Periodic Orbits and Trace Formula-Integrable Systems

SONG Jian-Jun; LI Xi-Guo; LIU Fang; LI Shu-Wei

Chinese Physics C> 2001, Vol. 25> Issue(9) : 872-876

Periodic Orbits and Trace Formula-Integrable Systems

Institute of Modern Physics, The Chinese Academy of Sciences, Lanzhou 730000, China2 Research Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Collision, Lanzhou 730000, China3 Institute of Modern Physics, Northwest University, Xi′an 710069, China

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Abstract：
The two-dimensional uncoupled quartic oscillator was chosen to test the validity of the Berry-Tabor trace formula. Periodic orbits of the system were calculated by integrating Hamiltonian equations of motion on reasonable tori and the curvatures on the energy surface were obtained by carefully constructing the orbits around the reasonable tori. Finally, the value of the semiclassical action function and that of the quantum action function in the case S<30 were compared, and the good agreement between the two functions indicates the validity of the trace formula. On the other hand, in the quantum action function R^QM(S,E)-S figuration, that peaks appeared at the action values corresponding to the periodic orbits of the classical system provide rich information on quantum-classical correspondence.

References

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. Gutzwiller M C .J.Math.Phys.,1967,8:1979—2000. 2. Berry,Tabor.Proc.R .Soc.Lond.,1976,A349:101—123. 3. Berry.Proc.R .Soc.Lond.,1985,A400:229—251. 4. Ozorio de Almeida A M ,Hannay J H .J.Phys.,1987,A20:5873—5883. 5. Ullmo Denis,Grinberg Maurice,Tomsovic Steven.Phys.Rev.,1996,E54:136—152. 6. Main J,Wunner Gunter.Phys.Rev.Lett.,1999,82:3038—3041. 7. Main J.Phys.Rep.,1999,316:233—338. 8. Berry M V ,Tabor M .J.Phys.,1977,A10:371—379. 9. Bohigas O ,Tomsovic S,Ullmo M .Phys.Rep.,1993,223(2):43—133 10. Quigg C ,Rosner Jonathan L .Phys.Rep.,1979,56(4):167—235

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SONG Jian-Jun, LI Xi-Guo, LIU Fang and LI Shu-Wei. Periodic Orbits and Trace Formula-Integrable Systems[J]. Chinese Physics C, 2001, 25(9): 872-876.

SONG Jian-Jun, LI Xi-Guo, LIU Fang and LI Shu-Wei. Periodic Orbits and Trace Formula-Integrable Systems[J]. Chinese Physics C, 2001, 25(9): 872-876. shu

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Received: 2000-08-16
Revised: 1900-01-01

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沈阳化工大学材料科学与工程学院沈阳 110142

Periodic Orbits and Trace Formula-Integrable Systems

Corresponding author: SONG Jian-Jun,

Institute of Modern Physics, The Chinese Academy of Sciences, Lanzhou 730000, China2 Research Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Collision, Lanzhou 730000, China3 Institute of Modern Physics, Northwest University, Xi′an 710069, China

Received Date: 2000-08-16
Accepted Date: 1900-01-01
Available Online: 2001-09-05

Abstract: The two-dimensional uncoupled quartic oscillator was chosen to test the validity of the Berry-Tabor trace formula. Periodic orbits of the system were calculated by integrating Hamiltonian equations of motion on reasonable tori and the curvatures on the energy surface were obtained by carefully constructing the orbits around the reasonable tori. Finally, the value of the semiclassical action function and that of the quantum action function in the case S<30 were compared, and the good agreement between the two functions indicates the validity of the trace formula. On the other hand, in the quantum action function R^QM(S,E)-S figuration, that peaks appeared at the action values corresponding to the periodic orbits of the classical system provide rich information on quantum-classical correspondence.