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研究了二维Sinai台球系统的经典与量子的对应关系，运用定态展开法和Gutzwiller的周期轨道理论对Sinai台球系统的态密度经傅里叶变换得到的量子长度谱进行分析，并把量子长度谱中峰的位置与其所对应的经典体系的周期轨道长度做对比，发现两者之间存在很好的对应关系. 观察到了一些量子态局域在短周期轨道附近形成量子scarred态或量子superscarred态. 还研究了同心与非同心Sinai 台球系统的能级最近邻间距分布，发现同心Sinai台球系统是近可积的，非同心Sinai台球系统在=3/8下，随两中心间距离的增加，能级最近邻间距分布将由近可积向维格那分布过渡.We study the classical and quantum correspondence for a two-dimensional Sinai billiard system. By using the Stationary state expansion method and Gutzwiller's periodic orbit theory, we analyze the quantum length spectrum obtained through the Fourier transformation of the quantum density of state for the Sinai billiard system, and by comparing the peak position with the length of the classical periodic orbit we find their excellent correspondence. We observe that some quantum states are localized near some short period orbits, forming the quantum scarred states or superscarred states. In this paper we also investigate the nearest-neighbor spacing distribution of levels for both concentric and nonconcentric Sinai billiard systems, and find that the concentric Sinai billiard system is nearintegrable, and for the nonconcentric Sinai billiard system with =3/8 its nearest-neighbor spacing distribution of levels transits from nearintegrable to the Wigner distribution as the distance between the two centers increases.
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Keywords:
- quantum billiard /
- periodic orbit theory /
- quantum length spectrum /
- level nearest neighbor spacing distribution
[1] Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer-Verlag) pp254-281, 282-321
[2] Robinett R W 1996 Am. J. Phys. 64 440
[3] Lu J, Du M L 2004 Acta Phys. Sin. 53 2450 (in Chinese) [陆军, 杜孟利 2004 物理学报 53 2450]
[4] Berry M V 1981 Eur. J. Phys. 2 91
[5] Heller E J, O'Connor P W, Gehlen J 1989 Physica Scripta 40 354
[6] Cheon T, Cohen T D 1989 Phys. Rev. Lett. 62 2769
[7] Tuan P H, Yu Y T, Chiang P Y, Liang H C, Huang K F, Chen Y F 2012 Phys. Rev. E 85 026202
[8] Shigehara T 1994 Phys. Rev. E 50 4357
[9] Nakamura K, Thomas H 1988 Phys. Rev. Lett. 61 247
[10] Wilkinson P B, Fromhold T M, Eaves L, Sheard F W, Miura N, Takamasu T 1996 Nature 380 606
[11] Marcus C M, Rimberg A J, Westervelt R M, Hopkins P F, Gossard A C 1992 Phys. Rev. Lett. 69 506
[12] Shudo A, Shimizu Y 1993 Phys. Rev. E 47 54
[13] Šeba P, Zyczkowski 1991 Phys. Rev. A 44 3457
[14] Kudroli A, Kidambi V, Sridhar S 1995 Phys. Rev. Lett. 75 822
[15] Stöckmann H J 1999 Quantum Chaos (Cambridge: Cambridge Vniversity Press) pp86-92
[16] Stöckmann H J, Stein J 1990 Phys. Rev. Lett. 64 2215
[17] Kaufman D L, Kosztin I, Schulten K 1999 Am. J. Phys. 67 133
[18] Heller E J 1984 Phys. Rev. Lett. 53 1515
[19] Bogomolny E, Schmit C 2004 Phys. Rev. Lett. 92 244102
[20] Bogomolny E, Dietz B, Friedrich T 2006 Phys. Rev. Lett. 97 254102
[21] Du M L, Delos J B 1988 Phys. Rev. A 38 1896
[22] Zhang Y H, Xu X Y, Lin S L 2009 Chin. Phys. B 18 35
[23] Du M L, Zhang Y H, Xu X Y 2005 Acta Phys. Sin. 54 4538 (in Chinese) [杜孟利, 张延惠, 徐学友 2005 物理学报 54 4538]
[24] Balian R, Bloch C 1974 Ann. Phys. 85 514
[25] Haak F 1991 Quantum Signature of Chaos (Berlin, Heidelberg: Springer) pp52-54
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[1] Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer-Verlag) pp254-281, 282-321
[2] Robinett R W 1996 Am. J. Phys. 64 440
[3] Lu J, Du M L 2004 Acta Phys. Sin. 53 2450 (in Chinese) [陆军, 杜孟利 2004 物理学报 53 2450]
[4] Berry M V 1981 Eur. J. Phys. 2 91
[5] Heller E J, O'Connor P W, Gehlen J 1989 Physica Scripta 40 354
[6] Cheon T, Cohen T D 1989 Phys. Rev. Lett. 62 2769
[7] Tuan P H, Yu Y T, Chiang P Y, Liang H C, Huang K F, Chen Y F 2012 Phys. Rev. E 85 026202
[8] Shigehara T 1994 Phys. Rev. E 50 4357
[9] Nakamura K, Thomas H 1988 Phys. Rev. Lett. 61 247
[10] