Foundations of Physics

, Volume 45, Issue 6, pp 591–610 | Cite as

The Wave Function as Matter Density: Ontological Assumptions and Experimental Consequences

  • Markku JääskeläinenEmail author


The wavefunction is the central mathematical entity of quantum mechanics, but it still lacks a universally accepted interpretation. Much effort is spent on attempts to probe its fundamental nature. Here I investigate the consequences of a matter ontology applied to spherical masses of constant bulk density. The governing equation for the center-of-mass wavefunction is derived and solved numerically. The ground state wavefunctions and resulting matter densities are investigated. A lowering of the density from its bulk value is found for low masses due to increased spatial spreading. A discussion of the possibility to experimentally observe these effects is given and the possible consequences for choosing an ontological interpretation for quantum mechanics are commented upon.


Foundations of quantum theory Wavefunction Newtonian gravity Bound states 


  1. 1.
    Born, M.: Zur Quantenmechanik der Stossvorgange. Z. Phys. 37, 863 (1926)CrossRefADSzbMATHGoogle Scholar
  2. 2.
    Born, M.: Zur Quantenmechanik der Stossvorgange. Z. Phys. 38, 803 (1926)CrossRefADSGoogle Scholar
  3. 3.
    Bacciagaluppi, G., Valentini, A.: Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. Cambridge University Press, Cambridge, UK (2009)CrossRefGoogle Scholar
  4. 4.
    Bohr, N.: The quantum postulate and the recent development of atomic theory. Nature 121, 580 (1928)CrossRefADSzbMATHGoogle Scholar
  5. 5.
    Schrödinger, E.: Quantisierung als Eigenwertproblem (Erste Mitteilung). A. Phys. 79, 361 (1926)zbMATHGoogle Scholar
  6. 6.
    Schrödinger, E.: Quantisierung als Eigenwertproblem (Zweite Mitteilung). A. Phys. 79, 489 (1926)zbMATHGoogle Scholar
  7. 7.
    Schrödinger, E.: Quantisierung als Eigenwertproblem (Dritte Mitteilung: Störungstheorie, mit Anwendung auf den Starkeffekt der Balmerlinien) III. A. Phys. 80, 437 (1926)Google Scholar
  8. 8.
    Schrödinger, E.: Quantisierung als Eigenwertproblem (Vierte Mitteilung) IV. A. Phys. 81, 109 (1926)Google Scholar
  9. 9.
    Przibram, K. (ed.): Letters on Wave Mechanics. Philosophical Library, New York (1967)Google Scholar
  10. 10.
    Moore, W.M.: Schrödinger: Life and Thought. Cambridge University Press, Cambridge, UK (1992)zbMATHGoogle Scholar
  11. 11.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-relativistic Theory. Pergamon Press, Oxford, UK (1976)Google Scholar
  12. 12.
    Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, New York (2010)Google Scholar
  13. 13.
    Bransden, B.H., Joachain, C.J.: Quantum Mechanics. Addison Wesley, New York (2000)Google Scholar
  14. 14.
    Einstein, A.: On the method of theoretical physics. Philos. Sci. 1, 163 (1934)CrossRefGoogle Scholar
  15. 15.
    Born, M.: Natural Philosophy of Cause and Chance. Dover Publications, New York (1964)Google Scholar
  16. 16.
    Peres, A.: Quantum Theory: Concepts and Metods. Kluwer, Dordrecht, the Netherlands (1995)Google Scholar
  17. 17.
    Bell, J.S.: Against ’measurement’. Phys. World 8, 33 (1990)Google Scholar
  18. 18.
    Bassi, A., Lochan, K., Satin, S., Singh, T.P., Ulbricht, H.: Models of wave-function collapse, underlying theories, and experimental results. Rev. Mod. Phys. 85, 471–526 (2013)CrossRefADSGoogle Scholar
  19. 19.
    Kiefer, C.: Quantum Gravity. Oxford University Press, Oxford (2007)CrossRefzbMATHGoogle Scholar
  20. 20.
    Rovelli, C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  21. 21.
    Einstein, A.: Spielen Gravitationsfelder im Aufber der materiellen Elementarteilchen eine wesentliche Rolle? Sitzungsberichte der Preussischen Akademie der Wissenschaften, pp. 349–356 (1919)Google Scholar
