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Journal of Mathematical Physics

Joint distribution indicated by the wave equations of quantum mechanics

Roderick I. Sutherland

Published Dec 1, 1982• 10 citations
Publisher

Abstract

By using the notion of conditional mean values and assuming that the classical relationship between momentum and energy remains valid, it is shown that both the Schrödinger and Klein–Gordon equations suggest a particular joint distribution expression for quantum mechanics, namely the Margenau and Hill distribution.
DOI
10.1063/1.525279

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The Schroedinger- and Klein-Gordon equations are directly derived from classical Lagrangians. The only inputs are given by the discreteness of energy (E=hbar.w) and momentum (p=hbar.k), respectively, as well as the assumed existence of a space-pervading field of "zero-point energy" (E_0=hbar.w/2 per spatial dimension) associated to each particle of energy E. The latter leads to an additional kinetic energy term in the classical Lagrangian, which alone suffices to pass from classical to quantum mechanics. Moreover, Heisenberg's uncertainty relations are also derived within this framework, i.e., without referring to quantum mechanical or other complex-numbered functions.
A derivation of the Klein-Gordon wave equation without substituting the corresponding quantum mechanical energy and momentum operators in Einstein's relativistic energy-momentum relationship is presented. An additional discussion is made to motivate readers to understand the distinction between the quantum wave equation and the governing law in the context of quantum mechanical formalisms. A resemblance of mathematical structure between the Maxwell wave equation and the deduced Klein-Gordan equation is shown. It is also noted that the wave velocity of the matter wave becomes equal to photon velocity for massless cases. This can be useful to explain why light equally shows matter and particle behaviour.
Derivation of Klein–Gordon equation Goals and prior knowledge The main wave equations of relativistic quantum mechanics are the Klein–Gordon, Dirac, Weyl , and Rarita–Schwinger equations. The principal aim of this chapter is to derive these, by means of the EPI principles (3.16) and (3.18). The derivations generally follow Frieden (1995), or Frieden and Soffer (1995), with numerous additions. In addition, the Schrödinger wave equation (SWE), which is non -relativistic, is also to be derived. Since rigorous use of EPI gives only relativistic wave equations (Chapter 3), EPI can only derive the SWE in some approximate sense. This is directly carried through in Appendix D, or by taking the well-known non-relativistic limit of the Klein–Gordon equation in Appendix G. Also, in Appendix D a simple EPI derivation of classical mechanics is obtained. In particular, in the limit h → 0 the EPI extremum condition (3.16) is found to give Newton's second law , and the zero-condition (3.18) gives the virial theorem , of classical mechanics. Other derived results are Eq. (4.17) – the equivalence of mass, momentum, and energy; the constancy of the Planck constant h (Sec. 4.1.14); the Compton wavelength as an ultimate resolution length (Sec. 4.1.17); and the Heisenberg uncertainty principle (Sec. 4.3). It is important to state what is assumed a priori . The relativistic properties of space and time coordinates were derived in Sec. 3.5. Hence, these are assumed here. Also, appropriate units for the concepts of mass and energy are assumed to be known, although the basic relation Eq. (4.17) between them is not presumed.
Following de Broglie, Bohm, and others, it is assumed that quantum mechanics may be interpreted causally and that the $\ensuremath{\psi}$ function plays the role of a generating function for particle trajectories. By arguing that the $\ensuremath{\psi}$ function should not be interpreted as a probability amplitude, a new method for generating particle trajectories is postulated. The four-momentum of a scalar particle is assumed to be given as the gradient of an unspecified function $F(\ensuremath{\psi})$, where $\ensuremath{\psi}$ is a pure real solution of the Klein-Gordon equation. Since the location of a particle is determined solely by its trajectory, the probability distribution differs from $\ensuremath{\psi}{\ensuremath{\psi}}^{*}$; and therefore, ordinary experimental results differing from the traditional theory may, in principle, be predicted. Particle motion and trajectories are discussed for three examples: a free particle, a particle in a box, and the double slit.