References
337
9.9 Conclusions
In this chapter, we have attempted to give a systematic derivation of the semiclassical path integrals that are now used to form a bridge between classically chaotic
systems and their quantum counterparts. We have only been able to discuss a few
applications. However, the field is rich and growing.
The problem begun by Selberg of finding semiclassical properties of a quantum
particle moving on a Riemann surface of constant negative curvature has been
studied by several authors. An extensive discussion of concepts and references can
be found in Gutzwiller (1990). See also Gutzwiller (1985), Balazs and Voros (1986),
and Balazs et al. (1987).
Another area that we should mention is scattering theory. There have been
several papers using semiclassical path integrals to describe scattering from twodimensional composite molecules. These are generally composed of three or four
scattering sites arranged in a symmetric manner. If the molecule is simple enough,
it is possible to systematically code all paths of an incident particle as it enters
the molecule and bounces around among the scattering sites before leaving the
molecule. For further discussion of this topic, see Gutzwiller (1985, 1990), Eckhardt
(1987), Jung and Scholz (1987), Blumel and Smilansky (1988), Gaspard and Rice
(1989), Bleher et al. (1989), and Blumel and Reinhardt (1997).
In all the applications we have mentioned for semiclassical path integrals, the
central reason for the success of the path integral approach is the fact that all the
unstable periodic orbits can be classified, their actions computed, and the path
integrals can be summed. The classification of periodic orbits is usually done with
symbolic dynamics. For this reason, we mention an excellent book on symbolic
dynamics by Hao (1989), even though the book does not discuss applications to
path integrals.
References
Balazs NL, Voros A (1986) Phys Rep 143:109
Balazs NL, Schmidt C, Voros A (1987) J Stat Phys 46:1067
Balian R, Bloch C (1972) Ann Phys 69:76
Balian R, Bloch C (1974) Ann Phys 85:514
Beims MW, Alber G (1993) Phys Rev A 48:3123
Berry MV, Mount KE (1972) Rep Prog Phys 35:315
Bleher S, Ott E, Grebogi C (1989) Phys Rev Lett 63:919
Blumel R, Reinhardt WP (1997) Chaos in atomic systems. Cambridge University Press, Cambridge
Blumel R, Smilansky U (1988) Phys Rev Lett 60:477
Born M (1960) The mechanics of the atom. Frederick Ungar, New York
Choquard Ph. (1955) Helv Phys Acta 28:89
Delande D, Gay JC (1986a) Phys Rev Lett 57:2006
Delande D, Gay JC (1986b) J. Phys. B 19:L173
Devaney RL (1978a) J. Differ Equ 29:253
Devaney RL (1978b) Invent Math 45:221
337
9.9 Conclusions
In this chapter, we have attempted to give a systematic derivation of the semiclassical path integrals that are now used to form a bridge between classically chaotic
systems and their quantum counterparts. We have only been able to discuss a few
applications. However, the field is rich and growing.
The problem begun by Selberg of finding semiclassical properties of a quantum
particle moving on a Riemann surface of constant negative curvature has been
studied by several authors. An extensive discussion of concepts and references can
be found in Gutzwiller (1990). See also Gutzwiller (1985), Balazs and Voros (1986),
and Balazs et al. (1987).
Another area that we should mention is scattering theory. There have been
several papers using semiclassical path integrals to describe scattering from twodimensional composite molecules. These are generally composed of three or four
scattering sites arranged in a symmetric manner. If the molecule is simple enough,
it is possible to systematically code all paths of an incident particle as it enters
the molecule and bounces around among the scattering sites before leaving the
molecule. For further discussion of this topic, see Gutzwiller (1985, 1990), Eckhardt
(1987), Jung and Scholz (1987), Blumel and Smilansky (1988), Gaspard and Rice
(1989), Bleher et al. (1989), and Blumel and Reinhardt (1997).
In all the applications we have mentioned for semiclassical path integrals, the
central reason for the success of the path integral approach is the fact that all the
unstable periodic orbits can be classified, their actions computed, and the path
integrals can be summed. The classification of periodic orbits is usually done with
symbolic dynamics. For this reason, we mention an excellent book on symbolic
dynamics by Hao (1989), even though the book does not discuss applications to
path integrals.
References
Balazs NL, Voros A (1986) Phys Rep 143:109
Balazs NL, Schmidt C, Voros A (1987) J Stat Phys 46:1067
Balian R, Bloch C (1972) Ann Phys 69:76
Balian R, Bloch C (1974) Ann Phys 85:514
Beims MW, Alber G (1993) Phys Rev A 48:3123
Berry MV, Mount KE (1972) Rep Prog Phys 35:315
Bleher S, Ott E, Grebogi C (1989) Phys Rev Lett 63:919
Blumel R, Reinhardt WP (1997) Chaos in atomic systems. Cambridge University Press, Cambridge
Blumel R, Smilansky U (1988) Phys Rev Lett 60:477
Born M (1960) The mechanics of the atom. Frederick Ungar, New York
Choquard Ph. (1955) Helv Phys Acta 28:89
Delande D, Gay JC (1986a) Phys Rev Lett 57:2006
Delande D, Gay JC (1986b) J. Phys. B 19:L173
Devaney RL (1978a) J. Differ Equ 29:253
Devaney RL (1978b) Invent Math 45:221