The Velocity Field of a Quantum Particle in the Case of Real Wavefunctions: The Tunnel Effect in a Box and a Three-Dimensional Generalization of the Model

In previous papers, a generalized kinematic momentum was introduced in standard quantum mechanics, in order to address Einstein’s objection concerning the vanishing Bohmian velocity related to real stationary wave functions. The model was applied successfully to the one-dimensional case of a particle in a box and of a bouncing ball solving a nonlinear Schrödinger equation of the Ermakov–Pinney type. In this paper we develop the formalism giving a three-dimensional generalization of the model. In the stationary regime, the associated kinematic current is divergence free and can be locally represented as the curl of an auxiliary vector field C. We consider, as an application of the three-dimensional generalization, the case of the Hydrogen atom. Finally, a possible experimental test of the model is proposed by applying the formalism to an infinite box with an internal finite potential step. This example suggests a possible experimental setting in which the presence of a non-vanishing Ermakov–Pinney kinematic current could produce deviations from the standard predictions of the linear quantum theory.