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Bulletin of the Lebedev Physics Institute

Sudden change in the equilibrium position and oscillator quantum transitions in the tomographic representation

Elena D. Zhebrak, Vladimir I. Man’ko

Published Nov 1, 2014• 1 citation
Publisher

Abstract

The transition probabilities during instantaneous change in the quantum oscillator equilibrium position are studied in the tomographic representation. A comparison with the known Franck-Condon factors obtained by calculating the wave function overlap integrals is performed. Explicit expressions for symplectic and optical tomograms are derived, as well as the generating function for overlap integrals of symplectic tomograms defining the transition probabilities in the oscillator, caused by the driving force.

DOI
10.3103/s1068335614110074

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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.
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The relation of theWigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner–Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of integrals with the state tomographic probability distributions is constructed having a standard form of averaging in the probability theory. New uncertainty relations for the position and momentum are written in terms of optical tomograms suitable for directexperimental check. Some recent experiments on checking the uncertainty relations including the entropic uncertainty relations are discussed.
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