Quantum-classical correspondence of the Dirac equation with a scalar-like potential

Antonio S. de Castro, Marcelo Hott

Ade Kurniawan, Suparmi Suparmi, C. Cari

An approximate solution of the Dirac equation for a spin-1/2 particle under the influence of q -deformed hyperbolic Pöschl–Teller potential combined with trigonometric Scarf II non-central potential is studied analytically. It is assumed that the scalar potential equals the vector potential in order to obtain analytical solutions. Both radial and angular parts of the Dirac equation are solved using the Nikiforov–Uvarov method. A relativistic energy spectrum and the relation between quantum numbers can be obtained using this method. Several quantum wave functions corresponding to several states are also presented in terms of the Jacobi Polynomials.

Wawrzyniec Guz

Yukihisa Nogami, F. Masafumi Toyama

The Dirac equation in one dimension with a Lorentz scalar potential is associated with a supersymmetric pair of Schr\"odinger Hamiltonians ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$. The ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$ share the same energy spectrum and scattering phases. The shared spectrum includes the lowest states unless the Dirac equation allows a zero mode (a zero-energy bound state). This situation is unlike the common examples of supersymmetric quantum mechanics. The Dirac equation admits a zero mode only if the scalar potential has certain ``topology.'' Various such features are illustrated through explicit examples. In particular, the phase equivalence between ${\mathit{H}}_{1}$ and ${\mathit{H}}_{2}$ is exploited to construct transparent potentials for the Dirac equation in one dimension.

Jens Bolte, Stefan Keppeler

Zhiguo Zhang

Exact solutions to the Dirac equations with a time-dependent mass and a static magnetic field or a time-dependent linear potential are given. Matrix elements of the coordinate, momentum and velocity operator are calculated. In the large quantum number limit, these matrix elements give the classical solution.