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Journal of Mathematical Physics

Green’s function for crossed time-dependent electric and magnetic fields. Phase-space quantum mechanics approach

Luis Miguel Nieto

Published Oct 1, 1992• 9 citations
Publisher

Abstract

In the present paper, the Moyal propagator is obtained for an electron inmersed in a constant uniform magnetic field crossed with a uniform electric field having arbitrary time dependence, using the techniques of phase space quantum mechanics. All the interesting magnitudes can be evaluated from it, and one can make quantum physics remaining in the context of phase space. Making use of these methods, the corresponding Green’s function is also obtained.

DOI
10.1063/1.529888

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In the present paper, the Moyal propagator is obtained for an electron inmersed in a constant uniform magnetic field crossed with a uniform electric field having arbitrary time dependence, using the techniques of phase space quantum mechanics. All the interesting magnitudes can be evaluated from it, and one can make quantum physics remaining in the context of phase space. Making use of these methods, the corresponding Green’s function is also obtained.
We give a rigorous proof for the following result that combines those discussed earlier by Kr\"uger and Poffyn [Physica 85A, 84 (1976)] and by Springborg [J. Phys. A 16, 535 (1983)]: The Wigner correspondence of a quantum operator can be obtained by simple substitutions under an arbitrary linear phase-space transformation. This property cannot be generalized to any other transformations or rules of phase-space correspondence. As an application of this result, the Wigner distribution functions of an anisotropic harmonic-oscillator system in crossed magnetic and electric fields are obtained.
The recently proposed Langevin formulation of quantum dynamics yields the propagator at imaginary time as a noise average involving the solutions to a certain Langevin equation with a Gaussian white noise. The main advantage of this approach is to permit a stochastic treatment of quantum systems even when no solutions to the Schrödinger equation are known. We show how the Langevin formulation works in the space of a (nonrelativistic) particle in a uniform magnetic field with arbitary time dependence. We are able to evaluate the quantum-mechanical propagator exactly. As far as we know, this is a new result of considerable interest for atomic and molecular physics.
We derive the retarded nonrelativistic Green's function for a Schrödinger electron confined to a plane and subject to a parabolic saddle potential and an electric field having arbitrary time dependence and orientation on the plane. This derivation is carried out using Heisenberg equations of motion for position and momentum operators, following an earlier analysis of Schwinger, to obtain the retarded Green's function in closed form, as an explicit function of position and time variables.
Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.