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Quantum Reports

A Normalization Condition for the Probability Current in Some Remarkable Cases

A. Feoli, Elmo Benedetto, Antonella Lucia Iannella

Published Apr 23, 2024• 3 citations
Publisher

Abstract

Starting from the dynamics of a bouncing ball in classical and quantum regime, we have suggested in a previous paper to add an arbitrary function of time to the standard expression of the probability current in quantum mechanics. In this paper, we suggest a way to determine this function: imposing a suitable normalization condition. The application of our proposal to the case of the harmonic oscillator is discussed.

DOI
10.3390/quantum6020012

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Starting from the dynamics of a bouncing ball in classical and quantum regime, we have suggested in a previous paper to add an arbitrary function of time to the standard expression of the probability current in quantum mechanics. In this paper, we suggest a way to determine this function: imposing a suitable normalization condition. The application of our proposal to the case of the harmonic oscillator is discussed.
We revisit the definition of the probability current for the Schrodinger equation. First, we prove that the Dirac probability currents of stationary wave functions of the hydrogen atom and of the isotrop harmonic oscillator are not nil and correspond to a circular rotation of the probability. Then, we recall how it is necessary to add to classical Pauli and Schrodinger currents, an additional spin-dependant current, the Gordan current. Consequently, we get a circular probability current in the Schrodinger approximation for the hydrogen atom and the isotrop harmonic oscillator.
An earlier study of quantum-mechanical time has suggested that, from the quantum-theoretic point of view, time is a random variable. The present work explores how such a description of time can be used to bring greater detail into the analysis of the one-dimensional harmonic oscillator. Specifically, the classical equations for harmonic motion are used as a point transformation between phase space and time-energy space, permitting probability distributions in the two spaces to be different representations of the same thing. The familiar energy-quantization hypothesis is supplemented with the hypothesis that quantized energy values are means (averages) of energy distributions characteristic of each energy state. With the probability distribution of time determined from the previously cited study, the desired quantized energy distributions follow from the requirements: (1) that they should be normalized, non-negative, and with means equal to the quantized energy values; (2) that the equivalent representation of distribution in position-momentum space should have marginal densities that are proportional (except near the origin) to the densities in position and momentum, ${|\ensuremath{\phi}(q)|}^{2}$ and ${|\ensuremath{\psi}(p)|}^{2}$, which are postulated in quantum theory. Requirement (2) is a weakening of the Born postulate. In the time-energy space, the probability distributions exclude energies smaller than the mode (energy value with maximum probability). In position-momentum space this exclusion requirement becomes the dynamical restriction: $\frac{{q}^{2}m{\ensuremath{\omega}}^{2}}{2}+\frac{{p}^{2}}{2m}\ensuremath{\ge}\frac{{\ensuremath{\Lambda}}_{n}\ensuremath{\hbar}\ensuremath{\omega}}{2}$, where, for the first three energy states, ${\ensuremath{\Lambda}}_{0}=0$, ${\ensuremath{\Lambda}}_{1}=\frac{3}{2}$, ${\ensuremath{\Lambda}}_{2}=2+\ensuremath{\surd}\frac{3}{2}$. In general terms it is thus shown that the quantized harmonic oscillator can be described as a statistical ensemble of classical oscillators. Some comparisons with work on phase-space formulations of quantum mechanics (especially with the work of Wigner) are made and some aspects of the theory that should submit to experimental verification are pointed out.
As an application of the recently proposed Langevin formulation of quantum dynamics, we quantize the time-dependent harmonic oscillator. In this way, we are able to explicitly compute the quantum-mechanical propagator. At variance with previous results, the expression found in this letter contains no auxiliary function to be determined by solving (as yet unsolved) non-linear differential equations, but it is written in terms of two series which arise quite naturally within the present formalism. Our strategy is compared with ordinary time-dependent perturbation theory.
Starting from the standard current density of probability for a quantum particle in a potential V(r), the total probability current in a stationary state is considered and investigated. Its physical interpretation leads to a natural definition of the time period for such a quantum stationary state. A simple and elegant formula relates the total current to the energy level of the state. This formula is shown to be consistent with the Correspondence Principle. The results are applied to the hydrogen atom and the harmonic oscillator.