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American Journal of Physics

Motion in a time-dependent, uniform electric field

Stavros Fallieros, James L. Friar

Published Nov 1, 1982• 5 citations
Publisher

Abstract

The quantal motion of a charged nonrelativistic particle in a uniform but time-varying electric field is discussed. It is pointed out that states of definite but variable momentum exist in this case and are, in a way, the analog of the coherent states of the harmonic oscillator. The gauge invariance of the theory is illustrated, and the results are related to the description of the motion in an accelerated reference frame. A brief discussion of the propagation function and of the evolution operator is also presented.

DOI
10.1119/1.12952

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•Jan 1, 1993•0 citations
The quantal motion of a charged nonrelativistic particle in a uniform but time-varying electric field is discussed. It is pointed out that states of definite but variable momentum exist in this case and are, in a way, the analog of the coherent states of the harmonic oscillator. The gauge invariance of the theory is illustrated, and the results are related to the description of the motion in an accelerated reference frame. A brief discussion of the propagation function and of the evolution operator is also presented.
When a system of particles in a bound state is placed in an external field the quantum state of the system depends on the manner in which the field is produced. The two limiting cases of interest are the adiabatic and impulse limits. Since the system does not approach a stationary state in either limit, in general, it is necessary to consider solutions of the time dependent Schrödinger equation. An example of this approach is provided by the problem of a harmonic oscillator in a uniform, time varying electric field for which an exact solution is easy to obtain using the Magnus expansion for the evolution operator. This operator contains the position and momentum in a particularly simple way that allows it to be interpreted in terms of position and momentum shifts produced by the field. The wavefunction of the oscillator is obtained in the adiabatic and impulse limits assuming the field to be switched on in an exponential fashion. Results obtained for these two limiting cases are then shown to be independent of the manner in which the field reaches its final steady value.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
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 nonrelativistic electron driven by a classical homogeneous time-varying field is given an exact wave-mechanical treatment. The problem is formulated and solved in two separate reference frames. One is a frame that accelerates like a classical electron that is in the same field, and here the wave mechanics is formally that of a free electron. The second frame is an inertial frame, and here the time-development operator is given two explicit forms. States of sharp but time-dependent momentum are found. Wave functions in both momentum and configuration spaces are given. Packets that move in the manner of a classical electron are produced conveniently. The physical interpretation of the wave-mechanical description is given graphical illustration in a phase-space plot. The circumstances where a purely classical description suffices are elucidated. It is shown that the wave-mechanical analysis based on a homogeneous electric field can be suitably applied to several physical situations as a good approximation. One situation is the electron in a propagating classical electromagnetic wave of arbitrary frequency. Another situation is an electron in transit through the fields in a microwave cavity. The wave mechanics of an electron plasma in a homogeneous time-dependent electric field is treated separately and yields some tangible information.
By successive applications of time-dependent rotation, translation and dilatation, it is shown that a charged time-dependent harmonic oscillator in a uniform time-dependent electromagnetic field is equivalent to a free particle. This equivalence is employed to present a simple derivation of the Feynman kernel as well as to obtain a concrete picture of the time evolution of a generic wave packet. A compact formula for an evolving anisotropic Gaussian packet is also derived. A special case of this formula gives a squeezed coherent state for a harmonic oscillator whose frequency is an arbitrary function of time. Finally we discuss an Aharonov-Bohm-Berry-like phase associated with a wave packet whose center traces a closed circuit.
We solve the quantum mechanical problem of the motion of charged particle in a quadrupole field that varies arbitrarily with time. The time-dependent wave functions are shown to be in a simple one-to-one correspondence with the wave functions of an ordinary static-field harmonic oscillator. We define and generate a set of coherent states for the charged particle that provide Gaussian localization about the classical particle trajectories. When the field variation is periodic in time, these minimum uncertainty states are shown to pulsate periodically in size and in phase. Under the action of external fields, coherent states retain their coherence. We discuss the way in which the Kapitza pseudopotential approach approximates the present exact results.