Path Integrals pp 163-200 | Cite as

# Path Integrals and The Relation Between Classical and Quantum Mechanics

Chapter

## Abstract

Our perception of the world outside ourselves can best be described in the terms of classical physics. The phenomena on the atomic scale require the ideas of quantum mechanics for their understanding which remain rather formal and abstract in our mind. Hence the need to tie the two realms together. Many problems can be treated successfully, although only approximately, with the help of classical mechanics, e.g. transition rates in chemistry and self-bound states in quantum field theory (solitons and instantons).

## Keywords

Periodic Orbit Canonical Transformation Position Space Conjugate Point Classical Trajectory
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.A. Einstein, Verhandlungen der Deutschen Physikalischen Gesellschaft
**19**, 82–92 (1917).Google Scholar - 2.Max Born, The Mechanics of the Atom, republished by Frederick Ungar Publishing Co., New York, 1960. Neither Born nor Sommerfeld in Atombau und Spektrallinien (first published 1919) mention Einstein’s article. It is hard to understand the extent to which this important contribution of Einstein to early quantum mechanics was ignored by his contemporaries. W. Pauli wrote two long review articles on the old quantum mechanics, “Quantentheorie” in Handbuch der Physik, H. Geiger and K. Scheel, eds., Vol. 23, Springer-Verlag 1926, pp. 1–278, and “Allgemeine Grundlagen der Quantentheorie des Atombaues”, Chap. 29 in Muller-Pouillet’s Lehrbuch, Vol. 2, part 2, F. Vieweg, Braunschweig, 1929, p.p. 1709–1842. He also wrote the article “Einstein’s Beitrag zur Quantentheorie” for “Allbert Einstein als Philosoph und Naturforscher”, P. A. Schilpp, ed., W. Kohlhammer Verlag, Stuttgart, 1955, pp. 74–83, which contains a list of Einstein’s contributions to quantum theory. Yet, the article of reference 1 is mentioned nowhere.Google Scholar
- 3.There is a wide variety of articles in this area which started with the paper by J. Hadamard, J. Math. Pure Appl.
**4**, 27–87 (1898).MATHGoogle Scholar - 3a.Further classic papers are by E. Artin, Abhandl. Math. Sem. Hamburg
**3**, 170–175 (1924);MathSciNetCrossRefMATHGoogle Scholar - 3b.G. A. Hediund, Bull. Am. Math. Soc.
**47**, 241–260 (1939);CrossRefGoogle Scholar - 3c.D. V. Anosov, Geodesic Flows on closed Riemamian manifolds with negative curvature, Proc. Steklov Inst. Math.
**97**, translated by Am. Math. Soc, Providence, 1969; E. Hopf, Bull. Am. Math. Soc.**77**,863–877(1971).MathSciNetCrossRefMATHGoogle Scholar - 4.The restricted three-body problem is the main topic in a monograph by V. Szebehely, Theory of Orbits, Academic Press, 1967.Google Scholar
- 4a.A lot of numerical exploration has been made since in order to establish various families of trajectories, e.g. a series of investigations by M. Henon, Ann. Astro.
**28**, 499 and 992 (1965);ADSGoogle Scholar - 4b.
- 4c.M. Henon, Astron. and Astrophys.
**1**, 223 (1969)ADSMATHGoogle Scholar - 4d.M. Henon, Astron. and Astrophys.
**9**, 24 (1970).ADSMATHGoogle Scholar - 5.The anisotropic Kepler problem has been investigated numerically in quantum mechanics by W. Kohn and J. M. Luttinger, Phys. Rev.
**96**, 1488 (1954);ADSCrossRefGoogle Scholar - 5a.R. A. Faulkner, Phys. Rev.
**184**, 713 (1969).ADSCrossRefGoogle Scholar - 6.The motion of the moon is the topic of E. W. Brown, An Introductory Treatise on Lunar Theory, Cambridge LTniversity Press 1896, republished by Dover Publications 1960; a less detailed account of lunar theory aud a very readable introduction to celestial mechanics is D. Brouwer and G. M. Clemence, Methods of Celestial Mechanics, Academic Press, 1961.MATHGoogle Scholar
- 7.These general properties are derived in many textbooks on classical mechanics, e.g. L. A. Pars, A Treatise on Analytical Dynamics, Heinemann, London, 1965;MATHGoogle Scholar
- 7a.C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer Verlay, 1971.CrossRefMATHGoogle Scholar
- 8.This important property was discovered by the astronomer G. W. Hill, Am. J. Math.
