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Open Quantum Systems Decoherence Quantum Simulation

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

arXiv
Authors: Richard L. Hall, Petr Zorin

Year

2013

Paper ID

32848

Status

Preprint

Abstract Read

~2 min

Abstract Words

139

Citations

N/A

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 xge 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 n1 and n2 in the upper and lower components of the Dirac spinor. For d=1, n2 = n1 + 1, whereas for d >1, n2 = n1 +1 if kd > 0, and n2 = n1 if kd < 0, where kd = τ\(j + frac{d-2}{2}\), and τ= 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 \(ψ1(r\), ψ2(r)), r ge 0.

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  • 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.

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References & Citation Signals

[1] DOI https://doi.org/arXiv:1309.1749 [2] arXiv https://arxiv.org/abs/1309.1749 [3] Publisher https://arxiv.org/abs/1309.1749

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