Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T23:29:23.311Z Has data issue: false hasContentIssue false

On stochastic theories of quantum mechanics

Published online by Cambridge University Press:  24 October 2008

J. G. Gilson
Affiliation:
Mathematics Department, Queen Mary College, London

Extract

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bartlett, M. S.An introduction to stochastic processes (Cambridge University Press, 1966).Google Scholar
(2)Gilson, J. G.Nuovo Cimento, Serie X, 40 (1965), 993, equation (1.5).CrossRefGoogle Scholar
(3)Furth, R.Z. Physik 81 (1933), 143.CrossRefGoogle Scholar
(4)Moyal, J. E.Proc. Cambridge Philos. Soc. 45 (1949), 99124.CrossRefGoogle Scholar
(5)Bartlett, M. S. and Moyal, J. E.Proc. Cambridge Philos. Soc. 45 (1949), 545.CrossRefGoogle Scholar
(6)Nelson, E.Phys. Rev. 150 (1966), 1079.CrossRefGoogle Scholar
(7)Kershaw, D.Phys. Rev. 136 (1964), 1850.CrossRefGoogle Scholar
(8)Feynman, R. P.Rev. Modern Phys. 29 (1948), 367.CrossRefGoogle Scholar
(9)Ming Chen, Wang and Uhlenbeck, G. E.On the theory of Brownian Motion 11. Rev. Modern Phys. 17 (1945), 323, sections 8b and c.Google Scholar
(10)Gilson, J. G.Feynman integral and phase space probability. To be published in J. Appl. Probability August 1968.CrossRefGoogle Scholar
(11)Gilson, J. G.Statistical relativistic particle. J. Appl. Probability (1967).CrossRefGoogle Scholar
(12)Cane, V.Random walks and physical processes. To be published in Bull. Inst. Internal. Statist.Google Scholar
(13)Gell-Mann, M.The eightfold way. Report CTSL-20.Google Scholar