Does Bohm’s Quantum Force Have a Classical Origin?

Abstract

In the de Broglie–Bohm formulation of quantum mechanics, the electron is stationary in the ground state of hydrogenic atoms, because the quantum force exactly cancels the Coulomb attraction of the electron to the nucleus. In this paper it is shown that classical electrodynamics similarly predicts the Coulomb force can be effectively canceled by part of the magnetic force that occurs between two similar particles each consisting of a point charge moving with circulatory motion at the speed of light. Supposition of such motion is the basis of the Zitterbewegung interpretation of quantum mechanics. The magnetic force between two luminally-circulating charges for separation large compared to their circulatory motions contains a radial inverse square law part with magnitude equal to the Coulomb force, sinusoidally modulated by the phase difference between the circulatory motions. When the particles have equal mass and their circulatory motions are aligned but out of phase, part of the magnetic force is equal but opposite the Coulomb force. This raises a possibility that the quantum force of Bohmian mechanics may be attributable to the magnetic force of classical electrodynamics. It is further shown that relative motion between the particles leads to modulation of the magnetic force with spatial period equal to the de Broglie wavelength.

Appendix: Evaluation of the Quantum Force in the Hydrogen Atom Ground State

The quantum potential and resulting quantum force is calculated from the Schrödinger wavefunction. In the application of the Schrödinger equation to the hydrogen-like atom, the classical potential term is simply the Coulomb potential of a point charge, while the quantum potential depends on the solution. For the ground state solution of the time-independent Schrödinger equation, and for all s-states, the electron momentum vanishes [33]. In these cases, the quantum potential is everywhere opposite the Coulomb potential, and so the quantum force exactly cancels the Coulomb force. However, the quantum force is not usually evaluated explicitly in the literature, since it is not needed to obtain the physical results of interest (e.g., that the electron momentum vanishes), so for completeness it is evaluated in the following for the hydrogen atom ground state. Also, it serves to illustrate that although the magnetic force component described here bears a similarity to the quantum force, it cannot be equated to it directly. More generally, the quantum force cannot in principle be equatable to the magnetic force, because while the quantum potential is scalar in both its non-relativistic and relativistic [34] forms, the magnetic force must be derived from a vector potential.

The quantum potential is [35]

$$\begin{aligned} Q = -\frac{\hbar ^2}{4m}\left[ \frac{\nabla ^2\rho }{\rho } - \frac{1}{2}\frac{(\nabla \rho )^2}{\rho ^2} \right] , \end{aligned}$$

(30)

where, if the Schrödinger wavefunction $\psi \equiv R \exp (iS/\hbar )$, then $\rho \equiv R^2$.

For the hydrogen ground state, the normalized Schrödinger wavefunction is [36]

$$\begin{aligned} \psi = \left( \frac{2}{{a_0}^{3/2}} \right) e^{-r/a_0}, \end{aligned}$$

(31)

or, equivalently,

$$\begin{aligned} \rho = \left( \frac{4}{{a_0}^{3}} \right) e^{-2r/a_0}, \end{aligned}$$

(32)

and the quantum potential evaluates as

$$\begin{aligned} Q = \frac{\hbar ^2}{2m a_0} \left[ \frac{2}{r} - \frac{1}{{a_0}} \right] , \end{aligned}$$

(33)

and the quantum force is

$$\begin{aligned} F_Q = -\nabla Q = \frac{\hbar ^2}{m a_0} \left[ \frac{\hat{\varvec{r}}}{r^2} \right] . \end{aligned}$$

(34)

With $a_0 = \hbar ^2/m e^2$,

$$\begin{aligned} F_Q = e^2 \left[ \frac{\hat{\varvec{r}}}{r^2} \right] . \end{aligned}$$

(35)

The quantum force thus exactly cancels the Coulomb attraction, in the Schrödinger hydrogen atom ground state.