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Revista Mexicana de Física

ON THE GENERAlIZED SCHROEDINGER EQUATION

L. de la Peña-Auerbach, L. S. Garcı́a-Colı́n

Published Jan 1, 1967• 0 citations
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Abstract

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).

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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).
An alternative method is proposed for deriving the time dependent Schroedinger equation from the pictures of wave and matrix mechanics. The derivation is of a mixed classical quantum character, since time is treated as a classical variable, thus avoiding any controversy over its meaning in quantum mechanics. The derivation method proposed in this paper requires no ad hoc assumption and avoids going through a second-order differential equation that can be reduced to the well known time-dependent Schroedinger equation only postulating a complex wavefunction with an exponential time dependence, as did by Schroedinger in its original paper of 1926.
Quantization processes generally assume the Hamiltonian formulation of classical mechanics. The notion of dissipation does not fit easily into the Hamiltonian structure. There appeared in the literature a quantization process that circumvents the use of the Hamiltonian approach and derives the Schr\"odinger equation from first principles. Thus, the usual approach of assuming a dissipative force of the type fd=k(t)v{f}_{d}=k(t)vfd​=k(t)v, where k(t)k(t)k(t) is a scalar depending only on the time and vvv is the velocity, can be approached using this quantization process to mathematically derive a Schr\"odinger equation. The derivation for this simple case was already performed by the authors elsewhere. The full generalization of a dissipative force in the context of a linear response theory, as is usual in classical mechanics, would be to write this force as fi(x,t)=Kij(x,t)vj{f}_{i}(\mathbf{x},t)={\mathcal{K}}_{ij}(\mathbf{x},t){v}_{j}fi​(x,t)=Kij​(x,t)vj​, where K(x,t)\mathcal{K}(\mathbf{x},t)K(x,t) is a tensor giving the nonhomogeneity and nonisotropy of the process. In this paper, we present this generalization and connect our results with nonlinear Schr\"odinger-like equations already known from the literature.
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.
The Schrödinger equation is the foundation of quantum mechanics and the starting point for any improvement to the description of submicroscopic physical and chemical systems. Although it cannot be proved or derived strictly, it has associated with it various formulations and ‘derivations’. In this work, we study the physical and mathematical evidence that make the Schrödinger equation plausible. First, we make an approach from the historical perspective, and study the methods of the pioneers Schrödinger, Heisenberg, and Dirac. Turning to modern treatments, we review numerous heuristic introductions to quantum mechanics and make a synthesis of the various methods into a coherent and meaningful whole. [Chem. Educ. Res. Pract. Eur.: 2001, 2, 203-213]
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)