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The Journal of Chemical Physics

Partial Differential Equation for the Probability Density of Quantum Statistical Thermodynamics

Morton D. Kostin

Published Dec 15, 1967• 9 citations
Publisher
DOI
10.1063/1.1701822

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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
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.
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.