By performing unitary transformations, the exact solution of the time-dependent singular oscillator is obtained. The invariant operator and the auxiliary equation are rigorously established. The non-adiabatic Berry's phase is calculated.
A generalisation of the Berry phase, which reduces to the usual Berry phase in the adiabatic and cyclic case is derived in a gauge-invariant way. This generalised Berry phase, unlike the Berry phase, is invariant under unitary transformations. A time-dependent generalised harmonic oscillator with a cross term in the Hamiltonian is taken as an example. The Hamiltonian of this system is not in general the energy. An energy, the time derivative of which is the power, is obtained from the equation of motion. When the system is quantised, the generalised Berry phase is zero, and remains invariant under unitary transformations. On the other hand, if the energy is chosen to be the Hamiltonian, a non-zero generalised Berry phase is obtained, but the power is given incorrectly. In this case the total phase, which includes the dynamical phase and the generalised Berry phase, is equal in the adiabatic limit to the correct total phase.
An approach to study the time-dependent quantum oscillator with the Schr\"odinger operator S(\ensuremath{\Omega}(t))\ensuremath{\equiv}1/2[p2+${\mathrm{\ensuremath{\Omega}}}^{2}$(t)x2]-i(\ensuremath{\partial}/\ensuremath{\partial}t) is presented. A family of unitary operators {W1z(t)} is found such that S(\ensuremath{\Omega}(t))=W1z(t)S(\ensuremath{\omega})${\mathbf{W}}_{1\mathit{z}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$(t), where z=z(t) is a certain function and \ensuremath{\omega} is a positive number. Wave vectors, the boson creation and destruction operators, the evolution operator, and invariants for the time-dependent oscillator are obtained by means of {W1z(t)}. The approach is applied to calculate transition probabilities, Berry phases, and to study coherent states of the time-dependent oscillator.
We study the time evolution of quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators. The invariant Hermitian operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. With the help of the invariant Hermitian operator we obtain not only the exact solutions of the Schr\"odinger equation but also the time-evolution operator. The adiabatic and nonadiabatic Berry phases are calculated with the exact solutions. \textcopyright{} 1996 The American Physical Society.