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arXiv:1209.4285 · Physica Scripta

Tomographic probability representation in the problem of transitions between Landau levels

E D Zhebrak

Published Mar 1, 2013• 6 citations
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Abstract

The problem of moving of a charged particle in electromagnetic field is considered in terms of tomographic probability representation. The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments. The obtained results are generalized in the present work and are applied to determining the transition probabilities between Landau levels. Transition probabilities are calculated using the symplectic tomograms instead of the wave functions. The same method is used in obtaining the transition probabilities between the Landau levels possessed by a charge moving in varying electromagnetic field.

DOI
10.1088/0031-8949/2013/t153/014063

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The problem of moving of a charged particle in electromagnetic field is considered in terms of tomographic probability representation. The coherent and Fock states of a charge moving in varying homogeneous magnetic field are studied in the tomographic probability representation of quantum mechanics. The states are expressed in terms of quantum tomograms. The Fock state tomograms are given in the form of probability distributions described by multivariable Hermite polynomials with time-dependent arguments. The obtained results are generalized in the present work and are applied to determining the transition probabilities between Landau levels. Transition probabilities are calculated using the symplectic tomograms instead of the wave functions. The same method is used in obtaining the transition probabilities between the Landau levels possessed by a charge moving in varying electromagnetic field.
The symplectic tomograms of coherent states of a charged particle moving in a constant uniform magnetic field are obtained in explicit form. The tomograms are shown to coincide with normal probability distributions of two random variables. The means and dispersions of the variables are found and expressed in terms of means and dispersions of the charged particle coordinates and momenta. The characteristic function of the tomographic probability distribution is found. The center of mass tomogram of the coherent state of charge in magnetic field is also found and the relation of the symplectic tomogram and the center of mass tomogram is established.
The gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated. Explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained. Gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions of states of quantum system are introduced, and their evolution equations having the Liouville equation in corresponding representations as the classical limit are found.
Abstract The gauge invariance of the evolution equations of tomographic probability distribution functions of quantum particles in an electromagnetic field is illustrated. Explicit expressions for the transformations of ordinary tomograms of states under a gauge transformation of electromagnetic field potentials are obtained. Gauge-independent optical and symplectic tomographic quasi-distributions and tomographic probability distributions of states of quantum system are introduced, and their evolution equations have the Liouville equation in corresponding representations as the classical limits are found.
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 W⃗(q,p)\vec W({\mathbf q},{\mathbf p})W(q,p), Q⃗(q,p)\vec Q({\mathbf q},{\mathbf p})Q​(q,p), P⃗(α⃗)\vec P(\vecα)P(α) 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.