Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Linear Differential Equation for the Radial Distribution Function of Quantum Mechanics

Morton D. Kostin

Published Mar 15, 1971• 5 citations
Publisher

Abstract

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.

DOI
10.1063/1.1675238

Related papers

On the probability density of quantum mechanics

Morton D. Kostin

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.

The Journal of Chemical Physics•May 15, 1974•3 citations

Partial Differential Equation for the Time-Dependent Probability Densities of Quantum Mechanics

Morton D. Kostin

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.

American Journal of Physics•Sep 1, 1969•0 citations

Partial differential equations for the probability density and charge density of quantum statistical mechanics

Morton D. Kostin

A partial differential equation is derived for the Slater sum of the Dirac equation. Normalization of the Slater sum gives the temperature-dependent probability density. Another partial differential equation is derived for the Slater sum of the Klein–Gordon equation. This Slater sum yields the temperature-dependent charge density. The transition from quantum theory to classical theory for the two partial differential equations is carried out and the relation between the two resulting classical equations is investigated.

Journal of Mathematical Physics•May 1, 1991•3 citations

Probability density for Bose–Einstein and Fermi–Dirac particles: Slater–Kahn functions

Morton D. Kostin

A nonlinear form of the Schrödinger equation is used to derive a partial differential equation for a Slater–Kahn function, which in turn yields the probability density of Bose–Einstein and Fermi–Dirac particles at a given temperature.

Journal of Mathematical Physics•Dec 1, 1986•1 citation

Wave Function for Classical Mechanics

Janne Mikael Karimäki

In this paper the relationship of classical physics and quantum physics is studied by introducing a partial differential equation, which describes classical mechanics, but looks very similar to the Schrodinger wave equation of quantum mechanics. This work is largely based on David Bohm's causal interpretation of quantum mechanics, but is in some sense complementary to it. In Bohm's theory the Schrodinger wave equation is used to derive classical looking equations of motion for quantum physics. Here exactly the opposite is done. The equations of classical physics are put into a form resembling the Schrodinger equation of quantum physics.

•Mar 1, 2012•0 citations

General solution of Kostin's equation arising in quantum mechanics

P.V. Ranganathan

For the non-linear ordinary differential equation in the radial component of the probability density discussed by Kostin (1971), a general solution is obtained. This solution is a homogeneous function of two linearly independent functions satisfying a certain associated linear homogeneous equation.

Journal of Physics A Mathematical and General•Dec 1, 1987•2 citations
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.
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.
A partial differential equation is derived for the Slater sum of the Dirac equation. Normalization of the Slater sum gives the temperature-dependent probability density. Another partial differential equation is derived for the Slater sum of the Klein–Gordon equation. This Slater sum yields the temperature-dependent charge density. The transition from quantum theory to classical theory for the two partial differential equations is carried out and the relation between the two resulting classical equations is investigated.
In this paper the relationship of classical physics and quantum physics is studied by introducing a partial differential equation, which describes classical mechanics, but looks very similar to the Schrodinger wave equation of quantum mechanics. This work is largely based on David Bohm's causal interpretation of quantum mechanics, but is in some sense complementary to it. In Bohm's theory the Schrodinger wave equation is used to derive classical looking equations of motion for quantum physics. Here exactly the opposite is done. The equations of classical physics are put into a form resembling the Schrodinger equation of quantum physics.