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Journal of Physics A Mathematical and Theoretical

Quantum power in de Broglie–Bohm theory

Donald H. Kobe

Published Apr 24, 2007• 48 citations
Publisher

Abstract

In the de Broglie–Bohm approach to quantum mechanics for a charged particle in a time-dependent electromagnetic field the time derivative of the energy is equal to the classical power plus a quantum power. We show that the average of the quantum power is zero. The de Broglie–Bohm energy is obtained from the quantum mechanical energy operator, which is the Hamiltonian with the gauge-dependent scalar potential subtracted. The time derivative of the average de Broglie–Bohm energy is shown to be equal to Ehrenfest's theorem for the quantum energy operator.

DOI
10.1088/1751-8113/40/19/015

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In the de Broglie–Bohm approach to quantum mechanics for a charged particle in a time-dependent electromagnetic field the time derivative of the energy is equal to the classical power plus a quantum power. We show that the average of the quantum power is zero. The de Broglie–Bohm energy is obtained from the quantum mechanical energy operator, which is the Hamiltonian with the gauge-dependent scalar potential subtracted. The time derivative of the average de Broglie–Bohm energy is shown to be equal to Ehrenfest's theorem for the quantum energy operator.
The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.
In the stochastic electrodynamics, various quantum effects on particles are supposed to emerge from the interactions of the particles with the fluctuating background field. We have adopted the same supposition and further supposed that the quantum potential arises from the velocity fluctuations of particles around their local averages. Based on these suppositions we have derived the expression of the quantum potential in the framework of de Broglie–Bohm theory, which is a reformulation of Schrödinger’s. Thus we are allowed to use the wave functions and Schrödinger operators explicitly so that we have been able to derive the quantum potential in simpler and more compact manner as compared with the stochastic electrodynamics’s. A new hydrodynamical version of Schrödinger equation is also presented, which has been found in the course of our study.
For a system consisting of a classical electromagnetic field and a charged quantum-mechanical particle, Poynting's theorem holds for the energy density of the electromagnetic field. For local energy conservation to hold, the power lost by the field is the power gained by the quantum-mechanical particle. The power per unit volume transferred from the field to the particle is $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}$, which is shown to be the density of the quantum-mechanical power operator. For the particle an energy theorem, similar to Poynting's theorem for the field, is satisfied in such a way that the energy of the total isolated system is conserved. The energy density of the particle is the density of the energy operator for the particle, which in this time-dependent problem is not the same as the Hamiltonian. The energy operator is gauge invariant, while the Hamiltonian is not. When the combined conditions of gauge invariance and local energy conservation are used, the appropriate basis functions to use are the eigenstates of the energy operator. Transitions between these eigenstates are determined by matrix elements of the power operator.
In this paper we suggested a natural interpretation of the de Broglie-Bohm quantum potential, as the energy due to the oscillating electromagnetic field (virtual photon) coupled with moving charged particle. Generalization of the Schrödinger equation is obtained. The wave function is shown to be the eigenfunction of the Sturm-Liouville problem in which we expand virtual photon to include it implicitly into consideration. It is shown that the non-locality of quantum mechanics is related only with virtual photon. As an example, the zero-energy of harmonic oscillator is obtained from classical equations.
The case of a single charged particle with potential energy is considered when the classical electromagnetic field is zero. The vector and scalar potentials describing this situation are not necessarily zero. When the conventional approach to radiation problems using the interaction A⋅p plus p⋅A plus A2 plus A0 (constants omitted) is used, time-dependent probabilities are predicted if the vector potential A=−∇Λ≠0 even though the electromagnetic field is zero. The reason for this unphysical behavior is that eigenfunctions of the unperturbed Hamiltonian, which is not gauge invariant, are used to calculate the probability amplitudes. When a manifestly gauge-invariant procedure is applied to this case the probabilities are constant in all gauges. The reason that physical results are obtained is that eigenstates of the energy operator are used to calculate the probability amplitudes.