Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 062, 12 pages      quant-ph/0511125

Quantum Potential and Symmetries in Extended Phase Space

Sadollah Nasiri a, b
a) Department of Physics, Zanjan University, Zanjan, Iran
b) Institute for Advanced Studies in Basic Sciences, IASBS, Zanjan, Iran

Received January 06, 2006, in final form May 19, 2006; Published online June 27, 2006

The behavior of the quantum potential is studied for a particle in a linear and a harmonic potential by means of an extended phase space technique. This is done by obtaining an expression for the quantum potential in momentum space representation followed by the generalization of this concept to extended phase space. It is shown that there exists an extended canonical transformation that removes the expression for the quantum potential in the dynamical equation. The situation, mathematically, is similar to disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates that changes the physical potential to an effective one. The representation where the quantum potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form, is one in which the dynamical equation turns out to be the Wigner equation.

Key words: quantum potential; Wigner equation; distribution functions; extended phase space.

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  1. Holland P.R., The quantum theory of motion, Cambridge University Press, 1993, 68-89.
  2. Sakurai J.J., Modern quantum mechanics, Addison-Wesley, 1985, 97-109.
  3. Madelung E., The hydrodynamical picture of quantum theory, Z. Phys., 1926, V.40, 322-326.
  4. Takabayasi T., The formulation of quantum mechanics in terms of ensemble in phase space, Progr. Theoret. Phys., 1954, V.11, 341-373.
  5. Muga J.G., Sala R., Snider R.F., Comparison of classical and quantum evolution of phase space distribution functions, Phys. Scripta, 1993, V.47, 732-739.
  6. Brown M.R., The quantum poptential: the breakdown of classical symplectic symmetry and the energy of localization and dispersion, quant-ph/9703007.
  7. Brown M.R., Hiley B.J., Schrödinger revisited: an algebraic approach, quant-ph/0005026.
  8. Bohm D., Hiley B.J., On the intuitive understanding of nonlocality and implied by quantum theory, Found. Phys., 1975, V.5, 93-109.
  9. Maroney O., Hiley B.J., Quantum teleportation understood through the Bohm interpretation, Found. Phys., 1999, V.29, 1403-1415.
  10. Bohm D., Hiley B.J., The undivided universe: the ontological interpretation of quantum theory, London, Routledge, 1993.
  11. Holland P.R., Quantum back-reaction and the particle law of motion, J. Phys. A: Math. Gen., 2006, V.39, 559-564.
  12. Shojai F., Shojai A., Constraints algebra and equation of motion in Bohmian interpretation of quantum gravity, Class. Quant. Grav., 2004, V.21, 1-9, gr-qc/0409035.
  13. Carroll R., Fluctuations, gravity, and the quantum potential, gr-qc/0501045.
  14. Carroll R., Some fundamental aspects of a quantum potential, quant-ph/0506075.
  15. Wigner E., On the quantum correction for thermodynamic equilibrium, Phys. Rev., 1932, V.40, 749-759.
  16. Moyal J.E., Quantum mechanics as a statistical theory, Proc. Camb. Philos. Soc., 1949, V.45, 99-124.
  17. Hillery M., O'Connell R.F., Scully M.O., Wigner E.P., Distribution functions in physics: fundamentals, Phys. Rep. C, 1984, V.106, 121-167.
  18. Mehta C.L., Classical functions corresponding to given quantum operators, J. Phys. A, 1968, V.1, 385-392.
  19. Agarval G., Wolf E., Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators, Phys. Rev. D, 1970, V.2, 2161-2186.
  20. Han D., Kim Y.S., Noz M.E., Linear canonical transformations of coherent and squeezed states in the Wigner phase space. II. Quantitative analysis, Phys. Rev. A, 1989, V.40, 902-912.
  21. Kim Y.S., Wigner E.P., Canonical transformation in quantum mechanics, Amer. J. Phys., 1990, V.58, 439-448.
  22. Jannussis A., Patargias N., Leodaris A., Phillippakis T., Streclas A., Papatheos V., Some remarks on the nonnegative quantum mechanical distribution functions, Preprint, Dept. Theor. Phys., Univ. of Patras, 1982.
  23. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization, Ann. Physics, 1978, V.111, 61-110.
  24. Torres-Vega G., Ferderick J.H., Quantum mechanics in phase space: new approaches to the correspondence principle, J. Chem. Phys., 1990, V.93, 8862-8874.
  25. Torres-Vega G., Ferderick J.H., A quantum mechanical representation in phase space, J. Chem. Phys., 1993, V.98, 3103-3120.
  26. de Gosson M.A., Schrödinger equation in phase espace and deformation quantization, math.SG/0504013.
  27. Bolivar A.O., Quantum-classical correspondence dynamical quantization and the classical limit series: the frontiers collection, 2000.
  28. de Gosson M.A., The principles of Newtonian and quantum mechanics, the need for Planck's constant, (h/2p), Blekinge Institute of Technology, Sweden, 2001.
  29. Sobouti Y., Nasiri S., A phase space formulation of quantum state functions, Internat. J. Modern Phys. B, 1993, V.7, 3255-3272.
  30. Goldstein H., Classical mechanics, London, Addison-Wesley, 1980.
  31. Merzbacher E., Quantum mechanics, New York, Wiley Interscience, 1970, 341-350.

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