  22. 22.
    Einstein, A., Rosen, N.: The particle problem in the general theory of relativity. Phys. Rev. 48, 73 (1935)CrossRefADSGoogle Scholar
  23. 23.
    Wheeler, J.A.: Geons. Phys. Rev. 97, 511 (1955)CrossRefADSzbMATHMathSciNetGoogle Scholar
  24. 24.
    Kaup, D.J.: Klein–Gordon geon. Phys. Rev. 172, 1331 (1968)CrossRefADSGoogle Scholar
  25. 25.
    Ruffini, R., Bonazzola, S.: Systems of self-gravitating particles in general relativity and the concept of an equation of state. Phys. Rev. 187, 1767 (1969)CrossRefADSGoogle Scholar
  26. 26.
    Liddle, A.R., Madsen, M.S.: The structure and formation of boson stars. Int. J. Mod. Phys. D 1, 101 (1992)CrossRefADSzbMATHGoogle Scholar
  27. 27.
    Mielke, E.W., Schunck, F.E.: Boson Stars: Early History and Recent Prospects. In: Piran, T., Ruffni, R. (eds.) The Eighth Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories, Proceedings of the meeting held at the Hebrew University of Jerusalem, 22–27 June 1997, pp. 1607–1626. World Scientic, Singapore (1999)Google Scholar
  28. 28.
    Mielke, E.W., Schunck, F.E.: Boson and Axion Stars. In: Gurzadyan, V.G., Jantzen, R.T., Ruffni, R. (eds.) The Ninth Marcel Grossmann Meeting: On recent developments in theoretical and experimental general relativity, gravitation, and relativistic eld theories, Proceedings of the MGIX MM meeting held at the University of Rome La Sapienza, 2(8), July 2000, pp. 581–591. World Scientic, Singapore (2002)Google Scholar
  29. 29.
    Schunck, F.E., Mielke, E.W.: General relativistic boson stars. Class. Quant. Grav. 20, R301 (2003)CrossRefADSzbMATHMathSciNetGoogle Scholar
  30. 30.
    Mielke, E.W., Schunk, F.E.: Boson stars: early history and recent prospects. Proceedings of The Eighth Marcel Grossman Meeting on General Relativity, World Scientific, Singapore (1999)Google Scholar
  31. 31.
    Møller, C.: The energy-momentum complex in general relativity and related problems. In: Lichnerowicz, A., Tonnelat, M.-A. (eds.) Les Theories Relativites de la Gravitation-Colloques Internationaux. CNRS, Paris (1962)Google Scholar
  32. 32.
    Rosenfeld, L.: On quantization of fields. Nucl. Phys. 40, 353–356 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    DeWitt, B.S.: The quantization of geometry. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research. Wiley, New York (1962)Google Scholar
  34. 34.
    Eppley, K., Hannah, E.: The necessity of quantizing the gravitational field. Found. Phys. 7, 51 (1977)CrossRefADSGoogle Scholar
  35. 35.
    Page, D.N., Geilker, C.D.: Indirect evidence for quantum gravity. Phys. Rev. Lett. 47, 979 (1981)CrossRefADSMathSciNetGoogle Scholar
  36. 36.
    Mattingly, J.: Why Eppley and Hannah’s thought experiment fails. Phys. Rev. D 73, 064025 (2006)CrossRefADSMathSciNetGoogle Scholar
  37. 37.
    Albers, M., Kiefer, C., Reginatto, M.: Measurement analysis and quantum gravity. Phys. Rev. D 78, 064051 (2008)CrossRefADSMathSciNetGoogle Scholar
  38. 38.
    Karolyhazy, F.: Gravitation and quantum mechanics of macroscopic objects. Il Nuovo Cimento 17, 390 (1966)CrossRefGoogle Scholar
  39. 39.
    Diosi, L.: Gravitation and quantum-mechanical localization of macro-objects. Phys. Lett. A 105, 199 (1984)CrossRefADSGoogle Scholar
  40. 40.
    Bialynicki-Birula, I., Mycielski, J.: Nonlinear wave mechanics. Ann. Phys. 100, 62 (1976)CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Penrose, R.: Gravity and state vector reduction. In: Penrose, R., Isham, C.J. (eds.) Quantum Concepts in Space and Time. Clarendon Press, Oxford (1986)Google Scholar
  42. 42.
    Penrose, R.: On Gravity’s role in quantum state reduction. Gen. Relat. Gravit. 28, 581 (1996)CrossRefADSzbMATHMathSciNetGoogle Scholar
