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American Journal of Physics

A representation of the Schrödinger and Klein–Gordon equations obtained using simple hyperbolic numbers

Paul Bracken, James Hayes

Published Jun 11, 2003• 6 citations
Publisher

Abstract

We write the Bohm–Landau wave function in terms of simple hyperbolic numbers and apply this representation to the Schrödinger and Klein–Gordon equations. It is shown that the Schrödinger equation can be separated in one space dimension. The radial part of the wave function and the derivatives of the radial function can be related to the potential and its derivative. We also apply this representation to the Klein–Gordon equation.

DOI
10.1119/1.1571839

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We write the Bohm–Landau wave function in terms of simple hyperbolic numbers and apply this representation to the Schrödinger and Klein–Gordon equations. It is shown that the Schrödinger equation can be separated in one space dimension. The radial part of the wave function and the derivatives of the radial function can be related to the potential and its derivative. We also apply this representation to the Klein–Gordon equation.
In this study, both the Klein-Gordon equation and time-dependent Schrodinger equation are directly derived from De Broglie's (\lamda=h/p). Besides that, a new formula of the Klein-Gordon equation was obtained by using two equations, i.e., the classical wave function and relativistic total energy relation [[hw] = [(h^2 k^2)/2m] + V]. Moreover, a new equation of time-dependent Schrodinger (TDSE) in three-dimension 3D was derived and evidently obtained. Thus, the previous result corresponds to (TDSE) in 3D calculated using two equations, (i.e., Einstein energy and Newton's second law of mechanics).
A de Broglie-Bohm like model of Klein-Gordon equation, that leads to the correct Schrodinger equation in the low-speed limit, is presented. Under this theoretical framework, that affords an interpretation of the quantum potential, the main assumption of the de Broglie-Bohm interpretation--that the local momentum of particles is given by the gradient of the phase of the wave function--is not but approximately correct. Also, the number of particles is not locally conserved. Furthermore, the representation of physical systems through wave functions wont be complete.
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.
The quantum hydrodynamic-like equations for two real variables (i.e., the phase and the amplitude of the wave function) of the relativistic Klein-Gordon equation are derived in the present paper. The paper also shows that in classical limit the hydrodynamic Klein-Gordon equations lead to the Madelung pseudo-potential [1] as well as to the quantum pseudo potential for a charged particle given by Janossy [2] and by Bialynicki et al [3]. The origin of the non-local interactions of quantum mechanics in the hydrodynamic model are discussed both for free and charged particles.
This paper deals with computational derivation of the Klein-Gordon quantum relativist equation and the Schrödinger quantum equation with forward and backward space-time shifts. The first part introduces forward and backward derivatives for discrete and continuous systems. Generalized complex discrete and continuous derivatives are deduced. In the second part, we apply these forward and backward derivatives to the harmonic oscillator. The third part deduces the Klein-Gordon equation from the space-time complex continuous derivatives. These derivatives take into account forward-backward space-time shifts related to a phase velocity u. Without time shift, the Schrödinger equation is deduced in part four. This quantum equation is interpreted as a Burgers fluid with a complex momentum in relation to the Bohm-Hiley quantum potential concept. Finally a time-dependent Schrödinger equation for a free particle in a constant potential is simulated. The quantum particle looks like an oscillator with two frequencies.