A remark on the integration of Schrödinger equation using quantum Itô's formula (Q1059330)

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.