Abstract
It is shown that dynamical equations for quantum tomograms retain the normalizations of their solutions during time evolution only if the solutions satisfy a set of special conditions. These conditions are found explicitly. On the contrary, it is also illustrated that the classical Liouville equation, Moyal equation for Wigner function, and evolution equation for Husimi function retain normalizations of any initially normalized and quickly decaying at infinity functions on the phase space. Necessary and sufficient conditions for optical and symplectic tomogram-like functions to be tomograms of physical states are also discussed.
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Mancini, S., Man’ko, V.I., Tombesi, P.: Symplectic tomography as classical approach to quantum systems. Phys. Lett. A 213, 1 (1996)
Ibort, A., Man’ko, V.I., Marmo, G., Simoni, A., Ventriglia, F.: An introduction to the tomographic picture of quantum mechanics. Phys. Scr. 79, 065013 (2009)
Landau, L.D.: The damping problem in wave mechanics. Z. Phys. 45, 430 (1927)
Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749 (1932)
Husimi, K.: Some formal properties of the density matrix. Proc. Phys.-Math. Soc. Japan 22, 264 (1940)
Glauber, R.J.: Photon correlations. Phys. Rev. Lett. 10, 84 (1963)
Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10, 277 (1963)
Bertrand, J., Bertrand, P.: A tomographic approach to Wigner’s function. Found. Phys. 17, 397 (1987)
Vogel, K., Risken, H.: Determination of quasiprobability distributions in terms of probability distributions for the rotated quadrature phase. Phys. Rev. A 40, 2847(R) (1989)
Mancini, S., Man’ko, V.I., Tombesi, P.: Wigner function and probability distribution for shifted and squeezed quadratures. J. Opt. B: Quantum Semiclass. Opt. 7, 615 (1995)
Narcowich, F.J., O’Connell, R.F.: Necessary and sufficient conditions for a phase-space function to be a Wigner distribution. Phys. Rev. A 34, 1 (1986)
Ibort, A., Man’ko, V.I., Marmo, G., Simoni, A., Ventriglia, F.: On the tomographic picture of quantum mechanics. Phys. Lett. A 374, 2614 (2010)
Mancini, S., Man’ko, V.I., Tombesi, P.: Classical-like description of quantum dynamics by means of symplectic tomography. Found. Phys. 27, 801 (1997)
Korennoy, Ya.A., Man’ko, V.I.: Probability representation of the quantum evolution and energy-level equations for optical tomograms. J. Russ. Laser Res. 32, 74 (2011)
Korennoy, Ya.A., Man’ko, V.I.: Evolution equation of the optical tomogram for arbitrary quantum Hamiltonian and optical tomography of relativistic classical and quantum systems. J. Russ. Laser Res. 32, 338 (2011)
Mancini, S., Man’ko, O.V., Man’ko, V.I., Tombesi, P.: The Pauli equation for probability distributions. J. Phys. A: Math. Gen. 34, 3461 (2001)
Korennoy, Ya.A., Man’ko, V.I.: Pauli equation for a joint tomographic probability distribution. J. Russ. Laser Res. 36, 534 (2015)
Korennoy, Ya.A., Man’ko, V.I.: Evolution equation for a joint tomographic probability distribution of spin-1 particles. Int. J. Theor. Phys. 55, 4885 (2016)
Korennoy, Ya.A., Man’ko, V.I.: Observables, evolution equation, and stationary states equation in the joint probability representation of quantum mechanics. Int. J. Theor. Phys. 56, 1183 (2017)
Korennoy, Ya.A., Man’ko, V.I.: Gauge transformation of quantum states in probability representation. J. Phys. A: Math. Theor. 50, 155302 (2017)
Korennoy, Y. a. A.: Gauge-independent Husimi functions of charged quantum particles in the electro-magnetic field. arXiv:1806.06443 [quant-ph] (2018)
Smithey, D.T., Beck, M., Raymer, M.G., Faridani, A.: Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. Phys. Rev. Lett. 70, 1244 (1993)
Lvovsky, A.I., Raymer, M.G.: Continuous-variable optical quantum-state tomography. Rev. Mod. Phys. 81, 299 (2009)