  43. 43.
    Penrose, R.: The Emperor’s New mind. Oxford University Press, Oxford (1989)Google Scholar
  44. 44.
    Penrose, R.: Shadows of the Mind. Oxford University Press, Oxford (1994)Google Scholar
  45. 45.
    Albert, D.Z., Ney, A.: The Wave Function: Essays on the Metaphysics of Quantum Mechanics. Oxford University Press, Oxford, UK (2013)Google Scholar
  46. 46.
    Ghirardi, G.C., Grassi, R., Benatti, F.: Describing the macroscopic world: closing the circle within the dynamical reduction program. Found. Phys. 25, 5 (1995)CrossRefADSzbMATHMathSciNetGoogle Scholar
  47. 47.
    Eckart, C.: The kinetic energy of polyatomic molecules. Phys. Rev. 46, 383–387 (1935)CrossRefADSGoogle Scholar
  48. 48.
    Barut, A.O., Kraus, J.: Nonperturbative quantum electrodynamics: the lamb shift. Found. Phys. 13, 189 (1983)CrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Dalibard, J., Dupont-Roc, J., Cohen-Tannoudji, C.: Vacuum fluctuations and radiation reaction: identification of their respective contributions. J. Phys. 43, 1617 (1982)CrossRefGoogle Scholar
  50. 50.
    Gao, S.: The wavefunction and quantum reality. AIP Conf. Proc. 1327, 334 (2011)CrossRefADSGoogle Scholar
  51. 51.
    Giulini, D., Grossardt, A.: The Schrödinger–Newton equation as non-relativistic limit of self-gravitating Klein–Gordon and Dirac elds. Class. Quant. Grav. 29, 215010 (2012)CrossRefADSMathSciNetGoogle Scholar
  52. 52.
    Giulini, D., Grossardt, A.: Gravitationally induced inhibitions of dispersion according to a modified Schrödinger–Newton equation for a homogeneous-sphere potential. Class. Quant. Grav. 30, 155018 (2013)CrossRefADSMathSciNetGoogle Scholar
  53. 53.
    Carstou, F., Lombard, R.J.: A new method of evaluating folding type integrals. Ann. Phys. 217, 279 (1992)CrossRefADSGoogle Scholar
  54. 54.
    Colin, S., Durt, T., Willox, R.: Can quantum systems succumb to their own (gravitational) attraction? arXiv:1403.2982v1 (2014)
  55. 55.
    Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  56. 56.
    Weideman, J.A.C., Reddy, S.C.: A Matlab differentiation matrix suite. ACM Trans. Math. Softw. 26, 465 (2000)CrossRefMathSciNetGoogle Scholar
  57. 57.
    Kythe, P.K., Schäferkotter, M.R.: Handbook of Computational Methods for Integration. Chapman and Hall/CRC, Boca Raton (2004)CrossRefGoogle Scholar
  58. 58.
    Chang, D.E., et al.: Cavity optomechanics using an optically levitated nanosphere. Proc. Natl. Acad. Sci. 107, 1005 (2010)CrossRefADSGoogle Scholar
  59. 59.
    Romero-Isart, O., Juan, M.L., Quidant, R., Cirac, J.I.: Towards quantum superposition of living organisms. New J. Phys. 12, 033015 (2010)CrossRefADSGoogle Scholar
  60. 60.
    Jääskeläinen, M.: Gravitational self-localization for spherical masses. Phys. Rev. A 86, 052105 (2012)CrossRefGoogle Scholar
  61. 61.
    Moroz, I., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger–Newton equations. Class. Quant. Grav. 15, 2733 (1998)CrossRefADSzbMATHMathSciNetGoogle Scholar
  62. 62.
    Bernstein, D.H., Giladi, E., Jones, K.R.W.: Eigenstates of the gravitational. Schrödinger equation. Mod. Phys. Lett. A 13, 2327 (1998)CrossRefADSGoogle Scholar
  63. 63.
    Colella, R., Overhauser, A.W., Werner, S.A.: Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472 (1975)CrossRefADSGoogle Scholar
  64. 64.
    Aminoff, C.G., et al.: Cesium atoms bouncing in a stable gravitational cavity. Phys. Rev. Lett. 71, 3083 (1993)CrossRefADSGoogle Scholar
  65. 65.
    Nezvizhevsky, V.V., et al.: Quantum states of neutrons in the Earth’s gravitational field. Nature 415, 297 (2002)CrossRefADSGoogle Scholar
  66. 66.
    Muller, H., Peters, A., Chu, S.: A precision measurement of the gravitational redshift by the interference of matter waves. Nature 463, 926 (2010)CrossRefADSGoogle Scholar
