近可积系统中经典与量子行为的对照
摘要
在经典力学里，“混沌”是指一类具有不可预测行为的确定性运动。混沌之所以表现出随机特性，其根本原因在于临近轨道的指数型分离。混沌的这一特征称作运动对初值的敏感性。细致地描述动力学系统的这种指数型局部不稳定性，需引入李雅普诺夫指数。
在认识到混沌在经典力学中的重要地位之后，人们很自然地会想到将确定性混沌的概念推广到量子力学中去。事实上，按照玻尔的对应原理，将量子力学应用到宏观运动上所得的结果，应该与经典力学的结果一致，故而力学系统的混沌特征，也必然在量子性质上有所表现。目前，人们对一具体量子系统的研究主要包括几方面：①量子系统所对应经典哈密顿动力学行为的研究。人们研究经典保守系统的可积性与混沌的一个简便而又有效的数值方法是计算相空间轨道的庞加莱截面。②经典混沌系统量子能谱和波函数的研究。人们发现一个经典的混沌系统的量子能谱的统计分布满足由随机矩阵理论所导出的分布，而可积系统满足无规谱的泊松分布。在研究经典混沌系统的量子能级随某一参数变化时，人们观察到了大量的“能级交叉与回避现象”，这种现象是量子混沌的一个重要标志。
本论文就已有的埃农海尔斯模型构造了一个新的近可积系统，发现这个近可积系统适合于做经典与量子行为的直接对照。主要工作包括以下几方面：
(1)经典可积系统在共振点对于微扰极端敏感，而对于量子系统有相应的
行为出现，观察到了能级由交叉到回避交又的转变过程。
(2)对于高于一维的经典系统，表现其非线性共振过程的最好方法是庞
加莱截面，而对于量子系统，我们在相干态的基础上，提出了一种构造高维
系统量子庞加莱截面的方法，从而实现了高维经典与量子相空间的直接对
比。利用相干态表象构造高维系统量子庞加莱截面的方法，我们研究了在高
能区域，当系统的能量很高时，自然接近于半经典或准经典的情况，发现了
在一个能区内，几条相互靠近的能级对应的量子态对应一个能量相同的经典
系综。
利用(2)中构造量子截面图的方法，我们在研究(l)中提到的近可积系统
时，发现了经典与量子截面图存在着明显的对应关系。在量子与经典情况下，
系统具有共同的性质—对于微扰具有极端的敏感性，但它们的表现形式不
同，在我们构造的量子截面中，它们的形式得到了统一。
(3)当对应的经典系统的动力学行为从典型的“规则”演化到“混沌”时，
随机矩阵理论预言，其能谱涨落统计特征发生了明显变化，紧邻间距分布
尸(s)从类possion分布向Wigner(高斯正交系综GOE和高斯酉系综GuE)分
布演变。
了
In classical mechanics, chaos means deterministic unpredictability .It leads to random in the chaotic motion that near orbits diverge exponently. The character of chaos is called "sensitivity to initial conditions". The more exact scientific term is exponently local unstability of the motion. Lyapunov characteristic exponent is defined to describe the local unstability.
With the extensive development of the chaotic theory in classical mechanics, it is natural that the notion of deterministic chaos is carried in quantum mechanics. In fact, according to Bohr corresponding theorem, the result achieved from carrying quantum mechanics to macroscopic motion corresponds with the one of classical mechanics, so chaotic characters of a dynamic system can be manifested in quantum mechanics. Now the studies of chaos in a quantum system include mainly the following several aspects:(1)the study of the dynamic behavior in the Hamitonian which corresponds with a certain quantum system. The numerical method which is used to analyze the integrable and chaotic system
is the Poincare section of the orbits which abides by the classical Hamitionian.(2) The study of quantum energy spectra and wavefunction in the classical chaotic system .It has been known that the energy spectra statistic of a chaotic system agrees with Wigner distribution which is achieved from Random Matrix Theory and the one of a integrable system is Possion distribution achieved originally from the Non-regular spectra. When the energy spectras of a chaotic system
change with a parameter, the phenomena of a lot of "energy avoiding crossing" are found and can be a signature of quantum chaos.
In this thesis, we construct a new quasi-integrable system which is based on the Henon-Heils model . we find the newly constructed system is quite suitable for the quantum-classical direct comparison.
Our important results include the following points:
(l)The classical integrable system is sensible to the external perturbation at the resonant points. To the quantum system , the evident correspondence has been found. We have also observed a transitional process of energy level structure from a type of intersection to a type of avoided crossing.
