Exactly Solvable Models of Stochastic Quantum Mechanics within the Framework of Langevin-Schroedinger Type Equations

Nonrelativistic Quantum Mechanics with Fundamental Environment

“…However, the consideration of a reduced density matrix on a semi-group [20] is quite an ambiguous procedure and, moreover, its technical realization is possible, as a rule, only by using the perturbation method. For investigation of the inseparably linked closed system QSE, a new mathematical scheme has been proposed [5][6][7][8] which allows one to construct all important parameters of the quantum system and environment in a closed form. The main idea of the developed approach is the following.…”

“…A crucially new approach to constructing the quantum mechanics of a closed non-relativistic system QS+FE has been developed recently by the authors of [5][6][7][8], based on the principle of local equivalence of Schrodinger representation. More precisely, it has been assumed that the evolution of a quantum system is such that it may be described by the Schrödinger equation on any small time interval, while the motion as a …”

“…Thus, first of all we need such a generalization of quantum mechanics which includes nonperturbative vacuum as fundamental environment (FE) of a quantum system (QS). As our recent theoretical works have shown [4][5][6][7][8][9], this can be achieved by naturally including the traditional scheme of nonrelativistic quantum mechanics if we define quantum mechanics in the limits of a nonstationary complex stochastic differential equation for a wave function (conditionally named a stochastic Schrödinger equation). Indeed, within the limits of the developed approach it is possible to solve the above-mentioned traditional difficulties of nonrelativistic quantum mechanics and obtain a new complementary criterion which differs from de-Broglie's criterion.…”

Nonrelativistic Quantum Mechanics with Fundamental Environment

“…Now, using the system of equations (2.10) and correlation properties (2.11), it is easy to obtain the Fokker-Planck equation for the joint probability distribution of fields {} ξ (see in particular [6,10] …”