Abstract
The relation between the density matrix obeying the von Neumann equation and the wave function obeying the Schrödinger equation is discussed in connection with the superposition principle of quantum states. The definition of the ray-addition law is given, and its relation to the addition law of vectors in the Hilbert space of states and the role of a constant phase factor of the wave function is elucidated. The superposition law of density matrices, Wigner functions, and tomographic probabilities describing quantum states in the probability representation of quantum mechanics is studied. Examples of spin-1/2 and Schrödinger-cat states of the harmonic oscillator are discussed. The connection of the addition law with the entanglement problem is considered.
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References
E. Schrödinger,Ann. Physik,79, 489 (1926).
P. A. M. Dirac,The Principles of Quantum, Mechanics, 4th edition, Pergamon, Oxford (1958).
J. von Neumann,Mathematische Grundlagen der Quantummechanik, Springer, Berlin (1932); Göttingenische Nachrichten,11 (Nov. 1927), S. 245–272.
L. D. Landau,Z. Physik,45, 430, (1930).
E. Wigner,Phys. Rev.,77, 711 (1932).
K. Husimi,Proc. Phys. Mat. Soc. Jpn.,23, 264 (1940).
Y. Kano,J. Math. Phys.,6, 1913 (1965).
E. C. G. Sudarshan,Phys. Rev. Lett.,10, 277 (1965); C. L. Mehta and E. C. G. Sudarshan,Phys. Rev. B138, 274 (1965).
R. J. Glauber,Phys. Rev.,131, 2766 (1963).
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).
S. Mancini, V. I. Man'ko, and P. Tombesi,Quantum Semiclass. Opt.,7, 615 (1995).
S. Mancini, V. I. Man'ko, and P. Tombesi,Phys. Lett. A,213, 1 (1996);Found. Phys.,27, 801 (1997).
V. V. Dodonov, I. A. Malkin, and V. I. Man'ko,Physica,72, 597 (1974).
B. Yurke and D. Stoler,Phys. Rev. Lett.,57, 13 (1986).
V. I. Man'ko and R. V. Mendes, “Noncommutative time-frequency tomography of analytic signals,” E-print LANL Physics/9712022 Data Analysis, Statistics, and Probability (1997);Phys. Lett. A (1999, in press).
Vladimir Man'ko, Marcos Moshinsky, and Anju Sharma,Phys. Rev. A,59, 1809 (1999).
V. I. Man'ko, L. Rosa, and P. Vitale,Phys. Lett. B,439, 328 (1998).
V. V. Dodonov and V. I. Man'ko,Phys. Lett. A,229, 335 (1997).
V. I. Man'ko and O. V. Man'ko,Zh. Éksp. Teor Fiz.,85, 430 (1997) [JETP,112, 796 (1997)];J. Russ. Laser Res.,18, 407 (1997).
L. D. Landau and E. M. Lifshits,Quantum Mechanics, 2nd edition Pergamon, Oxford (1965).
J. Klauder and E. C. G. Sudarshan,Fundamentals of Quantum Optics, W. A. Benjamin, New York-Amsterdam (1968).
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Man'ko, V.I., Marmo, G., Sudarshan, E.C.G. et al. On the relation between Schrödinger and von Neumann equations. J Russ Laser Res 20, 421–437 (1999). https://doi.org/10.1007/BF02508909
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DOI: https://doi.org/10.1007/BF02508909