We have previously developed a basic $U$ -matrix formalism and have been able to obtain explicit expressions for the $U$ matrix and the wave function of a quantum-mechanical system by solving the operator Schr\"odinger equation directly within the regime of the interaction picture [Phys. Rev. A 27, 1760 (1983)]. In our approach, we introduce the switching function ${S}_{w}$ which describes how the interaction Hamiltonian is introduced to the system. In the present investigation, we list the corresponding $U$ matrix and wave function pertaining to the Schr\"odinger picture. Under the special case where the unperturbed Hamiltonian and the Lagrangian commute, our wave function can be reduced to the form identical to that obtained from the path-integral method. A comparison of the path-integral theory and our $U$ -matrix theory is presented. The difference in the wave functions of the two theories lies in the fact that in the $U$ -matrix approach the Schr\"odinger equation is treated as an operator equation, whereas the key equation in the path-integral theory is taken as algebraic.We have previously developed a basic $U$-matrix formalism and have been able to obtain explicit expressions for the $U$ matrix and the wave function of a quantum-mechanical system by solving the operator Schr\"odinger equation directly within the regime of the interaction picture [Phys. Rev. A 27, 1760 (1983)]. In our approach, we introduce the switching function ${S}_{w}$ which describes how the interaction Hamiltonian is introduced to the system. In the present investigation, we list the corresponding $U$ matrix and wave function pertaining to the Schr\"odinger picture. Under the special case where the unperturbed Hamiltonian and the Lagrangian commute, our wave function can be reduced to the form identical to that obtained from the path-integral method. A comparison of the path-integral theory and our $U$-matrix theory is presented. The difference in the wave functions of the two theories lies in the fact that in the $U$-matrix approach the Schr\"odinger equation is treated as an operator equation, whereas the key equation in the path-integral theory is taken as algebraic.

Physical Review AMay 1, 19843 citations

Stochastic path integrals and open quantum systems

Walter T. Strunz

The general theory of path integral propagators for the solution of linear quantum state diffusion (LQSD) stochastic Schr\"odinger equations describing open quantum systems is developed. Both Hamiltonian and, where possible, Lagrangian path integrals are derived and their connection established. The Hamiltonian version turns out to be more suitable. The results also show how the stochastic terms in the LQSD equation introduce a weight functional under the path integral, thus restricting the set of contributing paths. The center of this weight functional is determined by the stochastic processes governing the LQSD equation. In general, this picture holds in a semiclassical limit only. Some peculiarities of stochastic path integrals are pointed out. We evaluate the stochastic path integral in closed form for soluble models, gaining further insight into the behavior of the solutions of the LQSD equation. \textcopyright{} 1996 The American Physical Society.The general theory of path integral propagators for the solution of linear quantum state diffusion (LQSD) stochastic Schr\"odinger equations describing open quantum systems is developed. Both Hamiltonian and, where possible, Lagrangian path integrals are derived and their connection established. The Hamiltonian version turns out to be more suitable. The results also show how the stochastic terms in the LQSD equation introduce a weight functional under the path integral, thus restricting the set of contributing paths. The center of this weight functional is determined by the stochastic processes governing the LQSD equation. In general, this picture holds in a semiclassical limit only. Some peculiarities of stochastic path integrals are pointed out. We evaluate the stochastic path integral in closed form for soluble models, gaining further insight into the behavior of the solutions of the LQSD equation. \textcopyright{} 1996 The American Physical Society.

Physical Review AOct 1, 199687 citations

Applications of path integrals

Eugene P. Gross

Springer eBooksNov 23, 20052 citations

METHOD OF PATH INTEGRALS

WORLD SCIENTIFIC eBooksJan 1, 19810 citations

Path integral solution of the Schrödinger equation in curvilinear coordinates: A straightforward procedure

J. LaChapelle

A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.A new axiomatic formulation of path integrals is used to construct a path integral solution of the Schrödinger equation in curvilinear coordinates. An important feature of the formalism is that a coordinate transformation in the variables of the wavefunction does not imply a change of variable of integration in the path integral. Consequently, a transformation from Euclidean to curvilinear coordinates is simple to handle; there is no need to introduce ‘‘quantum corrections’’ into the action functional. Furthermore, the paths are differentiable: hence, issues related to stochastic paths do not arise. The procedure for constructing the path integral solution of the Schrödinger equation is straightforward. The case of the Schrödinger equation in spherical coordinates for a free particle is presented in detail.

Journal of Mathematical PhysicsSep 1, 19960 citations

We have previously developed a basic

U

-matrix formalism and have been able to obtain explicit expressions for the

U

matrix and the wave function of a quantum-mechanical system by solving the operator Schr\"odinger equation directly within the regime of the interaction picture [Phys. Rev. A 27, 1760 (1983)]. In our approach, we introduce the switching function

{S}_{w}

which describes how the interaction Hamiltonian is introduced to the system. In the present investigation, we list the corresponding

U

matrix and wave function pertaining to the Schr\"odinger picture. Under the special case where the unperturbed Hamiltonian and the Lagrangian commute, our wave function can be reduced to the form identical to that obtained from the path-integral method. A comparison of the path-integral theory and our

U

-matrix theory is presented. The difference in the wave functions of the two theories lies in the fact that in the

U

-matrix approach the Schr\"odinger equation is treated as an operator equation, whereas the key equation in the path-integral theory is taken as algebraic.