References
235
Thus, classically chaotic quantum systems, while not sensitive to small changes in
their initial state, are sensitive to small changes in their Hamiltonian.
We have seen that there is a correspondence between the transition to chaos
in classical systems and a change in the spectral statistics in their quantum
counterparts. However, when the transition to chaos occurs, we lose the traditional
theoretical correspondence between a quantum system and its classical limit. The
Bohr-Sommerfeld correspondence principle is based on the existence of regular
tori in the phase space. It cannot be used in a chaotic system. However, it is
now possible to make a connection between classically chaotic systems and their
quantum counterpart by using path integrals. Path integrals also allow us to relate
the spectral properties of a quantum system to the behavior of unstable orbits in an
underlying classical system. We shall show how this is done in Chap. 9.
References
Agam O, Fishman S (1993) J Phys A Math Gen 26:2113
Agam O, Fishman S (1994) Phys Rev Lett 73:806
Akguc GB, Reichl LE (2000) J Stat Phys 98:813
Alt H, Dembowski C, Graf H-D, Hofferbert R, Rehfeld H, Richter A, Schmidt C (1999) Phys Rev
E 60:2851
Atkins PW, Child MS, Phillips CSG (1970) Tables for group theory. Oxford University Press,
Oxford
Balazs NL, Voros A (1989) Ann Phys 190:1
Barr AD, Barr AR, Porter MD, Reichl LE (2017) Chaos 27:104604
Berry MV (1981) Ann Phys 131:163
Berry MV (1986) In: Seligman TH, Nishioka H (eds) Quantum chaos and statistical nuclear
physics. Lecture Notes in Physics, vol 263. Springer, Berlin
Berry MV (1989) Proc Roy Soc London A 423:219
Berry MV, Keating JP (1992) Proc Roy Soc London A 437:151
Berry MV, Wilkenson M (1984) Proc Roy Soc London A 392:15
Bogomolny EB (1988) Physica D 31:169
Bohigas O, Giannoni MJ, Schmidt C (1984) Phys Rev Lett 52:1
Brun TA, Schack R (1999) Phys Rev A 59:2649
Bunimovich LA (1979) Commun Math Phys 65:295
Bunimovich L, Sinai YG (1980a) Commun Math Phys 78:247
Bunimovich L, Sinai YG (1980b) Commun Math Phys 78:479
Bunimovich L, Sinai YG (1986) Commun Math Phys 107:357(E)
Casati G, Vals-Gris F, Guarneri I (1980) Lett Nuovo Cimento 28:279
Casati G, Chirikov BV, Guarneri I (1985) Phys Rev Lett 54:1350
Chang S-J, Shi K-J (1986) Phys Rev A 34:7
Cheng Z, Lebowitz JL (1991) Phys Rev E 44:R3399
Colin de Verdiere Y (1985) Commun Math Phys 102:497
Cornfeld IP, Fomin SV, Sinai YG (1982) Ergodic theory. Springer, New York
Crespi B, Perez G, Chang S-J (1993) Phys Rev E 47:986
Dittes F-M, Doron E, Smilansky U (1994) Phys Rev E 49:R963
Eckhardt B (1988) Phys Rep 163:205
Eckhardt B, Haake F (1994) J Phys A Math Gen 27:4449
Feingold M, Peres A (1986) Phys Rev A 34:591
235
Thus, classically chaotic quantum systems, while not sensitive to small changes in
their initial state, are sensitive to small changes in their Hamiltonian.
We have seen that there is a correspondence between the transition to chaos
in classical systems and a change in the spectral statistics in their quantum
counterparts. However, when the transition to chaos occurs, we lose the traditional
theoretical correspondence between a quantum system and its classical limit. The
Bohr-Sommerfeld correspondence principle is based on the existence of regular
tori in the phase space. It cannot be used in a chaotic system. However, it is
now possible to make a connection between classically chaotic systems and their
quantum counterpart by using path integrals. Path integrals also allow us to relate
the spectral properties of a quantum system to the behavior of unstable orbits in an
underlying classical system. We shall show how this is done in Chap. 9.
References
Agam O, Fishman S (1993) J Phys A Math Gen 26:2113
Agam O, Fishman S (1994) Phys Rev Lett 73:806
Akguc GB, Reichl LE (2000) J Stat Phys 98:813
Alt H, Dembowski C, Graf H-D, Hofferbert R, Rehfeld H, Richter A, Schmidt C (1999) Phys Rev
E 60:2851
Atkins PW, Child MS, Phillips CSG (1970) Tables for group theory. Oxford University Press,
Oxford
Balazs NL, Voros A (1989) Ann Phys 190:1
Barr AD, Barr AR, Porter MD, Reichl LE (2017) Chaos 27:104604
Berry MV (1981) Ann Phys 131:163
Berry MV (1986) In: Seligman TH, Nishioka H (eds) Quantum chaos and statistical nuclear
physics. Lecture Notes in Physics, vol 263. Springer, Berlin
Berry MV (1989) Proc Roy Soc London A 423:219
Berry MV, Keating JP (1992) Proc Roy Soc London A 437:151
Berry MV, Wilkenson M (1984) Proc Roy Soc London A 392:15
Bogomolny EB (1988) Physica D 31:169
Bohigas O, Giannoni MJ, Schmidt C (1984) Phys Rev Lett 52:1
Brun TA, Schack R (1999) Phys Rev A 59:2649
Bunimovich LA (1979) Commun Math Phys 65:295
Bunimovich L, Sinai YG (1980a) Commun Math Phys 78:247
Bunimovich L, Sinai YG (1980b) Commun Math Phys 78:479
Bunimovich L, Sinai YG (1986) Commun Math Phys 107:357(E)
Casati G, Vals-Gris F, Guarneri I (1980) Lett Nuovo Cimento 28:279
Casati G, Chirikov BV, Guarneri I (1985) Phys Rev Lett 54:1350
Chang S-J, Shi K-J (1986) Phys Rev A 34:7
Cheng Z, Lebowitz JL (1991) Phys Rev E 44:R3399
Colin de Verdiere Y (1985) Commun Math Phys 102:497
Cornfeld IP, Fomin SV, Sinai YG (1982) Ergodic theory. Springer, New York
Crespi B, Perez G, Chang S-J (1993) Phys Rev E 47:986
Dittes F-M, Doron E, Smilansky U (1994) Phys Rev E 49:R963
Eckhardt B (1988) Phys Rep 163:205
Eckhardt B, Haake F (1994) J Phys A Math Gen 27:4449
Feingold M, Peres A (1986) Phys Rev A 34:591