  67. 67.
    Pusey, M.F., Barrett, J., Rudolph, T.: On the reality of the quantum state. Nature Phys. 8, 475 (2012)CrossRefADSGoogle Scholar
  68. 68.
    Colbeck, R., Renner, R.: Is a systems wavefunction in one-to-one correspondence with its elements of reality? Phys. Rev. Lett. 108, 150402 (2012)CrossRefADSGoogle Scholar
  69. 69.
    Adler, S.L.: Comments on proposed gravitational modifications of Schrödinger dynamics and their experimental implications. J. Phys. A 40, 755 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  70. 70.
    Tornay, S.C.: Ockham: Studies and Selections. Open Court Publishers, La Salle (1938)Google Scholar
  71. 71.
    Metcalf, H.J., van der Straten, P.: Laser Cooling and Trapping. Springer, New York (1999)CrossRefGoogle Scholar
  72. 72.
    Pethick, C.J., Smith, H.: Bose–Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  73. 73.
    Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A.: Observation of Bose–Einstein condensation in a dilute atomic vapor. Science 269, 198 (1995)CrossRefADSGoogle Scholar
  74. 74.
    Meystre, P.: Atom Optics. Springer, New York (2001)CrossRefGoogle Scholar
  75. 75.
    Neuman, K., Block, S.: Optical trapping. Rev. Sci. Instrum. 75, 2787 (2004)CrossRefADSGoogle Scholar
  76. 76.
    Dholakia, K., Reece, P., Gu, M.: Optical micromanipulation. Chem. Soc. Rev. 37, 42–55 (2008)CrossRefGoogle Scholar
  77. 77.
    Bowman, R.W., Padgett, M.J.: Optical trapping and binding. Rep. Prog. Phys. 76, 026401 (2013)CrossRefADSGoogle Scholar
  78. 78.
    Aspelmeyer, M., Kippenberg, T.J., Marquardt, F.: Cavity Optomechanics, arXiv:1303.0733 (2013)
  79. 79.
    Chen, Y.: Macroscopic quantum mechanics: theory and experimental concepts of optomechanics. J. Phys. B 46, 104001 (2013)CrossRefADSGoogle Scholar
  80. 80.
    Pikovski, I., Vanner, M.R., Aspelmeyer, M., Kim, M.S., Brukner, C.: Probing Planck-scale physics with quantum optics. Nature Phys. 8, 393 (2012)CrossRefADSGoogle Scholar
  81. 81.
    Romero-Isart, O.: Quantum superposition of massive objects and collapse models. Phys. Rev. A 84, 052121 (2012)CrossRefADSGoogle Scholar
  82. 82.
    Li, T., Kheifets, S., Reizen, M.G.: Millikelvin cooling of an optically trapped microsphere in vacuum. Nature Phys. 7, 527 (2011)CrossRefADSGoogle Scholar
  83. 83.
    Leanhardt, A.E., et al.: Cooling Bose–Einstein condensates below 500 picokelvin. Science 301, 1513 (2003)CrossRefADSGoogle Scholar
  84. 84.
    Arndt, M., Hornberger, K.: Quantum interferometry with complex molecules. In: Deveaud-Pledran, B., Quattropani, A., Schwendimann, P. (eds.) Quantum Coherence in Solid State Systems, International School of Physics “Enrico Fermi”, Course CLXXI, vol. 171. IOS Press, Amsterdam (2009)Google Scholar
  85. 85.
    Marshall, W., Simon, C., Penrose, R., Bouwmeester, D.: Towards quantum superpositions of a mirror. Phys. Rev. Lett. 91, 130401 (2003)CrossRefADSMathSciNetGoogle Scholar
  86. 86.
    Asenbaum, P., Kuhn, S., Nimmrichter, S., Sezer, U., Arndt, M.: Cavity cooling of free silicon nano particles in high vacuum. Nat. Commun. 4, 2743 (2013)CrossRefADSGoogle Scholar
  87. 87.
    Yang, H., Miao, H., Lee, D.-S., Helou, B., Chen, Y.: Macroscopic quantum mechanics in a classical spacetime. Phys. Rev. Lett. 110, 170401 (2013)CrossRefADSGoogle Scholar
  88. 88.
    Simon, C., Buzek, V., Gisin, N.: No-signalling condition and quantum dynamics. Phys. Rev. Lett. 87, 170405 (2001)CrossRefADSGoogle Scholar
  89. 89.
    Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G.: Photons and Atoms: Introduction to Quantum Electrodynamics. Wiley, Hoboken (1997)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Dalarna UniversityFalunSweden

Personalised recommendations