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

WORLD SCIENTIFIC eBooks

PROBABILITY IN QUANTUM MECHANICS

Internat J. Math, Math Sci, J. G. Gilson

Published Jun 1, 2000• 0 citations
Publisher

Abstract

ABSTRACT. By using a fluid theory which is an alternative to quantum theory but from which the latter can be deduced exactly, the long-standing problem of how quantum mechanics is related to stochastic processes is studied. It can be seen how the Schrodinger probability density has a relationship to time spent on small sections of an orbit, just as the probability density has in some classical contexts.

DOI
10.1142/9789812813459_0006

Related papers

PROBABILITY IN QUANTUMMECHANICS

James G. Gilson

By using a fluid theory which is an alternative to quantum theory but from which the latter can be deduced exactly, the long-standing problem of how quantum mechanics is related to stochastic processes is studied. It can be seen how the Schrödinger probability density has a relationship to time spent on small sections of an orbit, just as the probability density has in some classical contexts.

•Jan 1, 1982•0 citations

Probability in quantum mechanics

James G. Gilson

By using a fluid theory which is an alternative to quantum theory but from which the latter can be deduced exactly, the long‐standing problem of how quantum mechanics is related to stochastic processes is studied. It can be seen how the Schrödinger probability density has a relationship to time spent on small sections of an orbit, just as the probability density has in some classical contexts.

International Journal of Mathematics and Mathematical Sciences•Feb 17, 1981•1 citation

On stochastic theories of quantum mechanics

James G. Gilson

Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.

Mathematical Proceedings of the Cambridge Philosophical Society•Oct 1, 1968•36 citations

On Schrödinger's Search for a Stochastic Interpretation of Quantum Mechanics

Z. Marić

Abstract Following Schrödinger a stochastic interpretation of quantum mechanics is given based on the introduction of an intermediate probability in diffusion processes. The Schrödinger equation is derived following Nelson's approach and following a variational approach. Some problems of the quantum theory of measurement are discussed.

Annalen der Physik•Jan 1, 1988•1 citation

Time Dependent Density and Stochastic Approaches to Quantum Mechanics

Francesco R. Ruggeri

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).

•Nov 2, 2021•0 citations

QUANTUM MECHANICS AS QUANTUM MEASURE THEORY

Rafael D. Sorkin

The additivity of classical probabilities is only the first in a hierarchy of possible sum rules, each of which implies its successor. The first and most restrictive sum rule of the hierarchy yields measure theory in the Kolmogorov sense, which is appropriate physically for the description of stochastic processes such as Brownian motion. The next weaker sum rule defines a generalized measure theory which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed "as the squares of quantum amplitudes" is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a "realistic" interpretation of quantum mechanics.

arXiv:gr-qc/9401003•Modern Physics Letters A•Oct 30, 1994•406 citations
ABSTRACT. By using a fluid theory which is an alternative to quantum theory but from which the latter can be deduced exactly, the long-standing problem of how quantum mechanics is related to stochastic processes is studied. It can be seen how the Schrodinger probability density has a relationship to time spent on small sections of an orbit, just as the probability density has in some classical contexts.
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
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).
The additivity of classical probabilities is only the first in a hierarchy of possible sum rules, each of which implies its successor. The first and most restrictive sum rule of the hierarchy yields measure theory in the Kolmogorov sense, which is appropriate physically for the description of stochastic processes such as Brownian motion. The next weaker sum rule defines a generalized measure theory which includes quantum mechanics as a special case. The fact that quantum probabilities can be expressed "as the squares of quantum amplitudes" is thus derived in a natural manner, and a series of natural generalizations of the quantum formalism is delineated. Conversely, the mathematical sense in which classical physics is a special case of quantum physics is clarified. The present paper presents these relationships in the context of a "realistic" interpretation of quantum mechanics.