Nodal theorems for the Dirac equation in d >= 1 dimensions

TL;DR

This paper proves nodal theorems for the Dirac equation in one and higher dimensions, relating the number of nodes in the wave function components, and generalizes classic 3D results to arbitrary dimensions.

Contribution

It establishes new nodal theorems for the Dirac equation in any dimension, extending previous 3D results to $d eq 3$ and providing explicit relationships between wave function nodes.

Findings

Nodal theorems relate upper and lower component nodes in Dirac spinors.

Results apply to $d=1$ and $d>1$ dimensions with specific node relationships.

Includes graphical examples of Dirac spinor orbits.

Abstract

A single particle obeys the Dirac equation in $d \ge 1$ spatial dimensions and is bound by an attractive central monotone potential that vanishes at infinity. In one dimension, the potential is even, and monotone for $x\ge 0.$ The asymptotic behavior of the wave functions near the origin and at infinity are discussed. Nodal theorems are proven for the cases $d=1$ and $d > 1$, which specify the relationship between the numbers of nodes $n_1$ and $n_2$ in the upper and lower components of the Dirac spinor. For $d=1$, $n_2 = n_1 + 1,$ whereas for $d >1,$ $n_2 = n_1 +1$ if $k_d > 0,$ and $n_2 = n_1$ if $k_d < 0,$ where $k_d = \tau(j + \frac{d-2}{2}),$ and $\tau = \pm 1.$ This work generalizes the classic results of Rose and Newton in 1951 for the case $d=3.$ Specific examples are presented with graphs, including Dirac spinor orbits $(\psi_1(r), \psi_2(r)), r \ge 0.$