Foundations of Physics

, Volume 33, Issue 10, pp 1495–1541 | Cite as

Wavefunction Collapse and Random Walk

  • Brian Collett
  • Philip Pearle


Wavefunction collapse models modify Schrödinger's equation so that it describes the rapid evolution of a superposition of macroscopically distinguishable states to one of them. This provides a phenomenological basis for a physical resolution to the so-called “measurement problem.” Such models have experimentally testable differences from standard quantum theory. The most well developed such model at present is the Continuous Spontaneous Localization (CSL) model in which a universal fluctuating classical field interacts with particles to cause collapse. One “side effect” of this interaction is that the field imparts energy to the particles: experimental evidence on this has led to restrictions on the parameters of the model, suggesting that the coupling of the classical field to the particles must be mass-proportional. Another “side effect” is that the field imparts momentum to particles, causing a small blob of matter to undergo random walk. Here we explore this in order to supply predictions which could be experimentally tested. We examine the translational diffusion of a sphere and a disc, and the rotational diffusion of a disc, according to CSL. For example, we find that the rms distance an isolated 10−5 cm radius sphere diffuses is ≈(its diameter, 5 cm) in (20 sec, a day), and that a disc of radius 2 ⋅ 10−5 cm and thickness 0.5 ⋅ 10−5 cm diffuses through 2πrad in about 70 sec (this assumes the “standard” CSL parameter values). The comparable rms diffusions of standard quantum theory are smaller than these by a factor 10−3±1. It is shown that the CSL diffusion in air at STP is much reduced and, indeed, is swamped by the ordinary Brownian motion. It is also shown that the sphere's diffusion in a thermal radiation bath at room temperature is comparable to the CSL diffusion, but is utterly negligible at liquid He temperature. Thus, in order to observe CSL diffusion, the pressure and temperature must be low. At the low reported pressure of 5 ⋅ 10−17 Torr, achieved at 4.2°K, the mean time between air molecule collisions with the (sphere, disc) is ≈(80, 45)min. This is ample time for observation of the putative CSL diffusion with the standard parameters and, it is pointed out, with any parameters in the range over which the theory may be considered viable. This encourages consideration of how such an experiment may actually be performed, and the paper closes with some thoughts on this subject.