(2) The best method to manifest the classical nonlinear resonance is the classical Poincare section for the system of high dimensions. But to the quantum system, we suggest a scheme for the construction of quantum Poincare section plot for the system of high dementions which is based on so-called coherent states. We realized a direct comparison between the phase space structures of the classical and quantum version of a two-dimensional system.By using coherent states, we have constructed quantum Poincare section by which we researched the area of high energy levels. When the system's energy is very high , the system should be close to quasi-classical or standard-classical system naturally. I have found that the quantum corresponding states of several closer energies are corresponding to one classical state which has the average energy.
With the method suggested in (2) by which we construct quantum sections, when we study the quasi-integrable system, an obvious correspondance between quantum section plots and classical section plots has been found. Under the quantum and classical conditions, the two corresponding cases have the same property-sensitivity to the external perturbation, but they have different indications. Their implications are unified in our scheme of section plots.
(3) When the classical dynamic system is changed from the integrable
system to the chaotic one, according to Random Matrix Theory , the distribution of its spectra statistic has achieved the great transition from "similar Possion distribution"to "'Wigner distribution".
在认识到混沌在经典力学中的重要地位之后，人们很自然地会想到将确定性混沌的概念推广到量子力学中去。事实上，按照玻尔的对应原理，将量子力学应用到宏观运动上所得的结果，应该与经典力学的结果一致，故而力学系统的混沌特征，也必然在量子性质上有所表现。目前，人们对一具体量子系统的研究主要包括几方面：①量子系统所对应经典哈密顿动力学行为的研究。人们研究经典保守系统的可积性与混沌的一个简便而又有效的数值方法是计算相空间轨道的庞加莱截面。②经典混沌系统量子能谱和波函数的研究。人们发现一个经典的混沌系统的量子能谱的统计分布满足由随机矩阵理论所导出的分布，而可积系统满足无规谱的泊松分布。在研究经典混沌系统的量子能级随某一参数变化时，人们观察到了大量的“能级交叉与回避现象”，这种现象是量子混沌的一个重要标志。
本论文就已有的埃农海尔斯模型构造了一个新的近可积系统，发现这个近可积系统适合于做经典与量子行为的直接对照。主要工作包括以下几方面：
(1)经典可积系统在共振点对于微扰极端敏感，而对于量子系统有相应的
行为出现，观察到了能级由交叉到回避交又的转变过程。
(2)对于高于一维的经典系统，表现其非线性共振过程的最好方法是庞
加莱截面，而对于量子系统，我们在相干态的基础上，提出了一种构造高维
系统量子庞加莱截面的方法，从而实现了高维经典与量子相空间的直接对
比。利用相干态表象构造高维系统量子庞加莱截面的方法，我们研究了在高
能区域，当系统的能量很高时，自然接近于半经典或准经典的情况，发现了
在一个能区内，几条相互靠近的能级对应的量子态对应一个能量相同的经典
系综。
利用(2)中构造量子截面图的方法，我们在研究(l)中提到的近可积系统
时，发现了经典与量子截面图存在着明显的对应关系。在量子与经典情况下，
系统具有共同的性质—对于微扰具有极端的敏感性，但它们的表现形式不
同，在我们构造的量子截面中，它们的形式得到了统一。
(3)当对应的经典系统的动力学行为从典型的“规则”演化到“混沌”时，
随机矩阵理论预言，其能谱涨落统计特征发生了明显变化，紧邻间距分布
尸(s)从类possion分布向Wigner(高斯正交系综GOE和高斯酉系综GuE)分
布演变。
了
In classical mechanics, chaos means deterministic unpredictability .It leads to random in the chaotic motion that near orbits diverge exponently. The character of chaos is called "sensitivity to initial conditions". The more exact scientific term is exponently local unstability of the motion. Lyapunov characteristic exponent is defined to describe the local unstability.
With the extensive development of the chaotic theory in classical mechanics, it is natural that the notion of deterministic chaos is carried in quantum mechanics. In fact, according to Bohr corresponding theorem, the result achieved from carrying quantum mechanics to macroscopic motion corresponds with the one of classical mechanics, so chaotic characters of a dynamic system can be manifested in quantum mechanics. Now the studies of chaos in a quantum system include mainly the following several aspects:(1)the study of the dynamic behavior in the Hamitonian which corresponds with a certain quantum system. The numerical method which is used to analyze the integrable and chaotic system
is the Poincare section of the orbits which abides by the classical Hamitionian.(2) The study of quantum energy spectra and wavefunction in the classical chaotic system .It has been known that the energy spectra statistic of a chaotic system agrees with Wigner distribution which is achieved from Random Matrix Theory and the one of a integrable system is Possion distribution achieved originally from the Non-regular spectra. When the energy spectras of a chaotic system
change with a parameter, the phenomena of a lot of "energy avoiding crossing" are found and can be a signature of quantum chaos.
