New time-dependent invariants for the $N$-dimensional nonstationary harmonic oscillator and for a charged particle in a varying axially symmetric classical electromagnetic field are found. For these quantum systems, coherent states are introduced, and the Green's functions are obtained in closed form. For a special type of electromagnetic field which is constant in the remote past and future, the transition amplitudes between both arbitrary coherent states and energy eigenstates are calculated and expressed in terms of classical polynomials. The adiabatic approximation and adiabatic invariants are discussed. In the special case of a particle with time-dependent mass, the solution of the Schrödinger equation is found. The symmetry of nonstationary Hamiltonians is discussed, and the noncompact group $U(N, 1)$ is shown to be the group of dynamical symmetry for the time-dependent $N$-dimensional oscillator.

• Received 26 May 1970

DOI:https://doi.org/10.1103/PhysRevD.2.1371