Il Nuovo Cimento B (1971-1996)

, Volume 82, Issue 2, pp 214–224 | Cite as

Classical limit of quantum mechanics. A paradoxical example

  • D. Home
  • S. Sengupta


In the context of the general problem of equivalence between classical mechanics and quantum mechanics in the macroscopic limit, we point out that, for the particular case of the one-dimensional Coulomb potential, the quantum-mechanical result in the classical limit, corresponding to a certain superposition of odd- and even-parity energy eigenfunctions, leads to inconsistency with classical mechanics. It is shown that the contradiction persists even if the singularity of the Coulomb potential is treated as the limiting case of a modified Coulomb potential in which the singularity has been smoothed out. The possible implication of this paradoxical finding is briefly discussed.

PACS. 03.65 Quantum theory quantum mechanics 

Классический предел квантовой механики. Пример парадокса


В связи с общей проблемой эквивалентности между классической механикой и квантовой механикой в макроскопическом пределе, мы отмечаем, что для частного случая одномерного кулоновского потенциала квантовомеханический результат в классическом пределе, соответствующий определенной суперпозиции четных и нечетных собственных энергетических состояний, приводит к противоречию с классической механикой. Показывается, что это противоречие сохраняется, даже если сингулярность кулоновского потенциала рассматривается, как предельный случай модифицированного кулоновского потенциала, в котором сингулярность сглажена. Вкратце обсуждаются возможные следствия этого парадоксального результата.


Nel contesto del problema generale dell'equivalenza tra meccanica classica e quantistica nel limite macroscopico, si evidenzia che, per il caso particolare del potenziale di Coulomb unidimensionale, il risultato quantomeccanico nel limite classico, corrispondente a una certa sovrapposizione di autofunzioni di energia con parità dispari e pari, porta all'incoerenza con la meccanica classica. Si mostra che la contraddizione persiste anche se la singolarità del potenziale di Coulomb è trattata come caso limite di un potenziale di Coulomb modificato nel quale la singolarità è stata facilitata. Si discute brevemente la possibile implicazione di questa scoperta paradossale.


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  1. (1).
    A. Einstein: InScientific Papers Presented to Max Born (Oliver and Boyd, Edinburgh, London, 1953), p. 33.zbMATHGoogle Scholar
  2. (2).
    M. Born:Kg. Dnsk. Vid. Selsk. Mat. Medd.,30, 1 (1955);M. Born andW. Ludwig:Z. Phys.,150, 106 (1958).MathSciNetGoogle Scholar
  3. (3).
    W. Pauli: InThe Born-Einstein Letters (Macmillan, London, 1971), p. 221.Google Scholar
  4. (4).
    L. de Broglie:Nonlinear Wave Mechanics, a Causal Interpretation (Elsevier, Amsterdam, 1960), p. 136.Google Scholar
  5. (5).
    A. Peres andN. Rosen:Phys. Rev. B,135, 1486 (1964).MathSciNetADSCrossRefGoogle Scholar
  6. (6).
    L. S. Brown:Am. J. Phys.,40, 371 (1972).ADSCrossRefGoogle Scholar
  7. (7).
    D. Home andS. Sengupta:Am. J. Phys.,51, 265 (1983).ADSCrossRefGoogle Scholar
  8. (8).
    For detailed exposition of the ensemble interpretation of quantum mechanics see, for example,H. Margenau:Philos. Sci.,30, 6 (1963);J. L. Park:Am. J. Phys.,36, 211 (1968);J. B. Hartle:Am. J. Phys.,36, 704 (1968);L. E. Ballentine:Rev. Mod. Phys.,42, 358 (1970);F. J. Belinfante:Conventional Quantum Theory, Publications of the University of Joensuu, Series B1, No. 14, 17 (1978);R. G. Newton:Am. J. Phys.,48, 1029 (1980);D. Home andS. Sengupta:Physics News (Bulletin of the Indian Physics Association),12, 7 (1981).Google Scholar
  9. (9).
    See relevant remarks on this point byA. Einstein in ref. (1)..zbMATHGoogle Scholar
  10. (10).
    A. Einstein: inThe Born-Einstein Letters (macmillan, London, 1971), p. 208.Google Scholar
  11. (11).
    For a pertinent discussion regarding the classical-limit criterion see,e.g.,A. Messiah:Quantum Mechanics, Vol.1 (North-Holland, Amsterdam, 1961), p. 214.Google Scholar
  12. (12).
    M. Andrews:Am. J. Phys.,44, 1064 (1976).ADSCrossRefGoogle Scholar
  13. (13).
    M. W. Cole andM. H. Cohen:Phys. Rev. Lett.,23, 1238 (1969); for a comprehensive review of the related works seeF. Stern:Surf. Sci.,58, 383 (1976).ADSCrossRefGoogle Scholar
  14. (14).
    R. Loudon:Am. J. Phys.,27, 649 (1969); however, we have not considered the nondegenerate ground state obtained byLoudon because this state is physically irrelevant, which has been pointed out byM. Andrews:Am. J. Phys.,34, 1194 (1966). It may be also noted that the validity of the even-parity energy eigensolutions derived byLoudon (given by eq. (3) in the present paper) has been questioned byL. K. Haines andD. H. Roberts:Am. J. Phys.,37, 1145 (1969). However, there is fallacy in such an objection, which has been recently pointed out byD. Home andS. Sengupta:Am. J. Phys.,50 552 (1982).ADSCrossRefGoogle Scholar
  15. (15).
    See, for example,M. Abramowitz andI. A. Stegun (Editors):Handbook of Mathematical Functions (Dover, New York, 1970), p. 509.Google Scholar
  16. (16).
    E. T. Whittaker andG. N. Watson:Modern Analysis (Cambridge University Press, London, 1952), p. 337.Google Scholar
  17. (17).
    A. Erdelyi, W. Magnus, F. Oberhettinger andF. Tricomi:Higher Transcendental Functions, Vol.1 (McGraw-Hill, New York, 1953), p. 264.Google Scholar
  18. (18).
    G. Ludwig: inWerner Heisenberg und die Physik unserer Zeit (Berlin, 1961).Google Scholar
  19. (19).
    H. Kummel:Nuovo Cimento,1, 1057 (1955).MathSciNetCrossRefGoogle Scholar
  20. (20).
    N. Rosen:Am. J. Phys.,32, 597 (1964).ADSCrossRefGoogle Scholar
  21. (21).
    E. P. Wigner: inContemporary Research in the Foundations and Philosophy of Quantum Theory, edited byC. A. Hooker (D. Reidel, Dordrecht, Holland, 1973), p. 380.Google Scholar

Copyright information

© Società Italiana di Fisica 1984

Authors and Affiliations

  • D. Home
    • 1
  • S. Sengupta
    • 1
  1. 1.Solid State Physics Research Centre Physics DepartmentPresidency CollegeCalcuttaIndia

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