Foundations of Physics

, Volume 26, Issue 9, pp 1165–1199 | Cite as

Quantization by parts, self-adjoint extensions, and a novel derivation of the Josephson equation in superconductivity

  • K. Kong Wan
  • R. H. Fountain
Article

Abstract

There has been a lot of interest in generalizing orthodox quantum mechanics to include POV measures as observables, namely as unsharp obserrables. Such POV measures are related to symmetric operators. We have argued recently that only maximal symmetric operators should describe observables.1 This generalization to maximal symmetric operators has many physical applications. One application is in the area of quantization. We shall discuss a scheme, to he called quantization by parts,which can systematically deal with what may be called quantum circuits. As a specific application we shall present a novel derivation of the famous Josephson equation for the supercurrent through a Josephson junction in a superconducting circuit. An interesting effect emerges from our quantization scheme when applied to a superconducting Y-shape circuit configuration. We also propose an experimental test for this effect which is expected to shed light on some conceptual problems on the quantum nature of the condensate.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. K. Wan, R. H. Fountain, and Z. Y. Tao.J. Phys. A: Math. Gen. 28, 2379 (1995). Note that in this paper the condition: ℋ(F′φ) > ℋ() ∀φ∈Gin Definition 2 on p. 2382 should be replaced by the condition: ℋ(F′φ) ⩾ ℋ() ∀φ∈G and ℋ(F′φ) > ℋ() for some φ∈G.Google Scholar
  2. 2.
    J. Bardeen, L. N. Cooper, and J. R. Schreiffer.Phys. Rev. 108, 1175 (1957).Google Scholar
  3. 3.
    A. Barone and G. Paterno.Physics and Applications of the Josephson Effect (Wiley, New York, 1982). pp. 2.9 10. 18. 23.Google Scholar
  4. 4.
    D. R. Tilley and J. Tilley.Superfluidity and Superconductivity (Adam Hilger. Bristol. 1990). p. 39.Google Scholar
  5. 5.
    R. P. Feynman,Statistical Mechanics (Benjamin, Reading, Massachusetts, 1972). p. 304.Google Scholar
  6. 6.
    R. P. Feynman,The Feynman Lectures on Physics, VolIII (Addison-Wesley, New York, 1965). Sec. 21-9.Google Scholar
  7. 7.
    K. K. Wan and F. E. Harrison,Phys. Lett. A 174. 1 (1993).Google Scholar
  8. 8.
    K. K. Wan and F. E. Harrison, 1996 preprint entitled “Macroscopic quantum systems as measuring devices: de SQUIDS and superselection rules.”Google Scholar
  9. 9.
    Y. Srivastava and A. Widom,Phys. Rep. 148. 1 (1987).Google Scholar
  10. 10.
    T. P. Spiller, T. D. Clark, R. J. Prance, and A. Widom.Quantum Phenomena in Circuits at Low Temperature, in Progress in Low Temperature Physics. Vol XIII. E. D. Brewer, (North-Holland, Amsterdam, 1992).Google Scholar
  11. 11.
    B. D. Josephson,Phys. Lett. 1. 251 (1962).Google Scholar
  12. 12.
    P. Exner and P. Seba.J. Math. Phys. 28. 386 (1987).Google Scholar
  13. 13.
    P. Exner and P. Seba.Rep. Math. Phys. 28. 7 (1989).Google Scholar
  14. 14.
    P. Exner and P. Seba. “Quantum junctions and the self-adjoint extensions theory.” also P. Exner, P. Seba. and P. Stovicek, “Quantum waveguides.” both inApplications of Self-Adjoint Extensions in Quantum Physics. P. Exner and P. Seba. eds. (Springer, Berlin. 1989).Google Scholar
  15. 15.
    J. Blank. P. Exner. and M. Havlieek.Hilbert Space Operators in Quantum Physics (American Institute of Physics. New York. 1994). pp. 60. 471–489, 145, 149, 137, 122.Google Scholar
  16. 16.
    N. I. Akhiezer and I. M. Glazman.Theory of Linear Operators in Hilbert Space. Vol. 2 (Fiederick Ungar. New York. 1963). pp. 101, 97.Google Scholar
  17. 17.
    J. Weidman,Linear Operators in Hilbert Spaces (Springer, New York. 1980). pp. 160. 239, 240, 238, 162, 124.Google Scholar
  18. 18.
    R. D. Richtmyer,Principles of Advanced Mathematical Physics. Vol. 1 (Springer, New York. 1978). pp. 155, 141, 157.Google Scholar
  19. 19.
    G. Baym.Lectures on Quantum Mechanics (Benjamin, New York. 1969), pp. 58, 267.Google Scholar