**1**, 5–26, 129–147, 245–260 (1878); it was rediscovered by the mathematician Floquet, Ann. de l’Ecole norm, sup (2),**XII**,**43**(1883); and it entered physics through the work of F. Bloch, Zeits. f. Phys.**52**, 555 (1928).CrossRefMATHGoogle Scholar - 9.A good survey of the mathematical accomplishments in this area is given by Jurgen Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press, 1973.MATHGoogle Scholar
- 10.M. C. Gutzwiller, J. Math. Phys.
**14**, 139–152 (1973) contains the numerical investigations;MathSciNetADSCrossRefGoogle Scholar - 10a.while M. C. Gutzwiller, J. Math. Phys. 1
**8**, 806–823 (1977) gives the analytical arguments.MathSciNetADSCrossRefGoogle Scholar - 11.A particularly instructive mechanical example is worked out by M. Henon and C. Heiles, Astron. J.
**69**, 73 (1964).MathSciNetADSCrossRefGoogle Scholar - 11a.In contrast, a sequence of mathematical mappings to imitate the properties of a Poincare map is studied by M. Henon, Q. Appl. Math.
**27**, 291 (1969).MathSciNetMATHGoogle Scholar - 12.Bernoulli sequences are discussed in ref. 9. Another very readable account from a mathematician viewpoint can be found in V. I. Arnold and A. Avez, Problemes ergodignes de la mecanique classique, Gauthier-Villars, 1967; translated into English at Benjamin, New York, 1968.Google Scholar
- 13.C. Garrod, Rev. Mod. Phys.
**38**, 483 (1966).MathSciNetADSCrossRefMATHGoogle Scholar - 14.Cf. S. F. Edwards and Y. V. Gulayev, Proc. Roy. Soc. (London)
**279**, 229 (1964);ADSCrossRefGoogle Scholar - 14a.A. M. Arthurs, Proc. Roy. Soc. (London)
**318**, 523 (1970); also paper I in ref. 15.ADSCrossRefGoogle Scholar - 15.The developments in the remainder are mostly taken from a series of papers to be designated as I, II, III, IV by the author, M. C. Gutzwiller, J. Math. Phys.
**8**, 1979 (1967);ADSCrossRefGoogle Scholar - 15a.M. C. Gutzwiller, J. Math. Phys.
**10**, 1004 (1969);ADSCrossRefGoogle Scholar - 15b.M. C. Gutzwiller, J. Math. Phys.
**11**, 1791 (1971);ADSCrossRefGoogle Scholar - 15c.M. C. Gutzwiller, J. Math. Phys.
**12**, 343 (1971).ADSCrossRefGoogle Scholar - 16.M. Morse, The Calculus of Variations in the Large, Am. Math. Soc, Providence, Rhode Island, 1935. A more easily accessible account is given by N. Seifert and W. Threlfall, Variationsrechnung im Grossen (Theorie von Marston Morse) B. G. Teubner, 1928; republished by Chelsea, New York, 1951; J. Milnor, Morse Theory, Princeton University Press, 1962.Google Scholar
- 17.The simplicity of Kepler’s problem in momentum space has been widely recognized in quantum mechanics since the treatment of V. Fock, Z. Phys.
**98**, 145 (1935).ADSCrossRefGoogle Scholar - 17a.It rests on the connection with the group of rotations 0(4) which was first used by W. Pauli, Z. Phys.
**36**, 336 (1926).ADSCrossRefMATHGoogle Scholar - 18.This result was first derived by the author in I, but independently, by A. Norcliffe and I. C. Percival, J. Phys. B. Ser. 2,
**1**, 774, 784 (1968).ADSGoogle Scholar - 19.The classical quantization of non-separable mechanical systems with invariant tori has become a very active area of research, and only some early representative references can be given. R. A. Marcus, Faraday Discussions of the Chemical Society,
**55**, 34 (1973);CrossRefGoogle Scholar - 19a.R. A. Marcus, J. Chem. Phys.
**61**, 4301 (1974)ADSCrossRefGoogle Scholar - 19b.R. A. Marcus, J. Chem. Phys.
**62**, 2119 (1975);ADSCrossRefGoogle Scholar - 19c.M. V. Berry and M. Tabor, Proc. Roy. Soc. London A
**349**, 101–123 (1976);MathSciNetADSCrossRefGoogle Scholar - 19d.K. S. Sorbie, Molecular Physics,
**32**, 1577 and 1327 (1976);ADSCrossRefGoogle Scholar - 19e.K. S. J. Nordholm and S. A. Rice, J. Chem. Phys.
**61**, 203 and 768 (1974);ADSCrossRefGoogle Scholar - 19f.K. S. J. Nordholm and S. A. Rice, J. Chem. Phys.
**62**, 157 (1975).ADSCrossRefGoogle Scholar - 20.This argument has been put in doubt by K. F. Freed, Faraday Discussions of the Chemical Society
**55**, 68 (1973). He reasons that q’ and q” need only be inside the volume d q of integration which is assumed to be of linear extent δq, and the momenta may differ by an amount δp. The variation of S(q”q’E) is, therefore, of the order δp δq, and destructive interference does not occur as long as δp δq<h. Trajectories might be important when they close approximately, even if they cannot do it exactly. Our skill in treating mechanical systems, ergodic or not, does not seem sufficient at present to resolve this issue. For a start we shall assume the extreme position of excluding all except perfectly closed trajectories.Google Scholar - 21.The authors’ results have been misconstrued and misrepresented almost consistently in this respect. Cf. D. W. Noid and R. A. Marcus, J. Chem. Phys.
