Abstract
The von Neumann evolution equation for the density matrix and the Moyal equation for the Wigner function are mapped onto the evolution equation for the optical tomogram of the quantum state. The connection with the known evolution equation for the symplectic tomogram of the quantum state is clarified. The stationary states corresponding to quantum energy levels are associated with the probability representation of the von Neumann and Moyal equations written for optical tomograms. The classical Liouville equation for optical tomogram is obtained. An example of the parametric oscillator is considered in detail.
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References
S. Mancini, V. I. Man’ko and P. Tombesi, Phys. Lett. A, 213, 1 (1996).
S. Mancini, V. I. Man’ko, and P. Tombesi, Found. Phys., 27, 801 (1997).
A. Ibort, V. I. Man’ko, G. Marmo, et al., Phys. Scr., 79, 065013 (2009).
M. A. Man’ko and V. I. Man’ko, Found. Phys., doi:10.1007/s10701-009-9403-9 (2009).
J. Radon, Ber. Verh. Sachs. Akad., 69, 262 (1917).
J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1987).
K. Vogel and H. Risken, Phys. Rev. A, 40, 2847 (1989).
D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett., 70, 1244 (1993).
A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys., 81, 299 (2009).
S. Mancini, V. I. Man’ko, and P. Tombesi, Quantum Semiclass. Opt., 7, 615 (1995).
G. M. D’Ariano, S. Mancini, V. I. Man’ko, and P. Tombesi, J. Opt. B: Quantum Semiclass. Opt., 8, 1017 (1996).
O. V. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 18, 407 (1997).
V. I. Man’ko and R. V. Mendes, Physica D, 145, 330 (2000).
S. Mancini, O. V. Man’ko, V. I. Man’ko, and P. Tombesi, J. Phys. A: Math. Gen., 34, 3461 (2001).
J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin (1932).
J. E. Moyal, Proc. Cambridge Philos. Soc., 45, 99 (1949).
E. Wigner, Phys. Rev., 40, 749, (1932).
V. N. Chernega and V. I. Man’ko, J. Russ. Laser Res., 29, 43 (2008).
A. S. Arkhipov and V. I. Man’ko, J. Russ. Laser Res., 25, 468 (2004).
V. V. Dodonov, M. A. Marchiolli, Ya. A. Korennoy, et al., Phys. Scr., 58, 4087 (1998).
A. Erdélyi (ed.), Bateman Manuscript Project: Higher Transcendental Functions, McGraw-Hill, New York (1953).
G. Szegö, Orthogonal Polynomials, American Mathematical Society, Providence, RI (1959).
I. A. Malkin and V. I. Man’ko, Phys. Lett. A, 31, 243 (1970).
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Korennoy, Y.A., Man’ko, V.I. Probability representation of the quantum evolution and energy-level equations for optical tomograms. J Russ Laser Res 32, 74–85 (2011). https://doi.org/10.1007/s10946-011-9191-5
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DOI: https://doi.org/10.1007/s10946-011-9191-5