Pauli equation for joint tomographic probability distribution of spin 1/2 particle

TL;DR

This paper introduces a new vector optical tomogram for spin 1/2 particles, derives evolution equations, and applies these to analyze quantum systems in electromagnetic fields and quantum oscillators.

Contribution

It develops a comprehensive vector tomographic framework for spin 1/2 particles, including evolution equations and applications to magnetic fields and quantum oscillators.

Findings

Derived evolution equations for vector optical and symplectic tomograms.

Obtained the propagator for a charged spin 1/2 particle in a magnetic field.

Analyzed the evolution of entangled states in the vector optical representation.

Abstract

The positive vector optical tomogram fully describing the quantum state of spin 1/2 particle without any redundancy is introduced. Reciprocally the vector symplectic tomogram and vector quasidistributions $\vec W({\mathbf q},{\mathbf p})$, $\vec Q({\mathbf q},{\mathbf p})$, $\vec P(\vec\alpha)$ are introduced. The evolution equations for proposed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian are obtained. The quantum system of charged spin 1/2 particle in arbitrary electro-magnetic field is considered in proposed representations and evolution equations which are analogs of Pauli equation are obtained. The propagator of evolution equation in the case of homogeneous and stationary magnetic field in Landau gauge is found and the evolution of initial entangled superposition of lower Landau levels in the vector optical representation is considered. The system of linear quantum oscillator with spin in vector optical tomography representation is considered and the evolution of initial entangled superposition of two lower Fock states and spin-up, spin-down states is studied in this representation.