**62**, 2119 (1975);ADSCrossRefGoogle Scholar - 21a.M. V. Berry and M. Tabor, Proc. Roy. Soc. Lond. A.
**349**, 101 (1976); and others in ref. 19 and 22. On the contrary, the first applications of (84) in III show that all periodic orbits are needed to give rise to resonance, not only a single primitive one. In IV, however, an example was investigated where only one periodic orbit was known at the time, and it was used to construct approximate eigenvalues for the lack of more information about the system, the anisotropic Kepler motion in two dimensions. With the many more periodic orbits which have been discovered since (cf. ref. 10), the approximate treatment of IV will have to be reexamined.MathSciNetADSCrossRefGoogle Scholar - 22.The idea that the stability angle α includes the effect of the conjugate points was clearly stated in IV. Yet, Miller, J. Chem. Phys.
**63**, 996 (1975) states the contrary.CrossRefGoogle Scholar - 22a.Even more unfortunate is his assertion in J. Chem. Phys.
**64**, 502 (1976) that the author’s method based on (89) is just “an approximate version of Keller’s formalism”. The reference is to J. B. Keller, Ann. Phys. (N.Y.)**4**, 180 (1958);ADSCrossRefMATHGoogle Scholar - 22b.J. B. Keller and S. I. Rubinow, Ann. Phys. (N.Y.)
**9**, 24 (1960) whose work is based entirely on the construction of wave functions for a classical mechanical system with invariant tori in phase space. Ergodic systems are not considered there.ADSCrossRefMATHGoogle Scholar - 23.R. Balian and C. Bloch, Am. Phys. (N.Y.)
**60**, 401 (1970);MathSciNetADSMATHGoogle Scholar - 23a.R. Balian and C. Bloch, Am. Phys. (N.Y.)
**63**, 592 (1971);ADSGoogle Scholar - 23b.R. Balian and C. Bloch, Am. Phys. (N.Y.)
**64**, 271 (1971);MathSciNetADSMATHGoogle Scholar - 23c.R. Balian and C. Bloch, Am. Phys. (N.Y.)
**69**, 76 (1972);MathSciNetADSMATHGoogle Scholar - 23e.A. Voros, Am. Inst. H. Poincare
**24**, 31 (1976);MathSciNetGoogle Scholar - 23d.L. Hormander, Acta Math.
**127**, 79 (1971);MathSciNetCrossRefGoogle Scholar - 23f.J. J. Duistermaat, Comm. Pure Appl. Math.
**27**, 207 (1974);MathSciNetCrossRefMATHGoogle Scholar - 23g.J. Chazarain, Inventiones Math.
**24**, 65 (1974);MathSciNetADSCrossRefMATHGoogle Scholar - 23h.Y. Colin de Verdiere, Compositio math.
**27**, 83 and 159 (1973);MathSciNetMATHGoogle Scholar - 23i.J. J. Duistermaat and V. W. Guillemin, Inventiones Math.
**29**, 39 (1975);MathSciNetADSCrossRefMATHGoogle Scholar - 23k.A. Weinstein, in Fourier Integral Operators, Lecture Notes in Mathematics, Springer,
**459**, 341 (1975).CrossRefGoogle Scholar - 24.I. C. Percival, J. Phys. B
**6**L, 229 (1973).ADSCrossRefGoogle Scholar - 25.N. Pomphrey, J. Phys. B
**7**, 1909 (1974).ADSCrossRefGoogle Scholar - 26.W. H. Miller, J. Chem. Phys.
**64**, 2880 (1976).MathSciNetADSCrossRefGoogle Scholar - 27.Cf. ref. 22. Also W. H. Miller, J. Chem. Phys.
**56**, 38 (1972) and, in particular,ADSCrossRefGoogle Scholar - 27a.W. H. Miller, J. Chem. Phys.
**62**, 1899 (1975) are of interest in this connection. Path integral arguments are used in order to compute transition rates in quantum statistics. Periodic orbits now occur in the inverted potential and the stability exponents are indeed the frequencies for the transverse degrees of freedom. The expression (6) gives their occupation probability.ADSCrossRefGoogle Scholar - 28.This fact has been known for some time, in particular since the geodesics on a surface of negative curvature can be constructed quite explicitly, cf. ref. 3. A detailed numerical study in a system where no such complete description is available, was started by G. H. Lunsford and J. Ford, J. Math. Phys.
**13**, 700 (1972). They try to give a mathematical formulation (based on their empirical data) to the way in which periodic orbits occur in an ergodic system.MathSciNetADSCrossRefMATHGoogle Scholar

## Copyright information

© Springer Science+Business Media New York 1978