In this thesis, we construct a new quasi-integrable system which is based on the Henon-Heils model . we find the newly constructed system is quite suitable for the quantum-classical direct comparison.
Our important results include the following points:
(l)The classical integrable system is sensible to the external perturbation at the resonant points. To the quantum system , the evident correspondence has been found. We have also observed a transitional process of energy level structure from a type of intersection to a type of avoided crossing.
(2) The best method to manifest the classical nonlinear resonance is the classical Poincare section for the system of high dimensions. But to the quantum system, we suggest a scheme for the construction of quantum Poincare section plot for the system of high dementions which is based on so-called coherent states. We realized a direct comparison between the phase space structures of the classical and quantum version of a two-dimensional system.By using coherent states, we have constructed quantum Poincare section by which we researched the area of high energy levels. When the system's energy is very high , the system should be close to quasi-classical or standard-classical system naturally. I have found that the quantum corresponding states of several closer energies are corresponding to one classical state which has the average energy.
With the method suggested in (2) by which we construct quantum sections, when we study the quasi-integrable system, an obvious correspondance between quantum section plots and classical section plots has been found. Under the quantum and classical conditions, the two corresponding cases have the same property-sensitivity to the external perturbation, but they have different indications. Their implications are unified in our scheme of section plots.
(3) When the classical dynamic system is changed from the integrable
system to the chaotic one, according to Random Matrix Theory , the distribution of its spectra statistic has achieved the great transition from "similar Possion distribution"to "'Wigner distribution".
引文
1. M. C. Gutzwiller, chaos in classical and Quantum Mechanics, springer-Verlag, New York, 1992.
2. L. E. Reichel, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations, Springer-Verlag, New York, 1992.
3. F. hakke, Quantum Signatures of chaos, Springer-verlag, Berlin, 1992.
4．顾雁，《量子混沌》，上海科技教育出版社，1996．
5．刘秉正，《非线性动力学与混沌基础》，东北师范大学出版社，长春，1994．
6. E. Ott, Chaos in Dynamical Systems, Cambridge University, Cambridge, 1993.
7. M. C. Gutzwiller, chaos in classical and Quantum Mechanics, springer-Verlag, New York, 1990.
8．徐躬耦，《量子混沌运动》，上海科技教育出版社，1995．
9．陈式刚，《映象与混沌》，国防工业出版社，北京，1992．
10 R. L. Ingraham, A Survey of Nonlinear Dynamics, World Scientific, Singapore, 1992.
11. S. N. Rasband, Chaotic Dynamics of Nonlinear Systems, John Wiley & Sons, New York, 1990.
12. F. hakke, Quantum Signatures of chaos, Springer-verlag, Berlin, 1991.
13. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.
