Motion of two-dimensional quantum particle under a linear potential in the presence of Rashba and Dresselhaus spin–orbit interactions

The quantum mechanical problem of a particle moving in the presence of electric field and Rashba and/or Dresselhaus spin–orbit interactions (SOIs) is solved exactly. Existence of the SOI removes the spin degeneracy, yielding two coupled Schrödinger equations for spin-up and spin-down spinor eigenfunctions. Fourier-transform of these equations provides two coupled first-order differential equations, which are shown to be reduced to two decoupled Hermite-like equations with complex coefficients. The convergent solutions of the equations are found to be described in the form of so called Hermite series. The recurrence relations and the integral representation of the Hermite series are provided. In the absence of Rashba and Dresselhaus spin–orbit coupling constants, these interconnected equations are decoupled and reduced to two Schrödinger equations for a quantum particle moving in a linear potential, solution of which is described by Airy functions.