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Compt. rend.

THE SCHROEDINGER EQUATION APPLIED TO THE CALCULATION OF FLUCTUATIONS OF NUCLEAR REACTORS

Austin Blaquière, R. Pachowska

Published Dec 18, 1961• 0 citations
OpenAlex

Abstract

The formal analogy between the Fokker-Planck equation of nuclear reactors, obtained from the generating function of probabilities, and the Schroedinger equation utilized to describe the evolution of quantum systems has led to an attempt to define the relationship of the two theories. It was shown that the second method is also applicable to the study of reactor fluctuations. In this case, the Schroedinger equation appears as the matrix transcription of the FokkerPlanck reactor equation. (tr-auth)

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The formal analogy between the Fokker-Planck equation of nuclear reactors, obtained from the generating function of probabilities, and the Schroedinger equation utilized to describe the evolution of quantum systems has led to an attempt to define the relationship of the two theories. It was shown that the second method is also applicable to the study of reactor fluctuations. In this case, the Schroedinger equation appears as the matrix transcription of the FokkerPlanck reactor equation. (tr-auth)
The Fokker_Planck equation can be derived in a consistent manner through a microscopic approach based on a unified scheme of classical and quantum mechanics. Here we shall derive it through a purely quantum mechanical approach based on the reversible Schrodinger dynamics. We also give a brief discussion of the path integral representation of the Fokker_Planck equation in light of our derivation. We conclude that, because of the use of the representation of eigenstates of the time-independent Hamiltonian in our derivation, the thermodynamical entropy in this case must correspond to a coarse-graining of the quantum entropy.
SUMMARY 1967 In tbis paper a derivation of a genera/iud Schroedinger equation va lid lar velocity dependent forces. is presented. The physical assumptions invo/ved rest on the hypothesis that a more general quantum mechanical description of a partie/e may be constructed if we sJart from a Markoff process as described by FokkerPianck's equation in phase space. lt is also shown that Ihe asymptotic limil kads to the ardinary formalism of quantum mechanics in configuralion space togelher wilh a new equation in velocily space which al/ows us lo generate quanlum numbers which are tentatively associated with SOme of those characleristic ollhe portie/e. This formulation leads to a lirst es tima te ollbe parameler j3 introduc('d in pr(' vious papers. Consultont. Comision Nacional de Energia Nudear (Mexico).
A quantum mechanical form of the Langevin equation is derived from an explicit consideration of the molecule-medium interaction, as advocated by Simons in 1978, and by using two identities in the interaction picture. This can be easily reduced to the classical regime, and further simplified to the macroscopic Langevin equation by considering the stochastic Langevin force autocorrelation function. One of the so-called Einstein relations appears as a byproduct. By following the methodology proposed by Simons, an exact expression for the momentum autocorrelation function is obtained. The latter can be used to calculate the zero-frequency macroscopic diffusion coefficient that is observed to satisfy the second Einstein relation. The formalism described above gives rise to the possibility of explicitly computing the transport characteristics such as friction constant and diffusion coefficient from the corresponding quantum statistical mechanical expressions. A discussion on the Langevin equation becomes complete only when the corresponding Fokker-Planck equation is obtained. Therefore, the probability of the evolution of states with a particular absolute magnitude of linear momentum from those of another momentum eigenvalue is quantum mechanically defined. This probability appears as a special average value of a projection operator and as a special projection operator correlation function. A classical identity is introduced that is shown to be valid also for the quantum mechanically defined probability function. By using this identity, the so-called Fokker-Planck equation for the evolution probability is easily established.
We have advocated in a previous paper (Godart M. arXiv: 1206.2917v2[quant-ph] ) a version of the stochastic theory of quantum mechanics. It is indirectly based on a method proposed by Nelson to associate a Markov process with any solution of the Schroedinger equation. The debate began very soon on the question to know if the new theory based on that stochastic procees was equivalent to the orthodox Copenhagen version. We conclude in this paper that the answer is in the negative in agreement with the opinion of several physicists. We show however that the elementary solutions of the Schroedinger equation can be recovered formally from the stochastic theory in a great number of particular cases. We confirm also that this equation is no longer valid when a magnetic field is at work and that it must then give way to some other equation.