The Three-Dimensional Quantum Hamilton-Jacobi Equation and Microstates

Abstract

In a stationary case and for any potential, we solve the three-dimensional quantum Hamilton-Jacobi equation in terms of the solutions of the corresponding Schrödinger equation. Then, in the case of separated variables, by requiring that the conjugate momentum be invariant under any linear transformation of the solutions of the Schrödinger equation used in the reduced action, we clearly identify the integration constants successively in one, two and three dimensions. In each of these cases, we analytically establish that the quantum Hamilton-Jacobi equation describes microstates not detected by the Schrödinger equation in the real wave function case.

Dual wavefunctions in two–dimensional quantum mechanics

“…solves the same Schrödinger equation (6). For that quantum potential, the duality of the wavefunction solutions of Eq.…”

“…The first solution corresponds to a function f = √ 2mαz, whereas the second one is for f = −i √ 2mαz. It is straightforward to show by inspection that the dual wavefunctions (29) satisfy Schrödinger equation (6). These solutions are single-valued for θ → θ + 2π, describing non-relativistic spinless particles.…”

“…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

“…(12), it is not the only possible solution, and as it is stated in Ref. [16], another possible solution of the 3D-QSHJE is of the form…”

“…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.…”