Classical trajectories from the zeros of the quantum potential: the 2D isotropic harmonic oscillator

Abstract

In the first part of this work, using the quantum potential approach, we show that a solution to the time-independent Schrödinger equation determines a subset of classical solutions, only if the region corresponding to the zeroes of the quantum potential is tangent to the caustic region determined by the classical trajectories. Thus, the solutions of the time-independent Schrödinger equation, according to their caustic and the zeros of the quantum potential, can be classified in three different cases given by the following conditions: the two regions coincide, they are tangent at certain subset of points, and the two regions are not tangent at any point. In the second part, as examples of the first type of wave functions, we present the solutions of the Schrödinger equation for the 2D isotropic harmonic oscillator, which are eigenfunctions of both the Hamiltonian and the angular momentum operators. That is, we show that for this family of solutions, the zeroes of the quantum potential coincide with the caustic region. Furthermore, we find that the classical trajectories, determined from the quantum ones and the zeroes of the quantum potential, conform to a family of elliptical curves for a particle with energy, (2n + l + 1)ℏ ω, and orbital angular momentum l ℏ.