倾斜磁场下双势阱中的量子混沌
摘要
量子混沌作为一种新的力学现象，是上世纪70年代开始出现的量子力学中一个新的研究方向，但是至今仍然是一个有争议的问题。量子混沌的起源来自对经典混沌的量子对应的考虑。而经典混沌的概念打破了经典物理学中的确定性图像，指出运动的确定性方程给出的相空间的轨道也可能是随机的，不可预测的。
目前，量子混沌研究的主要工作是在揭示量子系统的半经典行为与经典混沌间的联系上，寻找量子不规则运动的特征，在这方面已经取得了一定的成绩，但是在阐明混沌现象的量子力学根源上没有什么进展。
近几年来，随着实验技术的发展，对凝聚态系统中的量子混沌的实验研究被提上了日程。例如，1995年，T.Fromhold和B.Willkinson等人所做的倾斜磁场作用下的共振隧穿二极管实验，以及后来由E.E.Narimanov和A.D.Stone等人采用方势阱模型进行的理论研究。本文考虑到实际的材料中的杂质问题，而引入了双势阱，考察势阱底部的起伏对系统混沌运动的影响，主要进行了以下两个方面的工作：
1．比较了倾斜磁场作用下的双势阱和含有4次方的势阱两种情况下系统的经典混沌运动特征。考虑到实际的实验数据，分别改变粒子的初始能量和磁场的倾斜角度，采用经典混沌研究的主要工具，即经典相空间的庞加莱截面方法来研究不同能量和倾角对系统中粒子运动的影响。
2．将系统的哈密顿量在一个适当的正交基矢下展开并对角化，得到系统的能谱，分别计算系统能谱的能级间距分布函数和谱刚度，研究了该系统量子能谱的统计特征随磁感应强度大小的改变而表现出的系统运动由规则到混沌的渐进变化。
Quantum chaos as a new dynamical phenomenon, which appeared in the 1970's , is a new research field of the quantum mechanics , but up to the present, it is a controversial topic . The origin of the quantum chaos come from the consideration of the quantum counterpart of dynamical chaos in classical mechanics. This conception destroy the deterministic image of classical physics and shows that the trajectories of deterministic equations of motion are in a sense random and unpredictable.
At present , the major research of quantum chaos is to show the relation between the semi-classical behavior of quantum system and the classical chaos , and to find the feature of quantum irregular motion . In this way , some development have been achived . But the quantum origin of chaos is still vague now .
Recently , along with the fast development of experiment technology , there are some experiment about the quantum chaos in the condensed-matter systems . In 1995 , the experiment of resonant tunneling diode in a magnetic field tiled by an angle with respect to the tunneling direction was accomplished by T.Fromhold and B.Willkinson etc . Subsequently , E.E.Narimanov and A.D.Stone have theoretically studied this system with the square-potential well model . In the present thesis , considering the impurity of the real material , we introduce the double-potential well and investigate that how the bottom fluctuation of the potential well influence the particle's chaos motion . My work includes the following points:
1. Compare the system's classical chaos motion between the
double-potential well in the tiled magnetic field and that of potential well including biquadratic term . Considering the actual experiment condition , we use the Poincare surface of section , which is the elementary method in the study of classical chaos , to investigate how the different relative energy and tiled angle influence on the particle's motion in the potential well.
2. Expanding the system's Hamiltonian to a real symmetric matrix in an appropriate orthogonal basis vector and then diagonalizing it , we get the energy spectrum of the system and calculate the energy level spacing distribution function and the spectral rigidity . Then we study the statistical character of the energy spectrum under the changing of magnetic field intension and find that the system's motion transfers from regular to chaos gradually.
