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arXiv:quant-ph/0304055

Revisiting the Schrodinger probability current

Michel Gondran, Alexandre Gondran

Published Apr 8, 2003• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0304055

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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.
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.
This paper analyzes how the existence of electron spin changes the equation for the probability current density in the quantum-mechanical continuity equation. A spinful electron moving in a potential energy field experiences the spin-orbit interaction, and that additional term in the time-dependent Schrödinger equation places an additional spin-dependent term in the probability current density. Further, making an analogy with classical magnetostatics hints that there may be an additional magnetization current contribution. This contribution seems not to be derivable from a non-relativistic time-dependent Schrödinger equation, but there is a procedure described in the quantum mechanics textbook by Landau and Lifschitz to obtain it. We utilize and extend their procedure to obtain this magnetization term, which also gives a second derivation of the spin-orbit term. We conclude with an evaluation of these terms for the ground state of the hydrogen atom with spin-orbit interaction. The magnetization contribution is generally the larger one, except very near an atomic nucleus.
Abstract We review the literature on the Pauli equation and its current density, discussing the progression from the original phenomenological version of Pauli to its derivation by Lévy-Leblond from a linearization of the Schrödinger equation. It was established conclusively by Lévy-Leblond’s work that the spin of a spin-1/2 particle such as an electron is non-relativistic in nature, contrary to what was often stated following Dirac’s derivation of a relativistic wave equation, and his subsequent demonstration that Pauli’s spin interaction term appeared in the non-relativistic limit. In this limit, the Gordon decomposition of the associated probability current density was found to contain a spin-dependent term. Such a term does not follow, however, from the usual derivation of the current density from the Pauli equation, although various physically motivated but otherwise ad hoc explanations were put forward to account for it. We comment on the only exception to these of which we are aware implying the spin term in the current was in fact non-relativistic in nature. However, the earlier work of Lévy-Leblond had already shown, with no additional assumptions, that this term was a prominent feature of the current density derived from his equation. Hence, just as with the spin itself, the spin current was non-relativistic, claims to the contrary notwithstanding. We present a somewhat simplified derivation of the Lévy-Leblond equation and its current density, commenting on possibilities for experimental work that might indicate measurable consequences of the spin term in the current density. The presentation is at a level of quantum mechanics, electromagnetic theory, and mathematical methods typical of courses taken by first-year graduate students in physics, assuming some familiarity with Pauli spin matrices and the Dirac equation. Calculations are presented in detail, making the material suitable for use by teachers as supplementary problems for assignments, and development of theoretical concepts peculiar to the Pauli equation and its probability current density.
This research examines the probability percentage of the existence of the hydrogen atom's electron in the space surrounding the nucleus of the mentioned atom. From the calculations of the semi-classical Bohr atomic model, a specific radius for the hydrogen atom's electron was obtained (with good approximation a_0=5.3×〖10〗^(-10) m); However, after the wave property of particles was introduced and this property was extended to the electron, Schrödinger formulated the wave equation of matter and its probabilistic physical interpretation was proposed by Born. In this paper, by applying the Schrödinger equation to the hydrogen atom, we aim to determine the probability of finding the electron of the hydrogen atom at the Bohr radius, as well as at distances smaller and greater than it.
Applying the concept of quantum hydrodynamics on a hydrogen atom, one can derive an equation of motion which describes the time derivative of the mass current density for the relative motion of the electron and the proton. Since this derivation can be performed with the Ehrenfest theorem, we call this equation the Ehrenfest equation of motion. As quantum hydrodynamics is consistent with Schrödinger's quantum mechanics, the stationary s-wave functions satisfy both the Schrödinger equation and the Ehrenfest equation of motion. It is the purpose of this paper to prove that the stationary s-wave functions satisfy the Ehrenfest equation of motion. From the quantum hydrodynamical point of view, this result for s-wave functions can be interpreted that the Coulomb force density that attracts the electron to the proton is compensated by a quantum force density that is related to the dispersion of the probability density.
Properties of observables in the Pauli and Schrödinger theories and first order relativistic approximations to them are derived from the Dirac theory. They are found to be inconsistent with customary interpretations in many respects. For example, failure to identify the ’’Darwin term’’ as the s−state spin−orbit energy in conventional treatments of the hydrogen atom is traced to a failure to distinguish between charge and momentum flow in the theory. Consistency with the Dirac theory is shown to imply that the Schrödinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schrödinger theory is discussed. Conservation laws of the Dirac theory are formulated in terms of relative variables, and used to derive virial theorems and the corresponding conservation laws in the Pauli−Schrödinger theory.