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Journal of Physics A Mathematical and General

Plane rotor with time-dependent magnetic flux in a multivalued representation

Donald H. Kobe

Published Feb 1, 1985• 6 citations
Publisher

Abstract

A quantum mechanical plane rotor with time-dependent magnetic flux on its axis is considered in a representation in which the wavefunction is multivalued. If the representation is chosen such that the new kinetic angular momentum is equal to the old canonical angular momentum, the Schrodinger equation is simplified. An energy eigenfunction expansion in the new representation is used to solve the Schrodinger equation exactly.

DOI
10.1088/0305-4470/18/2/013

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A quantum mechanical plane rotor with time-dependent magnetic flux on its axis is considered in a representation in which the wavefunction is multivalued. If the representation is chosen such that the new kinetic angular momentum is equal to the old canonical angular momentum, the Schrodinger equation is simplified. An energy eigenfunction expansion in the new representation is used to solve the Schrodinger equation exactly.
A plane rotor, which is an electron constrained to move in a circular orbit, has time-dependent magnetic flux on its axis. By Faraday's law, an induced electric field acts on the electron. The induced electric field exerts a torque on the electron and thus changes its kinetic angular momentum. The induced electric field also does work on the electron and changes its energy. These changes are in agreement with generalised Ehrenfest theorems. The kinetic angular momentum and the energy of the electron depend on the instantaneous flux through the orbit. The Shrodinger equation for the plane rotor is solved exactly in a manifestly gauge-invariant way. The probability that the rotor is in an eigenstate of its energy operator is contrasted with the probability that the system is in an eigenstate of the unperturbed Hamiltonian.
It is shown fields that cannot be represented over one complex plane can be further decomposed for representation over multiple complex planes. This finding is demonstrated here by solving of the Schrödinger equation for the hydrogen atom in a complex space containing three complex planes. The complex coordinate system is generated from real coordinates using an isometric transformation. One plane is applied to mix energy and time; the other two planes are used to represent the z-component of angular momentum of the electron. The eigensolutions of the Schrödinger equation are shown to be holomorphic in the complex planes.
Time-dependent unitary transformations are used to study the Schrödinger equation for explicitly time-dependent Hamiltonians of the form H(t)=R(t)⋅J, where R is an arbitrary real vector-valued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case R=(1,0,ν(t)). This corresponds to the problem of a driven two-level atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form H(t)=R(t)⋅J there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.
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.