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Physical Review A

Nonrelativistic Schrödinger Green’s function for crossed time-dependent electric and magnetic fields

Norman J. M. Horing, Hong‐Liang Cui, Giovanni Fiorenza

Published Jul 1, 1986• 14 citations
Publisher

Abstract

The retarded nonrelativistic Schr\"odinger Green's function for an electron in crossed electric and magnetic fields is derived here in closed form. The electric and magnetic fields are taken to be spatially uniform but the electric field may have arbitrary time dependence and arbitrary orientation with respect to the constant magnetic field.

DOI
10.1103/physreva.34.612

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The retarded nonrelativistic Schr"odinger Green's function for an electron in crossed electric and magnetic fields is derived here in closed form. The electric and magnetic fields are taken to be spatially uniform but the electric field may have arbitrary time dependence and arbitrary orientation with respect to the constant magnetic field.
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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 classification of electromagnetic fields using the equivalence relation on the set of all 4-potentials of the Schrodinger equation. In the general case we find the relations among the equivalent fields, currents, and charge densities. Particularly, we study the fields equivalent to the null field. We show that the non-stationary state function for a particle in arbitrary uniform time-dependent magnetic field is equivalent to a plane wave. We present that the known coherent states of a free particle are equivalent to the stationary states of an isotropic oscillator. We reveal that the only constant magnetic field is not equivalent to the null field (contrary to a constant electrical field) and we find other fields that are equivalent to the constant magnetic field. We establish that one particular transformation of the free Schrodinger equation puts a plane wave and Green's function in a equivalence relation. c
Abstract A quasi‐momentum representation for an arbitrary quantum system is constructed by the integrals‐of‐motion method and by the coherent‐states method. Explicit expressions in terms of the Jakobi ϑ‐functions are obtained for the Bloch functions of the following dynamical systems: a relativistic and non‐relativistic electron moving in a constant uniform magnetic field and a non‐relativistic electron moving in both, time‐dependent and constant crossed electromagnetic fields. New mathematical relations between the Jacobi ϑ‐functions are obtained.