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Journal of Mathematical Physics

Unitary transformation approach to the exact solution for a class of time-dependent nonlinear Hamiltonian systems

Mustapha Maamache

Published Jan 1, 1998• 28 citations
Publisher

Abstract

By performing unitary transformations on the time-dependent Schrödinger equation, the exact solution for a class of nonlinear Hamiltonian systems is obtained and it is shown that these time-dependent problems are related to the associated time- independent problems. In addition, the evolution operator is derived. Nonadiabatic Berry’s phase is calculated on the basis of the exact solution. The theory is applied to some illustrative examples.

DOI
10.1063/1.532341

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By performing unitary transformations on the time-dependent Schrödinger equation, the exact solution for a class of nonlinear Hamiltonian systems is obtained and it is shown that these time-dependent problems are related to the associated time- independent problems. In addition, the evolution operator is derived. Nonadiabatic Berry’s phase is calculated on the basis of the exact solution. The theory is applied to some illustrative examples.
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Solutions of the Schrödinger equation with an exact time dependence are derived as eigenfunctions of dynamical invariants which are constructed from time-independent operators using time-dependent unitary transformations. Exact solutions and a closed form expression for the corresponding time evolution operator are found for a wide range of time-dependent Hamiltonians in d dimensions, including non-Hermitean -symmetric Hamiltonians. Hamiltonians are constructed using time-dependent unitary spatial transformations comprising dilatations, translations and rotations and solutions are found in several forms: as eigenfunctions of a quadratic invariant, as coherent state eigenfunctions of boson operators, as plane wave solutions from which the general solution is obtained as an integral transform by means of the Fourier transform, and as distributional solutions for which the initial wavefunction is the Dirac δ-function. For the isotropic harmonic oscillator in d dimensions radial solutions are found which extend known results for d = 1, including Barut–Girardello and Perelomov coherent states (i.e., vector coherent states), which are shown to be related to eigenfunctions of the quadratic invariant by the ζ-transformation. This transformation, which leaves the Ermakov equation invariant, implements SU(1, 1) transformations on linear dynamical invariants. coherent states are derived also for the time-dependent linear potential. Exact solutions are found for Hamiltonians with electromagnetic interactions in which the time-dependent magnetic and electric fields are not necessarily spatially uniform. As an example, it is shown how to find exact solutions of the time-dependent Schrödinger equation for the Dirac magnetic monopole in the presence of time-dependent magnetic and electric fields of a specified form.
We study the time evolution of a class of exactly solvable time-dependent quantum systems with a time-dependent Hamiltonian given by a linear combination of SU(1,1) and SU(2) generators with the help of the invariant Hermitian operator. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the invariant operator. The adiabatic and non-adiabatic Berry phases are calculated with the exact solutions. Moreover, we derive an explicit time-evolution operator which is used to investigate the time-dependent two-photon squeezing states and SU(2) squeezing states. The squeezing properties of the time-dependent SU(1,1) coherent states are also discussed.