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The Journal of Chemical Physics

On the probability density of quantum mechanics

Morton D. Kostin

Published May 15, 1974• 3 citations
Publisher

Abstract

The Schrödinger equation is used to derive a nonlinear equation for the probability density of N independent spatial variables. It is shown that under certain conditions the equation is linear. In these cases a linear equation is then derived for the N-dimensional spatial distribution function of quantum statistical thermodynamics.
DOI
10.1063/1.1680849

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A nonlinear equation for the probability density of quantum theory is obtained from the Schrödinger equation for the case of complex wavefunctions. Motion of a particle in a centrally symmetric field is investigated and it is found that a linear differential equation for the radial distribution function can be derived. The linear equation is used to demonstrate the transition to classical theory and to derive a partial differential equation for the radial distribution function of statistical thermodynamics.
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A form of the Schr\"odinger equation is given in which the time variable is replaced by a complex time-temperature variable and the interaction with the heat bath is represented by a stochastic interaction operator with Gaussian white noise. It is shown that at large times the ensemble averaged solution yields the equilibrium spatial probability density of quantum statistical mechanics. As equilibrium is approached, we also show how the master equation for a canonical ensemble can be derived from this form of the Schr\"odinger equation.
The Schrödinger equation for the wave functions is used to derive a linear partial differential equation for the time-dependent probability densities, ħ24m δ3P(x, t)δx3 = 2V(x) δPδx + V′P(x, t) − 2δε(x, t)δx − mδj(x, t)δt, where P(x, t), ε(x, t), j(x, t) are the time-dependent position, energy, and current densities, respectively. In the limiting case where the system is in the nth quantum state, it is shown that this partial differential equation reduces to the previously derived third-order ordinary differential equation for the probability position density.
In the literature, stochastic approaches employed to derive the Schrodinger equation seem to focus on the idea of a classical spatial density which depends on time and space ie. density(x,t). As a result, d/dt (partial) density appears in various equations such as the continuity and Focker Planck equations. At one point in the derivation the quantum wavefunction is introduced “mathematically” as W(x,t)=sqrt(density(x,t)) exp(i S(x,t)) where S is related to a velocity field through u(x,t)=velocity= d/dx S. From quantum mechanics, -Et becomes a term of S the velocity field. The stochastic/hydrodynamical approaches employ the term du/dt partial = d/dt (partial) d/dx S and then “pull out”” d/dx leaving d/dt (partial) S which yields -E average energy from the -Et term. In this note we argue that for a bound quantum state density(x) so d/dt (partial) density = 0, yet the statistical ideas of quantum mechanics should be present in such a case. Furthermore for the free particle wavefunction exp(ipx-iEt), density=1. In fact, the Schrodinger equation should follow for a free particle even the case of density=1. Stochastic processes seem to be based on equations which consider a change in classical spatial density at a point x in time due to flow in or out or stochastic changes. Quantum processes seem to deal with periodic probability changes from one dimension to another exp(ipx) (2 dimensions) which are somehow related to a physical process of momentum flow. Ultimately one links this process to an energy conservation which does not seem to be the same as a classical stochastic/hydrodynamical approach. In this note we argue that a free particle with a constant momentum p has periodic stochastic processes associated with it through its complex probability change in the region of its wavelength, but that these are linked to creating the momentum p. Furthermore momentum is linked to energy so we argue that the Schrodinger equation is a stochastic equation linking energy and momentum and that time and space are separated i.e. there is no situation of following a change in spatial density in time for a free particle with momentum p. For a bound state, one has a distribution of free particles, but again time is removed from the problem in the sense that it only appears in exp(-iEt) in the wavefunction to again establish a relationship between energy and average momentum and potential. There is again no nonzero d/dt (partial) density term which seems to be key in stochastic equations. Only if one takes a linear combination of energy eigenstates Sum over En an(En) exp(-iEnt) Wn(x) does one obtain the form density(x,t), but we argue that quantum features should already appear in exp(ipx) and the bound state scenario with a zero d/dt (partial) density(x,t). We argue, however, that the Schrodinger equation and fundamental stochastic ideas of quantum mechanics should already be present at the level of exp(ipx).