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International Journal of Modern Physics A

BOSE OPERATOR HAMILTONIAN MODEL FOR ROTATING ELECTRIC DIPOLE AND GENERALIZED EHRENFEST'S THEOREM

Hong-Yi Fan

Published Jan 10, 2002• 2 citations
Publisher

Abstract

We present a Bose operator Hamiltonian model for totally quantum mechanically describing energy and eigenfunctions of a rotating dipole in the presence of electric field. The corresponding Heisenberg equations of motion are derived which leads us to the generalized Ehrenfest theorem for rotating system. Remarkably, a zero-point angular momentum ℏ/2 appears in the generalized Ehrenfest theorem corresponding to the classical equation Iω=L.

DOI
10.1142/s0217751x02006055

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We present a Bose operator Hamiltonian model for totally quantum mechanically describing energy and eigenfunctions of a rotating dipole in the presence of electric field. The corresponding Heisenberg equations of motion are derived which leads us to the generalized Ehrenfest theorem for rotating system. Remarkably, a zero-point angular momentum ℏ/2 appears in the generalized Ehrenfest theorem corresponding to the classical equation Iω=L.
Abstract Ehrenfest theorem is extended to the relativistic quantum theory of charged particles, moving under the influence of a classical electromagnetic field. In order to extend the original Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton’s second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.
A physical interpretation is given for the Heisenberg equations implied by the usual phenomenological Hamiltonian for a neutral, magnetic dipole in a radiation field. If the Heisenberg equations that characterize the quantum description are to be formally the same as their classical counterparts, a distinction must be made between the canonical and orbital momenta, even for a neutral system. This leads to a novel interpretation of the spin-orbit energy and a reidentification of the electric polarizability of the system. The general ideas are illustrated by considering the exactly soluble special cases of a neutral magnetic dipole with a nonvanishing electric polarizability in uniform, static fields and in the field of a monochromatic, plane wave. A Lagrangian formalism corresponding to these ideas is also presented.
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.
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.
We introduce the Poisson bracket operator, which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation, which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to ccc-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore, the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to nonstandard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where ccc-number and qqq-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations.