Dirac Hamiltonian

The Dirac Hamiltonian is the first-order time-evolution operator obtained from the Dirac equation. For a free massive particle,where the Dirac matrices satisfy $\{{\alpha }_i,{\alpha }_j\}=2{\delta }_{ij}I$, $\{{\alpha }_i,\beta \}=0$ and ${\beta }^2=I$. These identities give $H^2=c^2{P}^2+m^2{c}^4$, recovering the relativistic energy-momentum relation.

Minimal electromagnetic coupling replaces the canonical momentum by $\mathbf{P}-q\mathbf{A}$ and adds $q\varphi$, in SI units. The wavefunction has spinor components, and the free operator has positive- and negative-energy branches. The Dirac field interprets particle and antiparticle excitations with a stable vacuum; it is more complete than treating the negative branch as an ordinary one-particle spectrum with no lower bound.

The two-component model $H=c(p_x{\sigma }_x+p_y{\sigma }_y)+V{\sigma }_z$ has energies $E_s=s\sqrt{c^2{p}^2+V^2}$. The sign of $V$ changes eigenstates and spin polarization but not the uniform energy dispersion at fixed $|V|$. The Pauli matrices may represent actual spin or an effective pseudospin, depending on the physical realization.