The Quantum Reduced Action in Higher Dimensions

Abstract

The solution with respect to the reduced action of the one-dimensional stationary quantum Hamilton-Jacobi equation is well known in the literature. The extension to higher dimensions in the separated variable case was proposed in contradictory formulations. In this paper we provide new insights into the construction of the reduced action. In particular, contrary to the classical mechanics case, we analytically show that the reduced action constructed as a sum of one variable functions does not contain a complete information about the quantum motion. In the same context, we also make some observations about recent results concerning quantum trajectories. Finally, we will examine the conditions in which microstates appear even in the case where the wave function is complex.

Dual wavefunctions in two–dimensional quantum mechanics

“…For that quantum potential, the duality of the wavefunction solutions of Eq. ( 6) correspond to the interchangeability between the amplitude and the phase in wavefunctions ( 5) and (7). The family of potentials which allows this kind of wavefunction solutions is composed of couples of related potentials V 1 (x, y) and V 2 (x, y) [11], such that the general solutions to their respective Schrödinger equation ( 6) may be written in terms of the solutions to the other Schrödinger equation…”

“…The theory described by Eqs. ( 3) and ( 4), is also called Quantum Hamilton-Jacobi theory, and has been extensively studied in the context of their differences with respect to Hamilton-Jacobi equations [2][3][4][5][6][7][8][9][10].…”

Reduced quantum actions for symmetrical and some separated variables potentials cases

“…For this case, the symmetry of the potential is spherical and we proposed to separate the total reduced action into the sum of three 1D actions [13,14,15]. However, we have been criticized by A. Bouda [16,17] in our approach of the symmetrical potential problems. In fact we considered mainly that for these cases, the 3D reduced action can be written as the sum of three 1D reduced actions each one depending on one variable.…”

“…We totally agree with that. Furthermore, for the authors of [17] there are fifteen integration constants completely independents, one of them is an additive one. In Ref.…”

“…In Ref. [17], it was shown that if one considers a solution of the form given in Eq. ( 13), excluding the additive constant, not all of the sixteen constants do play the role of integration constants and two of them have to be fixed.…”