Pauli Equation for a Joint Tomographic Probability Distribution

Abstract

We introduce the positive vector optical tomogram fully describing the quantum states of spin-1/2 particles without any redundancy. Correspondingly, we introduce the vector symplectic tomogram and vector quasidistributions (Wigner, Husimi, and Glauber–Sudarshan vector functions). We obtain the evolution equations for the proposed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian. In the proposed representations, we consider the quantum system of a charged spin-1/2 particle in an arbitrary electromagnetic field and obtain the evolution equations, which are analogs of the Pauli equation. As an example of the use of the proposed approach, we find the propagator of the evolution equation in the case of homogeneous and stationary magnetic fields in Landau gauge and consider the evolution of the initial entangled superposition of lower Landau levels in the vector optical representation. Also, as an example, we consider the system of linear quantum oscillator with spin in vector optical tomography representation, and study in this representation the evolution of the initial entangled superposition of two lower Fock states, as well as spin-up and spin-down states.