Amosov, G.G., Korennoy, Ya.A., Man’ko, V.I.: Description and measurement of observables in the optical tomographic probability representation of quantum mechanics. Phys. Rev. A 85, 052119 (2012)
Man’ko, O.V., Man’ko, V.I.: Quantum states in probability representation and tomography. J. Russ. Laser Res. 18, 407 (1997)
Hillery, M., O’Connell, R.F., Scully, M.O., Wigner, E.P.: Distribution functions in physics: Fundamentals. Phys. Rep. 106, 121 (1984)
Man’ko, M.A., Man’ko, V.I.: Dynamic symmetries and entropic inequalities in the probability representation of quantum mechanics. AIP Conf. Proc. 1334, 217 (2011)
Man’ko, M.A., Man’ko, V.I., De Nicola, S., Fedele, R.: Probability representation and new entropic uncertainty relations for symplectic and optical tomograms. Acta Phys. Hung. B 26, 71 (2006)
De Nicola, S., Fedele, R., Man’ko, M.A., Man’ko, V.I.: New uncertainty relations for tomographic entropy: Application to squeezed states and solitons. Eur. Phys. J. B 52, 191 (2006)
Hirschman, I.I.: A note on entropy. Amer. J. Math. 79, 152 (1957)
Man’ko, O.V., Man’ko, V.I.: Classical mechanics is not the \(\hbar \to 0\) limit of quantum mechanics. J. Rus. Laser Res. 25, 477 (2004)
Korennoy, Ya.A., Man’ko, V.I.: Optical tomography of photon-added coherent states, even and odd coherent states, and thermal states. Phys. Rev. A 83, 053817 (2011)
Korennoy, Ya.A., Man’ko, V.I.: Entropic and information inequalities in the tomographic probability description of spin-1 particles. Bull. Lebedev Phys. Inst. 44, 106 (2017)
Kastler, D.: The C*-algebras of a free Boson field: I. Discussion of the basic facts. Commun. Math. Phys. 1, 14 (1965)
Loupias, G., Miracle-Sole, S.: C*-algèbres des systèmes canoniques. I. Commun. Math. Phys. 2, 31 (1966)
Loupias, G., Miracle-Sole, S.: C*-algèbre des systèmes canoniques. II. Ann. Inst. Henri Poincaré, 6, 39 (1967)
Ibort, A., Man’ko, V.I., Marmo, G., Simoni, A., Ventriglia, F.: A generalized Wigner function on the space of irreducible representations of the Weyl-Heisenberg group and its transformation properties. J. Phys. A: Math. Theor. 42, 155302 (2009)
Naimark, M.: Normed algebras. Wolters-Noordhoff, Gröningen (1972)
Korennoy, Ya.A., Man’ko, V.I.: Optical tomography of the distribution function of an ensemble of classical harmonic oscillators. J. Russ. Laser Res. 33, 84 (2012)
Korennoy, Ya.A., Man’ko, V.I.: Optical propagator of quantum systems in the probability representation. J. Russ. Laser Res. 32, 153 (2011)
Moyal, J.E.: Quantum mechanics as a statistical theory. Proc. Cambrige Philos. Soc. 45, 99 (1949)
Mizrahi, S.S.: Quantum mechanics in the Gaussian wave-packet phase space representation II: Dynamics. Physica A 135, 237 (1986)
Fehske, H., Schleedea, J., Schubert, G., Wellein, G., Filinov, V.S., Bishop, A.R.: Numerical approaches to time evolution of complex quantum systems. Phys. Lett. A 373, 2182 (2009)
Schubert, G., Filinov, V.S., Matyash, K., Schneider, R., Fehske, H.: Comparative study of semiclassical approaches to quantum dynamics. Int. J. Mod. Phys. C 20, 1155 (2009)
Lozovik, Yu.E., Filinov, A.V., Arkhipov, A.S.: Simulation of wave packet tunneling of interacting identical particles. Phys. Rev. E 67, 026707 (2003)
Lozovik, Yu.E., Sharapov, V.A., Arkhipov, A.S.: Simulation of tunneling in the quantum tomography approach. Phys. Rev. A 69, 022116 (2004)
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VIM is supported by Russian Science Foundation under Project No. 16-11-00084.
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Korennoy, Y.A., Man’ko, V.I. Conditions for Quantum and Classical Tomogram-Like Functions to Describe System States and to Retain Normalizations During Time Evolution. Int J Theor Phys 59, 574–595 (2020). https://doi.org/10.1007/s10773-019-04350-x
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DOI: https://doi.org/10.1007/s10773-019-04350-x