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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Quantization in curvilinear coordinates and construction of expansion formulas via eigenfunctions of Schrodinger operator

Alexander G. Chircov, Georgy A. Kazakov

Published Apr 5, 2004• 0 citations
Publisher

Abstract

In the present work the classical problem about a particle in a central field is quantized using Chirkov's method. According to this method, in contrast to classical one, quantization is carried out directly in spatial spherical coordinates. The generalization of Schrodinger equation for a case of spatial spherical coordinates is obtained. The expansion formulas of arbitrary functions via the eigenfunctions of this equation are obtained on the basis of Titchmarsh's method. It is proved that the Parseval equality is fulfilled directly in curvilinear coordinates.

DOI
10.1117/12.555368

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In the present work the classical problem about a particle in a central field is quantized using Chirkov's method. According to this method, in contrast to classical one, quantization is carried out directly in spatial spherical coordinates. The generalization of Schrodinger equation for a case of spatial spherical coordinates is obtained. The expansion formulas of arbitrary functions via the eigenfunctions of this equation are obtained on the basis of Titchmarsh's method. It is proved that the Parseval equality is fulfilled directly in curvilinear coordinates.
A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.
It is well-known that the coordinate as a continuous variable, consisting of a set of all points between 0 and LLL contradicts the observability of measurement. In other words there might exist a fundamental length in nature, such as the Planck length λPλ_PλP​, so that it is not possible to measure a position coordinate with accuracy smaller than this fundamental length. It is therefore necessary to investigate the formulation of quantum mechanics using only discrete variables as coordinates. To investigate all of quantum mechanics or any branch of physics from this approach is of course a daunting task and thus it is worthwhile to consider a specific simple problem in order to formulate the basic ideas. In this note we compare the solutions of Schrodinger equation for one-dimensional free particle under the usual space-time continuum with those that are obtained when space-time is assumed to be quantized using a simple model. For this purpose, we replace the derivatives occurring in Schrodinger equation with the corresponding discrete derivatives. We compute the probability density (and the probability current) under the two scenarios; they turn out to be quite different in the two cases. We also obtain the operator identity for the commutator [p,x]q[p,x]^q[p,x]q under the assumption of quantized space-time and contrast it with the usual commutator [p,x][p,x][p,x].
A simple procedure has been found for the general solution of the time-independent Schrödinger Equation (SE) with the help of quantization of potential area in one dimension without making any approximation.Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and anti symmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential well, the relativistic theory of Dirac as examples.