A remark on the integration of Schrödinger equation using quantum Itô's formula
type: Item
instance of: scholarly article
title: A remark on the integration of Schrödinger equation using quantum Itô's formula
zbMATH Open document ID: 0566.60064
DOI: 10.1007/BF00402238
author: K. R. Parthasarathy
published in: Letters in Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: The quantum mechanical method of taking expectations of observables with respect to density matrices leads in a natural manner to a theory of non- commutative integration and quantum stochastic differential equations which turns out to be simpler than and, in several ways, a generalization of the classical Itô calculus with respect to Brownian motion and Poisson process. Using the quantum version of Itô's formula in such a calculus an explicit formula for the Schrödinger one-parameter unitary group \(\exp -it(H_ 0(p)+V(q))\) is obtained when \(H_ 0\) is a function of the momentum variables \(p=(p_ 1,...,p_ n)\), \(p_ j=-i(\partial /\partial x_ j)\), \(1\leq j\leq n\), \(q=(q_ 1,...,q_ n)\), \(q_ j\) being multiplication by the position variables, \(1\leq j\leq n\), and V is the Fourier transform of a complex valued totally finite measure on \({\mathbb{R}}^ n.\) This formula is to be considered as a technique of separating the noncommuting variables p and q. Whereas the corresponding classical formula using Feynman integrals is not based on nonnegative integration the present one is based on nonnegative but noncommutative integration.
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 47D99
zbMATH DE Number: 3903690
zbMATH Keywords: non-commutative integration
zbMATH Keywords: quantum stochastic differential equations
zbMATH Keywords: quantum version of Itô's formula
zbMATH Keywords: Schrödinger one-parameter unitary group
zbMATH Keywords: Feynman integrals
MaRDI profile type: Publication
cites work: Quantum Ito's formula and stochastic evolutions
full work available at URL: https://doi.org/10.1007/bf00402238
OpenAlex ID: W2067248102
description: scientific article; zbMATH DE number 3903690