  20. 20.
    A. C. Rose-Innes and E. H. Rhoderick.Introduction to Superconductivity (Pergamon, London, 1969), p. 15.Google Scholar
  21. 21.
    F. Mandl.Quantum Mechanics (Wiley. Chichester, 1992). Sec. 2.2 10.Google Scholar
  22. 22.
    M. Reed and B. Simon.Fourier Analysis. Self-Adjointness (Academic New York, 1975). pp. 144 145.Google Scholar
  23. 23.
    P. G. de Gennes.Phys. Lett. 5. 22 (1963).Google Scholar
  24. 24.
    P. G. de Gennes.Rev. Mod. Phys. 36. 225 (1964).Google Scholar
  25. 25.
    C. G. Kuper.An Introduction to the Theory of Superconductivity (Clarendon. Oxford, 1968). p. 141.Google Scholar
  26. 26.
    K. K. Wan.Can. J. Phys. 58 (1980) 976.Google Scholar
  27. 27.
    E. Beltrametti and G. Cassinelli.The Logic of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts. 1981).Google Scholar
  28. 28.
    J. Dixmier.Von Neumann Algebras (North-Holland, Amsterdam, 1981). pp. 161 1172. 179 194.Google Scholar
  29. 29.
    R. H. Dicke and J. P. Wittke,Introduction to Quantum Mehcanics (Addison-Wesley. Reading, Massachusetts, 1963), p. 61.Google Scholar
  30. 30.
    R. A. Hegstrom and F. Sols.Found. Phys. 25. 681 (1995).Google Scholar
  31. 31.
    R. S. Longhurst.Geometrical and Physical Optics (Longmans, London. 1963), p. 294.Google Scholar
  32. 32.
    H. Rauch. “Tests of quantum mechanics by Neutron interferometry.” inOpen Questions in Quantum Physics. G. Torozzi and A. van der Merwe, eds. (Reidel. Dordrecht, 1985). pp. 345 476.Google Scholar
  33. 33.
    A. C. Rose-Innes and E. H. Rhoderick.Introduction to Superconductivity (Pergamon, London. 1994). pp. 170 178.Google Scholar
  34. 34.
    N. Bohr. “Discussions with Einstein on epistemological problems in atomic physics.” inAlbert Einstein; Philosopher-Scientist. P. A. Schilpp. ed. (Open Court, Evanston. 1949). pp. 200 241.Google Scholar
  35. 35.
    J. A. Wheeler. “Law without Law.” inQuantum Theory of Measurement. J. A. Wheeler and W. H. Zurek. eds. (Princeton University Press. Princeton. 1983). pp. 182 213.Google Scholar
  36. 36.
    T. Hellmuth. A. G. Zajonc. and H. Walther. “Realization of a «delayed-choice» Mach Zehnder interferometer, inSymposium in the Foundations of Modern Physics.” P. Lahti and P. Mittelstaedt. eds. I World Scientific, Singapore. 1985). pp. 417 422.Google Scholar
  37. 37.
    F. Selleri and G. Tarozzi,Riv. Nuovo Cimento 4. No. 2 (1981).Google Scholar
  38. 38.
    K. K. Wan,Found. Phys. 18. 887 (1988).Google Scholar
  39. 39.
    K. K. Wan and R. G. McLean.Found. Phys. 24 715 737 (1994).Google Scholar
  40. 40.
    K. K. Wan and F. E. Harrison.Found. Phys. 24. 831 853 (1994).Google Scholar
  41. 41.
    A. Widom.J. Low Temp Phys. 37. 449 466 (1979).Google Scholar
  42. 42.
    A. J. Leggett. “Quantum mechanics at the macroscopic level.” inThe Lesson oj Quantum Theory. J. de Boer. E. Dal. and O. Ullbeck. eds. (Elsevier. New York. 1986). pp. 395 506.Google Scholar
  43. 43.
    Y. Srivastava and A. Widom.Phys. Rep. 148, 1 65 (1987).Google Scholar
  44. 44.
    T. D. Clark. “Macroscopic quantum objects in quantum implications.” inEssays in Honour of David Bohm. B. J. Hiley and E. D. Peat. eds. (Routledge & Kegan, London. 1987). pp. 121 150.Google Scholar
  45. 45.
    A. Widom. “Implications of superconducting circuits for relativistic quantum electrodynamics.” inMacroscopic Quantum Phenomena. T. D. Clark. H. Prance. R. J. Prance, and T. P. Spiller, eds. (World Scientific. Singapore. 1991).Google Scholar
  46. 46.
    K. K. Wan.Can. J. Phys. 58. 976 (1980).Google Scholar
  47. 47.
    E. Beltrametti and G. Cassinelli.The Logic of Quantum Mechanics (Addison-Wesley. Reading. Massachusetts. 1981).Google Scholar
  48. 48.
    B. C. van Fraasscn.Quantum. Mechanics: An Empirieist View Clarendon. Oxford. 1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • K. Kong Wan
    • 1
  • R. H. Fountain
    • 1
  1. 1.Department of Physics and AstronomyUniversity of St. AndrewsFifeScotland, UK

Personalised recommendations