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数理解析研究所講究録

Relationship between quantum walks and differential equations (Progress in Variational Problems : Variational Problems Interacting with Probability Theories)

拓也 町田

Published Jun 1, 2013• 0 citations
OpenAlex

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Relationship between quantum walks and differential equations (Progress in Variational Problems : Variational Problems Interacting with Probability Theories)

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The stationary Schrodinger equation depending on spatial coordinates has been considered. The problem of obtaining a differential relationship for the wave function squared was posed. By extracting Schrodinger's equation itself from this relationship a differential equation for a physically interpretable quantity, i.e. the probability density (wave function squared), has been formulated. As an example the one-dimensional case admitting a simple analytic solution was considered. The solution obtained is shown to be a solution squared of the corresponding nonlinear differential equation for the probability density. In the final section a more general non-stationary case was considered for the potential involving a time-dependent term, such potentials are found in the non-stationary perturbation theory. The constant in separating the variables remains real. Thus the procedure considered proves to be similar to that presented above for the stationary equation.

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An external parameter is, by definition, a quantity which may be varied independently of the coordinate positions and of the momenta of the various particles composing the system. Thus, the Schr\"odinger equation, for a particular value of the external parameter, belongs to a family of differential equations obtained by varying the numerical value of the external parameter. An explicit example of an external parameter is an imposed magnetic field. The way in which the energy changes throughout the family of Schr\"odinger equations is given by the Hellmann-Feyman theorem. An exact integral relationship, involving a pair of external parameters, is derived from the Hellmann-Feynman theorem using assumptions which are not unduly restrictive. This relationship is used to arrive at trial (variational) wave functions for quantum mechanical problems which are not exactly soluble. The method is demonstrated in two sample problems. A technique for progressively improving variational calculations is discussed.
The limit theorems of discrete- and continuous-time quantum walks on the line have been intensively studied. We show a relation among limit distributions of quantum walks, Heun differential equations and Gauss differential equations. Indeed, we derive the second-order Fucksian differential equations which limit density functions of quantum walks satisfy. Moreover, using both differential equations, we discuss a relationship between discrete- and continuous-time quantum walks. Taking suitable limit, we can transform a Heun equation obtained from the limit density function of the discrete-time quantum walk to a Gauss equation given by that of the continuous-time quantum walk.
The stationary Schrodinger equation depending on spatial coordinates has been considered. The problem of obtaining a differential relationship for the wave function squared was posed. By extracting Schrodinger's equation itself from this relationship a differential equation for a physically interpretable quantity, i.e. the probability density (wave function squared), has been formulated. As an example the one-dimensional case admitting a simple analytic solution was considered. The solution obtained is shown to be a solution squared of the corresponding nonlinear differential equation for the probability density. In the final section a more general non-stationary case was considered for the potential involving a time-dependent term, such potentials are found in the non-stationary perturbation theory. The constant in separating the variables remains real. Thus the procedure considered proves to be similar to that presented above for the stationary equation.