Abstract
Formulation of the conventional quantum mechanics in which a state is described by probability instead of wave function and density matrix is presented. We consider the possibility of constructing the invertable map of spinors onto positive probability distributions. For any value of spin, the basis of the irreducible representation of a rotation group is realized by a family of probability distributions of the spin projection parametrized by points on a sphere. Quantum states of a symmetric top described by the probability distributions are discussed.
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References
E. Schrödinger, Ann. Phys. (Leipzig), 79, 489 (1926).
L. D. Landau, Z. Phys., 45, 430 (1927)
J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin (1932).
E. Wigner, Phys. Rev., 40, 749 (1932).
R. J. Glauber, Phys. Rev. Lett, 10, 84 (1963).
E. C. G. Sudarshan, Phys. Rev. Lett, 10, 277 (1963).
K. Husimi, Proc. Phys. Math. Soc. Jpn., 23, 264 (1940).
Y. Kano, J. Math. Phys., 6, 1913 (1986).
W. Heisenberg, Z. Phys., 43, 172 (1927).
E. Schrödinger, Sitzungsber. Preuss. Acad. Wiss., 24, 296 (1930).
H. P. Robertson, Phys. Rev., 35, 667 (1930); 46, 794 (1934).
E. Madelung, Z. Phys., 40, 332 (1926).
L. De Broglie, Compt. Rend., 183, 447 (1926); 184, 273 (1927); 185, 380 (1927).
D. Böhm, Phys. Rev., 85, 166; 180 (1952).
J. Bertrand and P. Bertrand, Found. Phys., 17, 397 (1989).
K. Vogel and H. Risken, Phys. Rev. A, 40, 2847 (1989).
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, Quantum Semiclass. Opt., 6, 1017 (1996).
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 ().
V. I. Man’ko, “Quantum mechanics and classical probability theory,” in: B. Gruber and M. Ramek (Eds.), Symmetries in Science IX (Bregenz, Austria, August 1996), Plenum, New York (1997), p. 215.
J. E. Moyal, Proc. Cambrige Philos. Soc., 99 (1949).
V. V. Dodonov and V. I. Man’ko, Phys. Lett. A, 229, 335 (1997).
V. I. Man’ko and O. V. Man’ko, JETP, 112, 796 (1997).
Olga Man’ko, “Tomography of spin states and classical formulation of quantum mechanics,” in: B. Gruber and M. Ramek (Eds.), Symmetries in Science X (Bregenz, Austria, July-August 1997), Plenum, New York (to appear, 1998).
U. Leonhardt, Phys. Rev. A, 53, 2998 (1996).
O. V. Man’ko, J. Russ. Laser Res., 17, 439 (1996); Phys. Lett. A, 228, 29 (1997).
L. C. Biedenhard and J. D. Louck, Angular Momentum in Quantum Physics. Theory and Application, in: Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Reading, MA (1981), Vol. 8.
D. P. Zhelobenko, Compact Lie Groups and Their Representation [in Russian], Nauka, Moscow (1970).
A. N. Leznov, Group Methods of Nonlinear Dynamic System Integration [in Russian], Nauka, Moscow (1985).
V. I. Man’ko, “Energy levels of quantum system in classical formulation of quantum mechanics,” in: I. M. Dremin and A. M. Semikhatov (Eds.), Proceedings of the Second International A. D. Sakharov Conference on Physics (Moscow, May 1996), World Scientific, Singapore (1997), p. 486; “Optical symplectic tomography and classical probability instead of wave function in quantum mechanics,” in: H.-D. Doebner, W. Scherer, and C. Schultz (Eds.), GROUP21. Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras (Goslar, Germany, June-July 1996), World Scientific, Singapore (1997), Vol. 2, p. 764; “Transition probability between energy levels in the framework of the classical approach,” invited lecture at the Inaguration Conference of APCTP (Seoul, June 1996), (to appear in the Proceedings of the Conference, World Scientific, 1998); “Classical description of quantum states and tomography,” in: D. Han, J. Janszky, Y. S. Kim, and V. I. Man’ko (Eds.), Fifth International Conference on Squeezed State and Uncertainly Relations (Balatonfured, Hungary, May 1997), NASA Conference Publication (1998), Vol. 206855, p. 523.
V. I. Man’ko and S. S. Safonov, Theor. Math. Phys., 112, 467 (1997).
V. A. Andreev and V. I. Man’ko, JETP (to appear, 1998).
O. V. Man’ko, V. I. Man’ko, and S. S. Safonov, Teor. Mat. Fiz., 115, 185 (1998).
V. I. Man’ko and S. S. Safonov, Phys. Atomic Nuclei, 61, 585 (1998).
L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon, New York (1958).
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Andreev, V.A., Man’ko, O.V., Man’ko, V.I. et al. Spin States and Probability Distribution Functions. J Russ Laser Res 19, 340–368 (1998). https://doi.org/10.1007/BF03380148
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DOI: https://doi.org/10.1007/BF03380148