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Journal of Physics A Mathematical and General

Time-dependent harmonic oscillators in an electromagnetic field

S -W Qian, Yayun Hu, J -S Wang

Published Jul 11, 1987• 1 citation
Publisher

Abstract

The authors consider the most general form of time-dependent charged harmonic oscillators in an electromagnetic field. The quantum mechanical solution is developed and the Green function is derived. Furthermore, they use gauge-invariant formulation of quantum mechanics to obtain the transition probabilities in the electric dipole approximation.

DOI
10.1088/0305-4470/20/10/027

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The authors consider the most general form of time-dependent charged harmonic oscillators in an electromagnetic field. The quantum mechanical solution is developed and the Green function is derived. Furthermore, they use gauge-invariant formulation of quantum mechanics to obtain the transition probabilities in the electric dipole approximation.
The gauge-invariant formulation of quantum mechanics is applied to the charged harmonic oscillator in an electromagnetic field in the electric dipole approximation to obtain the probability that the oscillator is in a particular state at time t. This probability is compared with the corresponding 'probability' calculated from the conventional approach to the interaction of radiation with matter using the interaction A.p. The probabilities do not agree with each other. From the principle of gauge invariance it is concluded that the probability calculated from the gauge-invariant formulation is correct in this case, and that the conventional approach in general is incorrect.
A manifestly gauge-invariant formulation of quantum mechanics is applied to a charged isotropic harmonic oscillator in a time-varying magnetic field in the magnetic dipole approximation. The energy operator for the problem is the sum of the kinetic and potential energies. The kinetic energy operator is the square of the gauge-invariant kinetic momentum operator divided by twice the mass. The energy eigenvalues and state probabilities are calculated and are shown to be the same in all gauges. In this problem there is no gauge in which the energy operator reduces to the unperturbed Hamiltonian, as there is in the electric dipole approximation. Consequently, eigenvalues of the unperturbed Hamiltonian and corresponding (gauge-dependent) state 'probabilities' are different from the gauge-invariant quantities in all gauges.
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.