Revisiting the Schrodinger probability current

TL;DR

This paper revisits the probability current in quantum mechanics, demonstrating non-zero currents in stationary states and emphasizing the importance of spin-dependent contributions for accurate descriptions.

Contribution

It shows that stationary wave functions have non-zero probability currents and highlights the necessity of including the Gordon current for a complete description.

Findings

Stationary wave functions of hydrogen and harmonic oscillator have non-zero probability currents.

The probability current exhibits circular rotation in these systems.

Adding the Gordon current accounts for spin-dependent effects in the probability current.

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.