Skip to main content
SpringerLink
Log in

Correspondence rules and path integrals

Правила соответствия и интегралы по траекториям

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

A path-integral representation is constructed for propagators corresponding to quantum Hamiltonian operators obtained from classical Hamiltonians by an arbitrary rule of correspondence. Each rule yields a unique way of defining the path integral in the context of a formalism which does not require a limiting process. This formalism is more reliable than the usual lattice definition in that all the expressions it entails are well defined for computational purposes and it allows the explicit evaluation of large classes of path integrals. Direct substitution in the Schrödinger equation shows that there are no restrictions on the Hamiltonian operator. Examples are given.

Riassunto

Si costruisce una rappresentazione dell’integrale del percorso per propagatori correspondenti ad operatori quanto-hamiltoniani ottenuti da hamiltoniane clamiche mediante una regola arbitraria di corrispondenza. Ogni regola fornisce un unico modo di definire l’integrale del percorso nel contesto di una formalismo che non richiede un processo limitante. Questo formalismo è più attendibile della solita definizione reticolare in quanto tutte le espressioni che esso comprende sono ben definite per fare calcoli ed esso permette una valutazione esplicita di grandi classi di integrali di percorso. La sostituzione diretta nell’equazione di Schrödinger mostra che non ci sono restrizioni sull’operatore hamiltoniano. Si danno esempi.

Резюме

Конструируется представление интегралов по траекториям для пропагаторов, соответствующих квантовым операторам Гамильтона, которые получаются из классических Гамильтонианов с помощью произвольного правила соответствия. Каждое правило дает единственный способ определения интеграла по траекториям в контексте формализма, который не требует предельного процесса. Этот формализм является более надежным, чем обычное решеточное определение, в том смысле, что все выражения точно определены для вычислительных целей, и что этот формалузм допускает точное вычисление больших классов интегралов по траекториям. Непосредственная подстановка в уравнение Шредингера показывает, что не существует ограничений на оператор Гамильтона. Поиводятся примеры.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. M. Mizrahi:J. Math. Phys. (N.Y.),16, 2201 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  2. L. Cohen:J. Math. Phys. (N.Y.),17, 597 (1976).

    Article  ADS  Google Scholar 

  3. J. S. Dowker:J. Math. Phys. (N.Y.),17, 1873 (1976).

    Article  MathSciNet  ADS  Google Scholar 

  4. M. M. Mizrahi:J. Math. Phys.,19, 298 (1978).

    Article  MathSciNet  ADS  Google Scholar 

  5. L. Cohen:J. Math. Phys. (N.Y.),7, 781 (1966).

    Article  ADS  Google Scholar 

  6. However, if one is interested only in a perturbation series in α of the propagator corresponding toH=[g(t) P 2/2m+f(t) Q 2/2+k(t)(PQ+QP)/2+αH 1(P, Q,t)], then the method described here applies directly.

  7. A similar formalism obtained by different methods can be found inC. DeWitt-Morette, A. Maheshwari andB. Nelson:Gen. Rel. Grav.,8, 581 (1977).

    Article  MathSciNet  ADS  Google Scholar 

  8. M. M. Mizrahi:J. Math. Phys. (N.Y.),17, 566 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. M. M. Mizrahi:J. Comp. Appl. Math.,1, 273 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  10. The proof of (19) is similar to the one given in ref. (8).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. This scheme for correspondence rules was also used inM. M. Mizrahi:J. Math. Phys. (N.Y.),18, 786 (1977), to determine the range of validity of a commonly used formula for the WKB approximation of the propagator <q b ,t b |q a ,t a >.

    Article  ADS  Google Scholar 

  12. This formula can be proved by left-multiplying both sides of (28) by exp [i(Q u’+P v’)/ħ], taking traces (note thatA ≡ ∫dq <q|A|q> = ∫dp <dp <p|A|p>), and inserting complete sets of states after using eq. (32).

  13. In thek-expansion of the left-hand side of (73), the vanishing of the 0th-order term results from the Schrödinger equation forK 0, the vanishing of the coefficient ofk results from (65) and the vanishing of the coefficients ofk 2 tok n+1 is what (74) (fromn=2 ton+1) proves.

Download references

Author information

Authors and Affiliations

  1. Center for Naval Analyses of the University of Rochester, 2000 N. Beauregard St., 22311, Alexandria, Va.

    M. M. Mizrahi

Authors
  1. M. M. Mizrahi
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Traduzione a cura della Redazione.

Перевебено ребакцией.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mizrahi, M.M. Correspondence rules and path integrals. Nuov Cim B 61, 81–98 (1981). https://doi.org/10.1007/BF02721705

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02721705

Search

Navigation