14. G. Casati and B. V. Chirikow. Phys. Rev. Lett. 1985, 54, 1350.
15．程崇庆，孙义燧，《哈密顿系统中的有序与无序运动》，上海科技教育出版社，上海，1996．
16．曾谨言，《量子力学卷Ⅰ Ⅱ卷》科学出版社，北京，1995．
17．曾谨言，《量子力学精选与剖析》，科学出版社，北京，1995．
18．曾谨言，《量子力学专题分析》，高等教育出版社，北京，1999．
19．徐在新，《高等量子力学》，华东师范大学出版社，上海，1994．
20．郑乐民，徐庚武，《原子结构与原子光谱》，北京大学出版社，北京，1998．
21．张启仁，《量子力学》，高等教育出版社，北京，1989．
22．胡岗，《非平衡统计讲义》，北京师范大学物理系，北京，1997．
23．喀兴林，《高等量子力学》，北京师范大学物理系，北京，1989．
24. L. E. Ballentine and J. P. Zibin, Phys. Rev. A 54, 3813, 1996
25. Y. Ashkenazy, L. P. Horwitz, J. Levitan, M. Lewkowicz, and Y. Rothschild, Phys. Rev. Lett. 75, 1070, 1995.
26. U. Schwengelbeck and F. H. M. Faisal, Phys. Lett. A 201, 1, 1995.
27. G. G. de Polavieja, Phys. Rev. A 53, 2059, 1996.
28. G. G. de Polavieja, Phys. Lett. A 220, 303, 1996.
29. H. Frisk, Phys. Lett. A 227, 139, 1997.
30. R. H. Parmenter and R. W. Valentine, Phys. Lett. A 201, 1, 1995.
31. G. Iacomelli and M. Pettini, Phys. Lett. A 212, 29, 1996.
32. S. Sengupta, P. K. Chattaraj, Phys. Lett. A 215, 119,, 1996.
33. G. G. de Polavieja and M. S. Child, Phys. Rev. E 55, 1451, 1997.
34. R. H. Parmenter and R. W. Valentine, Phys. Lett. A 227, 5, 1997.
35. P. Shukla, Phys. Rev. E 53, 1362, 1996.
36. M. Feinggold. Phys. Rev. Lett, 1985, 55, 2626.
37. T. H. Seligman and J. J. M. Vbaarschot. Phys. Rev. Lett, 1986, 56, 2767.
38. Yingxin Jin, Kaifen He, Phys. Rev. E. 63, 066210, 2001.
39．郭光灿，《量子力学》，高等教育出版社，北京，1990．
40．彭金生，《共振荧光与超荧光》，科学出版社，北京，1993．．
2. L. E. Reichel, The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations, Springer-Verlag, New York, 1992.
3. F. hakke, Quantum Signatures of chaos, Springer-verlag, Berlin, 1992.
4．顾雁，《量子混沌》，上海科技教育出版社，1996．
5．刘秉正，《非线性动力学与混沌基础》，东北师范大学出版社，长春，1994．
6. E. Ott, Chaos in Dynamical Systems, Cambridge University, Cambridge, 1993.
7. M. C. Gutzwiller, chaos in classical and Quantum Mechanics, springer-Verlag, New York, 1990.
8．徐躬耦，《量子混沌运动》，上海科技教育出版社，1995．
9．陈式刚，《映象与混沌》，国防工业出版社，北京，1992．
10 R. L. Ingraham, A Survey of Nonlinear Dynamics, World Scientific, Singapore, 1992.
11. S. N. Rasband, Chaotic Dynamics of Nonlinear Systems, John Wiley & Sons, New York, 1990.
12. F. hakke, Quantum Signatures of chaos, Springer-verlag, Berlin, 1991.
13. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.
14. G. Casati and B. V. Chirikow. Phys. Rev. Lett. 1985, 54, 1350.
15．程崇庆，孙义燧，《哈密顿系统中的有序与无序运动》，上海科技教育出版社，上海，1996．
16．曾谨言，《量子力学卷Ⅰ Ⅱ卷》科学出版社，北京，1995．
17．曾谨言，《量子力学精选与剖析》，科学出版社，北京，1995．
18．曾谨言，《量子力学专题分析》，高等教育出版社，北京，1999．
19．徐在新，《高等量子力学》，华东师范大学出版社，上海，1994．
20．郑乐民，徐庚武，《原子结构与原子光谱》，北京大学出版社，北京，1998．
21．张启仁，《量子力学》，高等教育出版社，北京，1989．
22．胡岗，《非平衡统计讲义》，北京师范大学物理系，北京，1997．
23．喀兴林，《高等量子力学》，北京师范大学物理系，北京，1989．
24. L. E. Ballentine and J. P. Zibin, Phys. Rev. A 54, 3813, 1996
25. Y. Ashkenazy, L. P. Horwitz, J. Levitan, M. Lewkowicz, and Y. Rothschild, Phys. Rev. Lett. 75, 1070, 1995.
26. U. Schwengelbeck and F. H. M. Faisal, Phys. Lett. A 201, 1, 1995.
27. G. G. de Polavieja, Phys. Rev. A 53, 2059, 1996.
28. G. G. de Polavieja, Phys. Lett. A 220, 303, 1996.
29. H. Frisk, Phys. Lett. A 227, 139, 1997.
30. R. H. Parmenter and R. W. Valentine, Phys. Lett. A 201, 1, 1995.
31. G. Iacomelli and M. Pettini, Phys. Lett. A 212, 29, 1996.
32. S. Sengupta, P. K. Chattaraj, Phys. Lett. A 215, 119,, 1996.
33. G. G. de Polavieja and M. S. Child, Phys. Rev. E 55, 1451, 1997.
34. R. H. Parmenter and R. W. Valentine, Phys. Lett. A 227, 5, 1997.
35. P. Shukla, Phys. Rev. E 53, 1362, 1996.
36. M. Feinggold. Phys. Rev. Lett, 1985, 55, 2626.
37. T. H. Seligman and J. J. M. Vbaarschot. Phys. Rev. Lett, 1986, 56, 2767.
38. Yingxin Jin, Kaifen He, Phys. Rev. E. 63, 066210, 2001.
39．郭光灿，《量子力学》，高等教育出版社，北京，1990．
40．彭金生，《共振荧光与超荧光》，科学出版社，北京，1993．．