We present a derivation of the Schr odinger equation for a path integral of a point particle in a space with curvature and torsion which is considerably shorter and more elegant than that commonly found in the literature. 1.
In this paper we present a simplification of the path integral solution of the Schrödinger equation in terms of coordinates which need not be Cartesian. After presenting the existing formula, we discuss the relationship between the distance and time differentials. Making this relationship precise through the technique of stationary phase, we are able to simplify the path integral. The resulting expression can be used to obtain a Hamiltonian path integral. Finally, we comment on a similar phenomenon involving differentials in the Itô integral.
The Schrödinger equation is derived classically assuming that particles present local \nrandom spatial fluctuations compatible with the presence of the zero-point field. With- \nout specifying the forces arising from this permanent matter-field interaction but ex- \nploring its fundamental properties (homogeneity, isotropy and random aspect) to justify \nthe emergence of the continuity equation in one-particle context, these fluctuations are \ndescribed in terms of the probability density. Specifically, the starting point is the as- \nsumption that the local activities, which turn the path followed by the particle totally \nunpredictable, must be associated with an energy proportional to \n@ \nP \n=@ \nt \n. The polar form \nof the wave function, which connects the obtained classical equations with the corre- \nsponding quantum equation, emerges as a by-product of the approach.
In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D 3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D 3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D 3d-II. I solve the path integral on D 3d-I in the (u, v, w) system and on D 3d-II in the (u, v, w) system and in spherical coordinates.