class of new canonical variables which are so chosen that the new momentum is any particular member of the class of invariants. The new coordinate is, of course, a cyclic variable. The meaning of the invariants is discussed, and the general solution for ρ(t) is given in terms of linearly independent solutions of the equations of motion for the classical oscillator. The general solution for ρ(t) is evaluated for some special cases. Finally, some aspects of the application of the invariants to quantum systems are discussed.