Abstract
We introduce the positive vector optical tomogram fully describing the quantum states of spin-1/2 particles without any redundancy. Correspondingly, we introduce the vector symplectic tomogram and vector quasidistributions (Wigner, Husimi, and Glauber–Sudarshan vector functions). We obtain the evolution equations for the proposed vector optical and symplectic tomograms and vector quasidistributions for arbitrary Hamiltonian. In the proposed representations, we consider the quantum system of a charged spin-1/2 particle in an arbitrary electromagnetic field and obtain the evolution equations, which are analogs of the Pauli equation. As an example of the use of the proposed approach, we find the propagator of the evolution equation in the case of homogeneous and stationary magnetic fields in Landau gauge and consider the evolution of the initial entangled superposition of lower Landau levels in the vector optical representation. Also, as an example, we consider the system of linear quantum oscillator with spin in vector optical tomography representation, and study in this representation the evolution of the initial entangled superposition of two lower Fock states, as well as spin-up and spin-down states.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1987).
K. Vogel and H. Risken, Phys. Rev. A, 40, 2847 (1989).
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).
V. V. Dodonov and V. I. Man’ko, Phys. Lett. A, 229, 335 (1997).
J.-P. Amiet and S. Weigert, J. Phys. A: Math. Gen., 31, L543 (1998).
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., 41, 330 (2011).
A. I. Lvovsky and M. G. Raymer, Rev. Mod. Phys., 81, 299 (2009).
R. G. Newton and B. Young, Ann. Phys., 49, 393 (1968).
S. Weigert, Phys. Rev. A, 45, 7688 (1992).
J.-P. Amiet and S. Weigert, J. Phys. A: Math. Gen., 32, 2777 (1999).
J.-P. Amiet and S. Weigert, J. Opt. B: Quantum Semiclass. Opt., 1, L5 (1999).
J.-P. Amiet and S. Weigert, J. Phys. A: Math. Gen., 32, L269 (1999).
J.-P. Amiet and S. Weigert, J. Opt. B: Quantum Semiclass. Opt., 2, 118 (2000).
S. Heiss and S. Weigert, Phys. Rev. A, 63, 012105 (2000).
S. N. Filippov and V. I. Man’ko, J. Russ. Laser Res., 31, 32 (2010).
V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, Phys. Lett. A, 327, 353 (2004).
O. V. Man’ko and V. I. Man’ko, J. Russ. Laser Res., 18, 407 (1997).
Ya. A. Korennoy and V. I. Man’ko, J. Russ. Laser Res., 32, 74 (2011).
Ya. A. Korennoy and V. I. Man’ko, J. Russ. Laser Res., 32, 338 (2011).
S. Mancini, O. V. Man’ko, V. I. Man’ko, and P. Tombesi, J. Phys. A: Math. Gen., 34, 3461 (2001).
G. G. Amosov, Ya. A. Korennoy, and V. I. Man’ko, Phys. Rev. A, 85, 052119 (2012).
F. Lizzi and P. Vitale, SIGMA, 10, 086 (2014).
E. Wigner, Phys. Rev., 40, 749 (1932).
K. Husimi, Proc. Phys.-Math. Soc. Jpn., 22, 264 (1940).
S. S. Mizrahi, Physica A, 127, 241 (1984).
D. M. Davidović and D. Lalović, J. Phys. A: Math. Gen., 26, 5099 (1993).
R. J. Glauber, Phys. Rev. Lett., 10, 84 (1963).
E. C. G. Sudarshan, Phys. Rev., 10, 277 (1963).
J. E. Moyal, Proc. Cambrige Philos. Soc., 45, 99 (1949).
Ya. A. Korennoy and V. I. Man’ko, J. Russ. Laser Res., 32, 153 (2011).
V. I. Man’ko, L. Rosa, and P. Vitale, Phys. Rev. A, 57, 3291 (1998).
V. V. Dodonov and V. I. Man’ko, Invariants and the Evolution of Nonstationary Quantum Systems, Proceedings of the Lebedev Physical Institute, Nova Science, Commack, New York (1989), Vol. 183.
L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Nonrelativistic Theory, Course of Theoretical Physics, Pergamon, Oxford (1997), Vol. 3.
I. A. Malkin and V. I. Man’ko, Zh. Éksp. Teor. Fiz., 55, 1014 (1968).
L. S. Brown and G. Gabrielse, Rev. Mod. Phys., 58, 233 (1986).
M. Massini, M. Fortunato, S. Mancini, and P. Tombesi, Phys. Rev. A, 62, 041401(R) (2000).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Korennoy, Y.A., Man’ko, V.I. Pauli Equation for a Joint Tomographic Probability Distribution. J Russ Laser Res 36, 534–549 (2015). https://doi.org/10.1007/s10946-015-9532-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10946-015-9532-x