Position Representation of Quantum Mechanics over Riemannian Configuration Space

  • Maciej Błaszak


The last chapter of the book is devoted to two very important issues of the developed quantum theory. The first one is related with systematic construction of the so called position representation of quantum mechanics over an appropriate class of Riemaniann spaces in any admissible local curvilinear coordinates. In particular, for a flat space and Cartesian coordinates we reconstruct the standard quantization procedure from textbooks of quantum mechanics. The second issue of that chapter is related with quantum integrability (quantum superintegrability) and quantum separability. Actually, we present the reader a class of quantizations of classical Stäckel systems considered in previous chapters, which preserve quantum integrability, quantum superintegrability and quantum stationary separability of related quantum Hamiltonian operators.


  1. 2.
    Agarwal, G.S., Wolf, E.: Calculus for functions of noncommuting operators and general phase-space method in quantum mechanics I. Phys. Rev. D 2, 2161 (1970)MathSciNetCrossRefADSGoogle Scholar
  2. 7.
    Baker, G.: Formulation of quantum mechanics based on the quasi-probability distribution induced on phase space. Phys. Rev. 109, 2198 (1958)MathSciNetCrossRefADSGoogle Scholar
  3. 17.
    Benenti, S., Chanu, C., Rastelli, G.: Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. I. The completeness and Robertson conditions. J. Math. Phys. 43, 5183 (2002)Google Scholar
  4. 18.
    Benenti, S., Chanu, C., Rastelli, G.: Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operators. J. Math. Phys. 43, 5223 (2002)Google Scholar
  5. 33.
    Błaszak, M., Domański, Z.: Phase space quantum mechanics. Ann. Phys. 327, 167 (2012)MathSciNetCrossRefADSGoogle Scholar
  6. 36.
    Błaszak, M., Domański, Z.: Canonical quantization of classical mechanics in curvilinear coordinates. Invariant quantization procedure. Ann. Phys. 339, 89 (2013)Google Scholar
  7. 37.
    Błaszak, M., Domański, Z.: Natural star-products on symplectic manifolds and related quantum mechanical operators. Ann. Phys. 344, 29 (2013)MathSciNetCrossRefADSGoogle Scholar
  8. 47.
    Błaszak, M., Domański, Z., Sergyeyev, A., Szablikowski, B.: Integrable quantum Stäckel systems. Phys. Lett. A 377, 2564 (2013)MathSciNetCrossRefADSGoogle Scholar
  9. 48.
    Błaszak, M., Domański, Z., Silindir, B.: Flat minimal quantization of Stäckel systems and quantum separability. Ann. Phys. 351, 152 (2014)CrossRefADSGoogle Scholar
  10. 49.
    Błaszak, M., Marciniak, K., Domański, Z.: Quantizations preserving separability of Stäckel systems. Ann. Phys. 371, 460 (2016)CrossRefADSGoogle Scholar
  11. 61.
    Carter, B.: Killing tensor quantum numbers and conserved currents in curved space. Phys. Rev. D 16, 3395 (1977)MathSciNetCrossRefADSGoogle Scholar
  12. 62.
    Caux, J.S., Mossel, J.: Remarks on the notion of quantum integrability. J. Stat. Mech Theory Exp. 2011, P02023 (2011)Google Scholar
  13. 63.
    Chanu, C., Rastelli, G.: Fixt energy R-separation for Schrödinger equation. Int. J. Geom. Methods Mod. Phys. 3, 489 (2006)MathSciNetCrossRefGoogle Scholar
  14. 67.
    Chanu, C., Degiovanni, L., Rastell, G.: Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion. J. Math. Phys. 58, 033509 (2017)MathSciNetCrossRefADSGoogle Scholar
  15. 78.
    Darboux, G.: Leçons Sur les Systèmes Orthogonaux et les Coordonnées Curvilignes. Gauthier-Villars, Paris (1910)zbMATHGoogle Scholar
  16. 88.
    De Witt, B.S.: Point transformations in quantum mechanics. Phys. Rev. 85, 653 (1952)CrossRefADSGoogle Scholar
  17. 89.
    De Witt, B.S.: Dynamical theory in curved spaces. I. A review of the classical and quantum action principles. Rev. Modern Phys. 29, 377 (1957)Google Scholar
  18. 98.
    Domański, Z.: Admissible invariant canonical quantizations of classical mechanics. PhD Thesis (2014)Google Scholar
  19. 102.
