Rigorous Derivation of Traveling Dirac Wave Packets Within an Attractive Coulomb

We construct exact, positive-energy, normalizable wave-packet solutions of the Dirac equation in the axisymmetric potential $V=-\,v_0/ρ$ -- to our knowledge, the first such solutions in any external potential. Remarkably, one family comprises only elementary functions whose longitudinal profiles reproduce the free-Schrödinger Hermite--Gauss wave packets in the nonrelativistic limit. All packets share two striking features: (i) a probability density that is pointwise decoupled from spin orientation -- despite the inherent spin-orbit coupling of the Dirac equation -- and (ii) a complete freezing of their time evolution at the critical coupling $v_0\to\hbar c/2$ . We also present a simple scheme that maps solutions of the 2D Helmholtz equation to further exact Dirac wave packets.We construct exact, positive-energy, normalizable wave-packet solutions of the Dirac equation in the axisymmetric potential $V=-\,v_0/ρ$ -- to our knowledge, the first such solutions in any external potential. Remarkably, one family comprises only elementary functions whose longitudinal profiles reproduce the free-Schrödinger Hermite--Gauss wave packets in the nonrelativistic limit. All packets share two striking features: (i) a probability density that is pointwise decoupled from spin orientation -- despite the inherent spin-orbit coupling of the Dirac equation -- and (ii) a complete freezing of their time evolution at the critical coupling $v_0\to\hbar c/2$. We also present a simple scheme that maps solutions of the 2D Helmholtz equation to further exact Dirac wave packets.

quant-pharXiv:2606.18470Jun 18, 20260 citations

COHERENT WAVE-PACKETS AND SEMICLASSICAL APPROXIMATION IN THE COULOMB POTENTIAL

Michael Nauenberg

May 1, 19940 citations

We construct exact, positive-energy, normalizable wave-packet solutions of the Dirac equation in the axisymmetric potential

V=-\,v_0/ρ

-- to our knowledge, the first such solutions in any external potential. Remarkably, one family comprises only elementary functions whose longitudinal profiles reproduce the free-Schrödinger Hermite--Gauss wave packets in the nonrelativistic limit. All packets share two striking features: (i) a probability density that is pointwise decoupled from spin orientation -- despite the inherent spin-orbit coupling of the Dirac equation -- and (ii) a complete freezing of their time evolution at the critical coupling

v_0\to\hbar c/2

. We also present a simple scheme that maps solutions of the 2D Helmholtz equation to further exact Dirac wave packets.