A remark on the integration of Schrödinger equation using quantum Itô's formula
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.
quantum stochastic differential equations
scientific article; zbMATH DE number 3903690