wavefunction collapse continuous spontaneous localization Brownian motion random walk diffusion Paul trap 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Schrödinger, Naturwiss. 23, 807(1935).Google Scholar
  2. 2.
    P. Pearle, Phys. Rev. A 39, 2277(1989).Google Scholar
  3. 3.
    G. C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A 42, 78(1990).Google Scholar
  4. 4.
    G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 34, 470(1986); G. C. Ghirardi, A. Rimini, and T. Weber, Phys. Rev. D 36, 3287 (1987); G. C. Ghirardi, A. Rimini, and T. Weber Found. Phys. 18, l(1988). F. Benatti, G. C. Ghirardi, A. Rimini, and T. Weber, Nuovo Cimento B 100, 27(1987); F. Benatti, G. C. Ghirardi, A. Rimini, and T. Weber, Nuovo Cimento B 101, 333(1988). J. Bell, in Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987), p. 201.Google Scholar
  5. 5.
    P. Pearle, Phys. Rev. D 13, 857(1976); Int. Theor. Phys. 48, 489(1979; Found. Phys. 12, 249(1982); Phys. Rev. D 29, 235(1984); in The Wave-Particle Dualism, S. Diner et al., eds. (Reidel, Dordrecht, 1984); J. Stat. Phys. 41, 719(1985); in Quantum Concepts in Space and Time, R. Penrose and C. J. Isham, eds. (Clarendon, Oxford, 1986); Phys. Rev. D 33, 2240(1986); in New Techniques in Quantum Measurement Theory, D. M. Greenberger, ed. (N.Y. Acad. Sci., New York, 1986), p. 539.Google Scholar
  6. 6.
    For some other collapse models, see N. Gisin, Helv. Phys. Acta 62, 363(1989). V. P. Belavkin, Phys. Lett. A 140, 355(1989). I. C. Percival, Proc. Roy. Soc. A 451, 503(1995) I. C. Percival and Quantum State Diffusion (Cambridge University Press, Cambridge, 1998). L. P. Hughston, Proc. Roy. Soc. A 452, 953(1995). R. Penrose, Gen. Rel. and Grav. 28, 581(1996). D. Fivel, Phys. Rev. A 56, 146(1997). S. L. Adier and L. P Horwitz, J. Math. Phys. 41, 2485(2000).Google Scholar
  7. 7.
    For a recent review of CSL, see P. Pearle in Open Systems and Measurement in Relativistic Quantum Theory, H. P. Breuer and F. Petruccione, eds. (Springer, Heidelberg, 1999), p. 195.Google Scholar
  8. 8.
    G. C. Ghirardi and A. Rimini, in Sixty-Two Years of Uncertainty, A. Miller, ed. (Plenum, New York, 1990), p. 167.Google Scholar
  9. 9.
    E. J. Squires, Phys. Lett. A 158, 431(1991).Google Scholar
  10. 10.
    L. E. Ballentine, Phys. Rev. A 43, 9(1991).Google Scholar
  11. 11.
    P. Pearle and E. Squires, Phys. Rev. Lett. 73, 1(1994).Google Scholar
  12. 12.
    B. Collett, P. Pearle, F. Avignone, and S. Nussinov, Found. Phys. 25, 1399(1995).Google Scholar
  13. 13.
    P. Pearle, James Ring, J. I. Collar, and F. T. Avignone, III, Found. Phys. 29, 465(1999).Google Scholar
  14. 14.
    F. Karolyhazy, Nuovo Cimento A 42, 1506(1966). F. Karolyhazy, A. Frenkel, and B. Lukacs, in Physics as Natural Philosophy, A. Shimony and H. Feshbach, eds. (M.I.T. Press, Cambridge 1982), p. 204; in Quantum Concepts in Space and Time, R. Penrose and C. J. Isham, eds. (Clarendon, Oxford, 1986), p. 109; A. Frenkel, Found. Phys. 20, 159(1990).Google Scholar
  15. 15.
    F. Benatti, G. C. Ghirardi, and R. Grassi, Found Phys. 35, 5(1995).Google Scholar
  16. 16.
    A. Bassi and G. C. Ghirardi, Brit. J. Phil. Sci. 50, 719(1999).Google Scholar
  17. 17.
    L. Diosi, Phys. Lett. A 132, 233(1988).Google Scholar
  18. 18.
    L. Diosi, Phys. Rev. A 40, 1165(1989).Google Scholar
  19. 19.
    G. Gabrielse et al., Phys. Rev. Lett. 65, 1317(1990).Google Scholar
  20. 20.
    For a nice treatment, see R. M. Mazo in Stochastic Processes in Nonequilibrium Systems, Lecture Notes in Physics, Vol. 84, L. Garrido, P. Seglar, and P. J. Shepard, eds. (Springer, Berlin, 1978), p. 53.Google Scholar
  21. 21.
    R. A. Millikan, Phys. Rev. 32, 349(1911); Phys. Rev. 22, 1 (1923).1Google Scholar
  22. 22.
    M. D. Allen and O. G. Raabe, Aerosol Sci. and Tech. 4, 269(1985).Google Scholar
  23. 23.
    E. Cunningham, Proc. Roy. Soc. 83, 357(1910).Google Scholar
  24. 24.
    P. S. Epstein, Phys. Rev. 23, 710(1924).Google Scholar
  25. 25.
    A. Einstein and L. Hopf, Ann. Phys. 33, 1105(1910).Google Scholar
  26. 26.
    A. Einstein, Phys. Z. 10, 185(1909).Google Scholar
  27. 27.
    A. Einstein, Ann. Phys. 17, 549(1905).Google Scholar
  28. 28.
    E. N. daC. Andrade and R. C. Parker, Proc. Roy. Soc. 159, 507(1937).Google Scholar
  29. 29.
    H. Lamb, Hydrodynamics (Dover, New York, 1945), p. 605.Google Scholar
  30. 30.
    A. Einstein, Ann. Phys. 19, 371(1906).Google Scholar
  31. 31.
    H. Lamb, op. cit., p. 589.Google Scholar
  32. 32.
    P. Pearle, Found. Phys. 30, 1145(2000).Google Scholar
  33. 33.
    We are indebted to Frank Avignone for supplying recent data and to Jim Ring for analyzing it.Google Scholar
  34. 34.
    Q. Fu, Phys. Rev. A 56, 1806(1997).Google Scholar
  35. 35.
    G. C. Ghirardi, R. Grassi, and A. Rimini, Phys. Rev. A 42, 1057(1990).Google Scholar
  36. 36.
    P. Pearle and E. Squires, Found. Phys. 26, 291(1996).Google Scholar
  37. 37.
    F. Aicardi, A. Borsellino, G. C. Ghirardi, and R. Grassi, Found. Phys. Lett. 4, 109(1991).Google Scholar
  38. 38.
    B. T. Chen et al., J. Aerosol. Sci. 24, 181(1993).Google Scholar
  39. 39.
    D. M. Tanenbaum et al., J. Vac. Sci., submitted; Cornell Project 789-99.Google Scholar
  40. 40.
    W. Paul, Rev. Mod. Phys. 60, 531(1990).Google Scholar
  41. 41.
    R. F. Wuerker et al., J. Appl. Phys. 30, 342(1958).Google Scholar
  42. 42.
    S. Arnold, J. H. Li, S. Holler, A. Korn, and A. F. Izmailov, J. Appl. Phys. 78, 3566(1995).Google Scholar
  43. 43.
    S. Arnold, L. M. Foley, and A. Korn, J. Appl. Phys. 74, 4291(1993).Google Scholar
  44. 44.
    J. S. Hoye and I. Brevik, Physica A 196, 241(1993). We would like to thank Peter Milonni for calling our attention to this paper.Google Scholar
  45. 45.
    J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 414.Google Scholar
  46. 46.
    J. D. Jackson, ibid., p. 805.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Brian Collett
    • 1
  • Philip Pearle
    • 1
  1. 1.Department of PhysicsHamilton CollegeClinton

Personalised recommendations