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Quantum Reports

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

A. Feoli, Elmo Benedetto, Alessandro Cesta, Luca D’Errico, Antonella Lucia Iannella

Published Oct 26, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.3390/quantum8040110

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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.
When the solution of the time independent Schrödinger equation is a real function, the velocity of the associated quantum particle is vanishing in Bohmian Mechanics. We can obtain a moving particle if we consider that its velocity can be related not to the phase of the wave function but to its amplitude. By using the rules of standard quantum mechanics, we have constructed a model generalizing the momentum operator and modifying the corresponding Schrödinger equation. We obtain a new nonlinear equation of the form studied by Ermakov and Pinney and we apply the model to the cases of the particle in a box and of a bouncing ball. The new formula of quantum velocity and the corresponding motion of quantum particles have an immediate correspondence with the behavior of the bodies in classical mechanics.
In the de Broglie–Bohm interpretation of quantum mechanics, the momentum of a particle in a box of perfectly reflecting walls is vanishing. On the contrary, the classical physics predicts an oscillatory motion between the walls. The resulting difference between the behavior of a body in classical and quantum regime is unsatisfactory for Einstein and many other scientists. In order to solve the problem, we show that it is possible to obtain a motion of the body using the rules of quantum mechanics. The result holds for the quantum systems described by solutions of the time independent Schrödinger equation that are real functions.
Many-body quantum-mechanical stationary states that have real valued wavefunctions are shown to satisfy a classical conservation of energy equation with a kinetic energy function. The terms in the equation depend on the probability distribution, and, in addition, pressure and velocity functions, but these functions also depend on the probability distribution. There are two possible directions of the velocity that satisfy the energy equation. A linear momentum function is defined that integrates to zero, and this property is consistent with the expectation value of the linear momentum for stationary states with real-valued wave functions. The energy equation is integrated to obtain a version of the well known energy equation involving reduced density matrices, where the kinetic energy functional of the one-particle density matrix is replaced by a function of the electron density and a velocity function. Also, the noninteracting kinetic energy functional from the Hohenberg--Kohn theorem is given as an explicit functional of the orbital densities. For the purpose of describing the behavior of particles in a stationary state, a model based on the energy equation is constructed. The model is evaluated for the two different velocity directions using the grounds state of the particle in a one-dimensional box and the hydrogen atom. For one velocity direction, equations of motions with contradictory properties are obtained, and, in the other, an unstable system is found. A discussion is given with suggestions of additional elements that might improve the model.
Abstract In this chapter, the mathematical machinery of quantum mechanics is further developed in order to address real-world 3-dimensional physics. 3-dimensional vector notation is used for quantum mechanical operators and the Schrödinger equation is presented in this notation. The density of states of a particle in a box is considered. The angular momentum operators are defined. The eigenfunctions of the Laplacian are found. The Schrödinger equation with a spherical potential is analysed and solved for a Coulomb potential. The spectroscopy of the hydrogen atom is discussed. The spin operators are introduced. The Stern–Gerlach experiment and the Zeeman effect are discussed. The quantum mechanics of identical particles is considered and fundamental particles are shown to behave as either bosons or fermions depending on their spin. The action and the Feynman path integral are shown to offer an alternative approach to quantum mechanics that elucidates the connection between quantum and classical physics.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.
The Ermakov Pinney equation and its associated invariant are shown to arise naturally in stationary quantum mechanics when the Schrodinger equation is expressed in Bohm Madelung form and the Hamiltonian is diagonal and separable. Under these conditions, the stationary continuity constraint induces a nonlinear amplitude equation of Ermakov Pinney type in each degree of freedom, revealing a hidden invariant structure that is independent of whether the evolution parameter is time or space. By reformulating the separated stationary equations in Sturm Liouville form and applying Liouville normalization, we demonstrate that the quantum potential is encoded as a curvature contribution of the self adjoint operator rather than appearing as an additional dynamical term. This correspondence preserves the standard probabilistic predictions of quantum mechanics while yielding exact stationary Bohmian amplitudes and their associated invariants. The resulting invariant-based formulation provides stationary guiding fields and clarifies the ontological status of Bohmian amplitudes as geometrically encoded structures rather than auxiliary dynamical additions. The results further show that stationary constrained Bohm Madelung systems naturally admit variational formulations whose extremals preserve the Ermakov Lewis invariant.