Stationary path integrals formalism for the one-dimensional time-independent Schroedinger equation

Ashmead, John

In non-relativistic quantum mechanics, path integrals are normally derived from the Schroedinger equation. This assumes the two formalisms are equivalent. Since time plays a very different role in the Schroedinger equation and in path integrals, this may not be the case. We here derive path integrals directly by imposing two requirements: correct behavior in the classical limit and the most complete practicable symmetry between time and space. With these requirements, the path integral formalism predicts quantum fluctuations over the time dimension analogous to the quantum fluctuations seen over the three space dimensions. For constant potentials there is no effect. But the coupling between rapidly varying electromagnetic fields and the quantum fluctuations in time should be detectable. We consider a variation on the Stern-Gerlach experiment in which a particle with a non-zero electric dipole moment is sent through a rapidly varying electric field, oriented parallel to the particle's trajectory. The Schroedinger equation predicts changes to the precession frequency of the wave function about the trajectory but no physical splitting of the beam. With the approach here, path integrals predict the changes to the precession frequency and in addition that the beam will be split in velocity and time.

Atsushi Inoue

Hagen M. Kleinert

Sonia Mazzucchi

Infinite Dimensional Oscillatory Integrals Feynman Path Integrals and The Schroedinger Equation The Stationary Phase Method and the Semiclassical Limit of Quantum Mechanics Quantum Open Systems Alternative Approaches and Further Results.

Ewan M. Wright

Robert Alicki

The solution of the Markovian master equation for the quantum open system of n degrees of freedom is formally written in terms of a path integral and the stationary phase approximation is discussed. The exactly soluble models with generators quadratic in position and momentum operators are investigated and the explicit expressions for the space–time propagators for one-dimensional systems are derived.