Graphene's honeycomb lattice structure is quite remarkable in the sense that it leads, in the long wavelength limit, to a massless Dirac equation description of nonrelativistic quasiparticles associated with electrons and holes present in the two dimensional crystallite. In the case of cold bosonic atoms trapped in a honeycomb optical lattice, Haddad and Carr (2009) have recently shown, by taking into account binary contact interactions, that the dynamics of these Bose-Einstein condensates is governed by a nonlinear Dirac equation (NLDE). In this paper, we study exact stationary solutions of such a NLDE. After proving that the energy eigenvalues are real, we show that the sum of orbital angular momentum and pseudospin angular momentum normal to the crystal commutes with the nonlinear Hamiltonian whenever magnitudes of the pseudospin components do not depend on the polar angle $ϕ$ . We obtain some exact stationary and localized solutions of the NLDE.Graphene's honeycomb lattice structure is quite remarkable in the sense that it leads, in the long wavelength limit, to a massless Dirac equation description of nonrelativistic quasiparticles associated with electrons and holes present in the two dimensional crystallite. In the case of cold bosonic atoms trapped in a honeycomb optical lattice, Haddad and Carr (2009) have recently shown, by taking into account binary contact interactions, that the dynamics of these Bose-Einstein condensates is governed by a nonlinear Dirac equation (NLDE). In this paper, we study exact stationary solutions of such a NLDE. After proving that the energy eigenvalues are real, we show that the sum of orbital angular momentum and pseudospin angular momentum normal to the crystal commutes with the nonlinear Hamiltonian whenever magnitudes of the pseudospin components do not depend on the polar angle $ϕ$. We obtain some exact stationary and localized solutions of the NLDE.

arXiv:1012.0976Dec 5, 20101 citation

Exact propagating Dirac wave packets in an attractive Coulomb-like potential

Siddhant Das

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

A comment on the deduction of the Dirac equation

J. Buitrago

European Journal of PhysicsApr 1, 19881 citation

The Dirac Equation

WORLD SCIENTIFIC eBooksApr 5, 20220 citations

The Dirac Equation

UMBERTO LOMBARDO, G. Giuliani, Yifei Niu

UNITEXT for physicsJan 1, 20240 citations

ϕ

. We obtain some exact stationary and localized solutions of the NLDE.

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.