目前，量子混沌研究的主要工作是在揭示量子系统的半经典行为与经典混沌间的联系上，寻找量子不规则运动的特征，在这方面已经取得了一定的成绩，但是在阐明混沌现象的量子力学根源上没有什么进展。
近几年来，随着实验技术的发展，对凝聚态系统中的量子混沌的实验研究被提上了日程。例如，1995年，T.Fromhold和B.Willkinson等人所做的倾斜磁场作用下的共振隧穿二极管实验，以及后来由E.E.Narimanov和A.D.Stone等人采用方势阱模型进行的理论研究。本文考虑到实际的材料中的杂质问题，而引入了双势阱，考察势阱底部的起伏对系统混沌运动的影响，主要进行了以下两个方面的工作：
1．比较了倾斜磁场作用下的双势阱和含有4次方的势阱两种情况下系统的经典混沌运动特征。考虑到实际的实验数据，分别改变粒子的初始能量和磁场的倾斜角度，采用经典混沌研究的主要工具，即经典相空间的庞加莱截面方法来研究不同能量和倾角对系统中粒子运动的影响。
2．将系统的哈密顿量在一个适当的正交基矢下展开并对角化，得到系统的能谱，分别计算系统能谱的能级间距分布函数和谱刚度，研究了该系统量子能谱的统计特征随磁感应强度大小的改变而表现出的系统运动由规则到混沌的渐进变化。
Quantum chaos as a new dynamical phenomenon, which appeared in the 1970's , is a new research field of the quantum mechanics , but up to the present, it is a controversial topic . The origin of the quantum chaos come from the consideration of the quantum counterpart of dynamical chaos in classical mechanics. This conception destroy the deterministic image of classical physics and shows that the trajectories of deterministic equations of motion are in a sense random and unpredictable.
At present , the major research of quantum chaos is to show the relation between the semi-classical behavior of quantum system and the classical chaos , and to find the feature of quantum irregular motion . In this way , some development have been achived . But the quantum origin of chaos is still vague now .
Recently , along with the fast development of experiment technology , there are some experiment about the quantum chaos in the condensed-matter systems . In 1995 , the experiment of resonant tunneling diode in a magnetic field tiled by an angle with respect to the tunneling direction was accomplished by T.Fromhold and B.Willkinson etc . Subsequently , E.E.Narimanov and A.D.Stone have theoretically studied this system with the square-potential well model . In the present thesis , considering the impurity of the real material , we introduce the double-potential well and investigate that how the bottom fluctuation of the potential well influence the particle's chaos motion . My work includes the following points:
1. Compare the system's classical chaos motion between the
double-potential well in the tiled magnetic field and that of potential well including biquadratic term . Considering the actual experiment condition , we use the Poincare surface of section , which is the elementary method in the study of classical chaos , to investigate how the different relative energy and tiled angle influence on the particle's motion in the potential well.
2. Expanding the system's Hamiltonian to a real symmetric matrix in an appropriate orthogonal basis vector and then diagonalizing it , we get the energy spectrum of the system and calculate the energy level spacing distribution function and the spectral rigidity . Then we study the statistical character of the energy spectrum under the changing of magnetic field intension and find that the system's motion transfers from regular to chaos gradually.
引文
[1]．顾雁，量子混沌，上海科技出版社，上海，1996．
[2]．徐躬耦，量子混沌运动，上海科技出版社，上海，1995．
[3]．陈式刚，映象与混沌，国防工业出版社，北京，1992．
[4]．徐躬耦，杨亚天，微观世界中的混沌，东北师范大学出版社，长春，1999．
[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，1992．
[6]．曾谨言，量子力学卷Ⅰ、卷Ⅱ，科技出版社，北京，1995．
[7]. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York, 1992.
[8]. Katsuhiro Nakamura, Quantum Chaos, 1993.
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[10]. Liu. Bo, Dai Jian-hua, Zhang Guang-cai, Zhang Hong_jun, Chin. Phy. Lett. 16, 140(1999).
[11]. E. E. Narimanov, A. D. Stone, Phys. Rew. B57, 9807(1998).
[12]. Richard L. Liboff, Phys. Lett. A269, 230(2000).
[13]. T. M. Fromhold, L. Eaves, etc, Phys. Rev. Lett. 72, 2608(1994).