    Duval, C., Ovsienko, V.: Conformally equivariant quantum Hamiltonians. Selecta Math. (NS) 7, 291 (2001)Google Scholar
  20. 103.
    Duval, C., Valent, G.: Quantum integrability of quadratic Killing tensors. L. Math. Phys. 46, 053516 (2005)MathSciNetCrossRefADSGoogle Scholar
  21. 104.
    Eisenhart, L.P.: Separable systems of Stäckel. Ann. Math. 35, 284 (1934)MathSciNetCrossRefGoogle Scholar
  22. 109.
    Essen, H.: Quantization and independent coordinates. Amer. J. Phys. 46, 983 (1978)MathSciNetCrossRefADSGoogle Scholar
  23. 111.
    Fairlie, D.B.: The formulation of quantum mechanics in terms of phase space functions. Math. Proc. Camb. Philos. Soc. 60, 581 (1964)MathSciNetCrossRefADSGoogle Scholar
  24. 131.
    Gervais, J.L., Jevicki, A.: Point canonical transformations in the path integral. Nuclear Phys. B 110, 93 (1976)MathSciNetCrossRefADSGoogle Scholar
  25. 132.
    Glauber, R.J.: Coherent and Incoherent States of the Radiation Field. Phys. Rev. 131, 2766 (1963)MathSciNetCrossRefADSGoogle Scholar
  26. 133.
    Glauber, R.J. In: Quantum Optics and Electronics (eds), by DeWitt, C., Blandin, A., Cohen-Tannoudji, C., p. 63. Gordon and Breach, New York (1965)Google Scholar
  27. 137.
    Gradstheyn, I.S., Ryzhik, M.: Table of Integrals, Series and Products, p. 838. Academic, New York (1980)Google Scholar
  28. 138.
    Gravel, S., Winternitz, P.: Superintegrability with third-order integrals in quantum and classical mechanics. J. Math. Phys. 56, 5902 (2002)MathSciNetCrossRefADSGoogle Scholar
  29. 145.
    Harnad, J., Winternitz, P.: Classical and quantum integrable systems in \(\widetilde {\mathfrak {gl}}(2)^{+\ast }\) and separation of variables. Commun. Math. Phys. 172, 263 (1995)Google Scholar
  30. 148.
    Hietarinta, J.: Classical versus quantum integrability. J. Math. Phys. 25, 1833 (1984)MathSciNetCrossRefADSGoogle Scholar
  31. 149.
    Hietarinta, J., Grammaticos, B.: On the ħ 2 correction terms in quantum integrability. J. Phys. A 22, 1315 (1989)MathSciNetCrossRefADSGoogle Scholar
  32. 151.
    Hillery, M., O’Connell, R.F., Scully, M.O., Wigner, E.P.: Distribution functions in physics: fundamentals. Phys. Rep. 106, 121 (1984)MathSciNetCrossRefADSGoogle Scholar
  33. 155.
    Husimi, K.: Some formal properties of the density matrix. Proc. Phys. Math. Soc. Jpn 22, 264 (1940)zbMATHGoogle Scholar
  34. 159.
    Kalnins, E.G., Miller, Jr. W.: R-separation of variables for the four-dimensional flat space Laplace and Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 244, 241 (1978)MathSciNetzbMATHGoogle Scholar
  35. 160.
    Kalnins, E.G., Miller, Jr. W.: Some remarkable R-separable coordinate systems for the Helmholtz equation. Lett. Math. Phys. 4, 469 (1980)MathSciNetCrossRefADSGoogle Scholar
  36. 161.
    Kalnins, E.G., Miller, Jr. W.: The theory of orthogonal R-separation for Helmholtz equations. Adv. Math. 51, 91 (1984)MathSciNetCrossRefGoogle Scholar
  37. 165.
    Kalnins, E.G., Kress, J.M., Miller, Jr. W.: Second-order superintegrable systems in conformally flat spaces. V. Two- and three-dimensional quantum systems. J. Math. Phys. 47, 093501 (2006)Google Scholar
  38. 167.
    Kalnins, E.G., Kress, J., Miller, Jr. W.: Tools for verifying classical and quantum superintegrability. SIGMA 6, 066 (2010)MathSciNetzbMATHGoogle Scholar
  39. 169.
    Kalnins, E.G., Kress, J.M., Miller, Jr. W.: A recurrence relation approach to higher order quantum superintegrability. SIGMA 7, 031 (2011)MathSciNetzbMATHGoogle Scholar
  40. 171.
