The Analytic Eigenvalue Structure of the 1+1 Dirac Oscillator

Abstract

We study the analytic structure for eigenvalues of the one-dimensional Dirac oscillator, by analytically continuing its frequency on the complex plane. A twofold Riemann surface is found connecting the two states of a pair of particle and antiparticle. One can, at least in principle, accomplish the transition from a positive energy state to its antiparticle state, by moving the frequency continuously on the complex plane, without changing the Hamiltonian after transition. This result provides a visual explanation for the absence of a negative energy state with the quantum number n = 0.

Dynamical Algebras in the 1+1 Dirac Oscillator and the Jaynes-Cummings Model

“…where a † = mω 2 (x − i mω p) and a = mω 2 (x + i mω p) are the raising and lowering operators of a nonrelativistic harmonic oscillator. The 1D DO can be regarded as a coupled quantum system composed by the harmonic oscillator and a spin, and the coupling vanishes in the nonrelativistic limit [27]. This interaction leads to the case that the number operator N = a † a, which is a good quantum number of the harmonic oscillator, is no longer conserved.…”

Dynamical Algebras in the 1+1 Dirac Oscillator and the Jaynes–Cummings Model*

“…The 1D DO can be regarded as a coupled quantum system composed by the harmonic oscillator and a spin, and the coupling vanishes in the nonrelativistic limit. [27] This interaction leads to the case that the number operator 𝑁 = 𝑎 † 𝑎, which is a good quantum number of the harmonic oscillator, is no longer conserved. It is noteworthy that the matrix (7) can be obtained by replacing the first momentum with the coordinate corresponding to the second momentum.…”