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

拓也町田

数理解析研究所講究録Jun 1, 20130 citations

Exact Integral Relationship in Quantum Mechanics: Application to the Variational Method

Edwin M. Roberts

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.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.

Physical ReviewNov 1, 19622 citations

Quantum Stochastic Paths and Stochastic Differential Equations

Kazuo Takatsuka

Springer eBooksJan 1, 20260 citations

A random walk model for the Schrödinger equation

Wolfgang Wagner

Mathematics and Computers in SimulationAug 9, 20164 citations

The Heun differential equation and the Gauss differential equation related to quantum walks

Norio Konno, Takuya Machida, Tohru Wakasa

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 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.

arXiv:1109.2662Sep 13, 20116 citations

Lagrangian form of stochastic equations and quantum theory

L. de la Peña-Auerbach, Ana Marı́a Cetto

Physics Letters AJul 1, 196910 citations

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

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