[14]. G. Muller, G. S. Boebinger, H. Mathur, L.N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 75, 2875(1995).
[15]. E. E. Narimanov, A. Douglas Stone, and G. S. Boebinger, Phys. Rev. Lett. 80, 4024(1998).
[16]. Wen Yong Su, Qian Shui Li, Modern Phys. Lett. B 14, 147(2000).
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[22] .D.L.Shepelyansky,A.D.Stone,Phys.Rev.Lett.74,2098(1995) .
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[24] .O.Bohigas,M.J.Giannoni and C.Schmit,Phys.Rev.Lett.60, 971(1984) .
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[28] .T.M.Fromhold,F.W.Sheard,L.Eaves,P.B.Wilkinson, Nature 387,864(1997) .
[29] .M.Robnik,M.V.Berry,J.Phys.A.Math.Gen.19,669 (1986) .
[2]．徐躬耦，量子混沌运动，上海科技出版社，上海，1995．
[3]．陈式刚，映象与混沌，国防工业出版社，北京，1992．
[4]．徐躬耦，杨亚天，微观世界中的混沌，东北师范大学出版社，长春，1999．
[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，1992．
[6]．曾谨言，量子力学卷Ⅰ、卷Ⅱ，科技出版社，北京，1995．
[7]. M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer-Verlag, New York, 1992.
[8]. Katsuhiro Nakamura, Quantum Chaos, 1993.
[9]. G. Casati, B. Chirikov, Physica D86, 220(1995).
[10]. Liu. Bo, Dai Jian-hua, Zhang Guang-cai, Zhang Hong_jun, Chin. Phy. Lett. 16, 140(1999).
[11]. E. E. Narimanov, A. D. Stone, Phys. Rew. B57, 9807(1998).
[12]. Richard L. Liboff, Phys. Lett. A269, 230(2000).
[13]. T. M. Fromhold, L. Eaves, etc, Phys. Rev. Lett. 72, 2608(1994).
[14]. G. Muller, G. S. Boebinger, H. Mathur, L.N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 75, 2875(1995).
[15]. E. E. Narimanov, A. Douglas Stone, and G. S. Boebinger, Phys. Rev. Lett. 80, 4024(1998).
[16]. Wen Yong Su, Qian Shui Li, Modern Phys. Lett. B 14, 147(2000).
[17] .R.K.Hayden,P.B.Wilkinson,T.M.Fromhold,L.Eaves, F.W.Sheard,M,Henini,N.Miura,PhysicaE 7,735(2000) .
[18] .V.Ya.Demikhovskii,D.J.Kamenev,Phys.Rev.E 59,294(1999) .
[19] .Xu Gong-ou,Xing Yong-zhong,Yang Ya-tian,Chin.Phys.Lett.16, 82(1999) .
[20] .Analabha Roy,J.K.Bhattacharjee,Phys.Lett.A 288,1(2001) .
[21] .T.M.Fromhold,P.B.Wilkinson,F.W.Sheard,L.Eaves, J.Miao,and G.Edwards,Phys.Rev.Lett.75,1142(1995)
[22] .D.L.Shepelyansky,A.D.Stone,Phys.Rev.Lett.74,2098(1995) .
[23] .E.E.Narimanov ,A.D.Stone,and G.S.Boebinger, Phys.Rev.Lett.80,4024(1998) .
[24] .O.Bohigas,M.J.Giannoni and C.Schmit,Phys.Rev.Lett.60, 971(1984) .
[25] .E.J.Heller,Phys.Rev.Lett.53,1515(1984) .
[26] .A.Holle,et al,Phys.Rev.Lett.61,161(1988) .
[27] .L.Faccioli,F .Finelli,G.P .Vacca,and G.Venturi, Phys.Rev.Lett.81,240(1998) .
[28] .T.M.Fromhold,F.W.Sheard,L.Eaves,P.B.Wilkinson, Nature 387,864(1997) .
[29] .M.Robnik,M.V.Berry,J.Phys.A.Math.Gen.19,669 (1986) .