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Journal of Physics A Mathematical and General

Non-adiabatic parametric excitation of oscillator-type systems

R. G. Agaeva

Published May 1, 1980• 17 citations
Publisher

Abstract

A harmonic quantum oscillator and the motion of charged particles in a magnetic field have been considered, the dependence on time of the magnetic field and the oscillator frequency being such that they permit us to keep an exact record of non-adiabaticity. For these systems, exact solutions, motion integrals, coherent states and Green functions have been constructed and proper amplitudes and transition probabilities calculated. One of the possible practical applications of the results obtained has been pointed out, namely their use for the interpretation of experiments with electronic vibrational transitions in molecules.

DOI
10.1088/0305-4470/13/5/026

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A harmonic quantum oscillator and the motion of charged particles in a magnetic field have been considered, the dependence on time of the magnetic field and the oscillator frequency being such that they permit us to keep an exact record of non-adiabaticity. For these systems, exact solutions, motion integrals, coherent states and Green functions have been constructed and proper amplitudes and transition probabilities calculated. One of the possible practical applications of the results obtained has been pointed out, namely their use for the interpretation of experiments with electronic vibrational transitions in molecules.
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This paper deals with non-adiabatic processes (i.e. processes excluded by the adiabatic theorem) from the geometrical (group-theoretical) point of view. An approximate formula for the probabilities of the non-adiabatic transitions is derived in the adiabatic regime for the case when the parameter-dependent Hamiltonian represents a smooth curve in the Lie algebra and the quantal dynamics is determined by the corresponding Lie group evolution operator. We treat the spin precession in a time-dependent magnetic field and the over-barrier reflection problem in a uniform way using the first-order dynamical equations on SU (2) and SU (1.1) group manifolds, respectively. A comparison with analytic solutions for simple solvable models is provided.
New time-dependent invariants for the NNN-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 Schr\"odinger equation is found. The symmetry of nonstationary Hamiltonians is discussed, and the noncompact group U(N,1)U(N, 1)U(N,1) is shown to be the group of dynamical symmetry for the time-dependent NNN-dimensional oscillator.
The theory of explicitly time-dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrödinger equation. As a specific well-posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time-dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time-dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time-dependent uniform charge distribution. A class of explicitly time-dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrödinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.
The present review is devoted to the study of certain aspects of anharmonic, time-dependent and damped oscillator(s) system using different theoretical techniques. A theoretical understanding of these systems is important for application in many problems in physics, mechanics and other fields. We discuss in detail the difficulties in the theoretical analysis of the problem. In particular we discuss here the regular, well-behaved perturbative solution, the large quantum number behaviour of anharmonic oscillator(s) using the technique of coherent states, exact solution of quantum anharmonic oscillators, the electromagnetic radiation emitted by a charged particle executing damped anharmonic oscillator motion using Krylov-Bogoliubov approximation method, use of invariants to obtain solution and coherent states of time-dependent oscillator(s), the derivation of perturbative frequencies of a damped coupled anharmonic oscillators system using suitable canonical transformation in the framework of Hamilton-Jacobi formalism and the quantisation and construction of coherent states of a damped oscillator using time-dependent operators.