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Journal of Physics A Mathematical and General

The Schrodinger equation in terms of probability densities

Peter Rastall

Published Mar 7, 1994• 0 citations
Publisher

Abstract

The probability density of a non-relativistic particle is written as the sum of two non-negative density functions. The Schrodinger equation is equivalent to continuity equations (with source terms) for these functions. The equations are interpreted as describing two modes of the particle, which continually transform into each other.
DOI
10.1088/0305-4470/27/5/008

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T he necessity for the treatment of the properties of quantum particles in terms of their most likely values is indicated in the previous chapter. Schrödinger’s equation, which provides the means to do this, is developed by examining the classical relationships describing the properties of a particle in motion and their transformation into quantum mechanical equivalents. The probable properties of quantum particles are obtained from the wavefunctions that are meaningful solutions to Schrödinger’s equation.
The Schrödinger equation: the quantum description of one massive, slow-moving particle To establish a probability wave equation for one photon, it was useful to note that probability density and electromagnetic energy density were proportional. The governing equation for EM radiation fields is the Maxwell wave equation, which is also the governing equation for photon probability wave functions. Converting from EM radiation to photon information is made plausible by identifying energy and momentum operators with time- and space-derivatives, respectively. Thus, the Maxwell wave equation can be interpreted as being equivalent to the energy-momentum relation for photons. Though we don’t have a “Maxwell wave equation” to start with for massive particles, we do have an energy-momentum relation, so borrowing the energy and momentum operators from photons leads to an analogous wave equation for massive particles.
Starting with a polar decomposition of the wavefunction, it is shown that the probability and energy density currents are explicit functionals of the probability and energy densities. At a non-reduced level and for a (n 1 +n 2 )-particle system, the expression of these functionals is exactly calculated. This leads to a system of coupled equations equivalent to the Schrodinger equation. The probability and energy densities are unknown. Some properties of the functionals are analysed and the particular case of a one-body Hamiltonian studied.