    Kirkwood, J.G.: Quantum statistics of almost classical assemblies. Phys. Rev. 44, 31 (1933)CrossRefADSGoogle Scholar
  41. 176.
    Lee, H.W.: Theoty and application of the quantum phase-space distribution functions. Phys. Rep. 259, 147 (1995)MathSciNetCrossRefADSGoogle Scholar
  42. 182.
    Liu, Z.J.: Quantum integrable systems constrained on the sphere. Lett. Math. Phys. 20, 151 (1990)MathSciNetCrossRefADSGoogle Scholar
  43. 183.
    Liu, Z.J., Quian, M.: Guage invariant quantization on Riemannian manifolds. Trans. Amer. Math. Soc. 331, 321 (1992)MathSciNetCrossRefGoogle Scholar
  44. 199.
    Mehta, C.L.: Phase-space formulation of the dynamics of canonical variables. J. Math. Phys. 5, 677 (1964)MathSciNetCrossRefADSGoogle Scholar
  45. 200.
    Miller, Jr. W., Post, S., Winternitz, P.: Classical and quantum superintegrability with applications. J. Phys. A Math. Theor. 46, 423001 (2013)MathSciNetCrossRefADSGoogle Scholar
  46. 206.
    Mykytiuk, I.V., Prykarpatsky, A.K., Andrushkiw, R.I., Samoilenko, V.H.: Geometric quantization of Neumann-type completely integrable Hamiltonian systems. J. Math. Phys. 35, 1532 (1994)MathSciNetCrossRefADSGoogle Scholar
  47. 220.
    Podolsky, B.: Quantum-mechanically correct form of Hamiltonian function for conservative systems. Phys. Rev. 32, 812 (1928)CrossRefADSGoogle Scholar
  48. 221.
    Post, S., Winternitz, P.: A nonseparable quantum superintegrable system in 2D real Euclidean space. J. Phys. A Math. Theor. 44, 152001 (2011)MathSciNetCrossRefADSGoogle Scholar
  49. 222.
    Prus, R., Sym A.: Non-regular and non-Stäckel R-separation for 3-dimensional Helmholtz equation and cyclidic solitons of wave equation. Phys. Lett. A 336, 459 (2005)MathSciNetCrossRefADSGoogle Scholar
  50. 227.
    Rihaczek, A.W.: Signal energy distribution in time and frequency. IEEE Trans. Inf. Theory 14, 369 (1968)CrossRefGoogle Scholar
  51. 228.
    Robertson, H.P.: Bemerkung über separierbare Systeme in der Wellenmechanik. Math. Ann. 98, 749 (1927)CrossRefGoogle Scholar
  52. 229.
    Rodriguez, M.A., Winternitz, P.: Quantum superintegrability and exact solvability in n dimensions. J. Math Phys. 43, 1309 (2002)MathSciNetCrossRefADSGoogle Scholar
  53. 243.
    Sudarshan, E.C.G.: Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams. Phys. Rev. Lett. 10, 277 (1963)MathSciNetCrossRefADSGoogle Scholar
  54. 244.
    Sym, A., Szereszewski, A.: On Darboux’s approach to R-Separability of variables. SIGMA 7, 95 (2011)MathSciNetzbMATHGoogle Scholar
  55. 245.
    Szereszewski, A., Sym, A.: On Darboux’s approach to R-separability of variables. Classification of conformally flat 4-dimensional binary metrics. J. Phys. A Math. Theor. 48, 385201 (2015)Google Scholar
  56. 246.
    Takabayasi, T.: The formulation of quantum mechanics in terms of ensemble in phase space. Prog. Theor. Phys. 11, 341 (1954)MathSciNetCrossRefADSGoogle Scholar
  57. 248.
    Toth, J.A.: Various quantum mechanical aspects of quadratic forms. J. Funct. Anal. 130, 1 (1995)MathSciNetCrossRefGoogle Scholar
  58. 254.
    von Neumann, J.: Mathematical Foundations of Quantum Mechanics, p. 173. Princeton University Press, Princeton (1955)Google Scholar
  59. 259.
    Weingert S.: The problem of quantum integrability. Physica D 56, 117 (1992)MathSciNetADSGoogle Scholar
  60. 263.
    Wigner, E.P.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749 (1932)CrossRefADSGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Maciej Błaszak
    • 1
  1. 1.Division of Mathematical PhysicsFaculty of Physics UAMPoznańPoland

Personalised recommendations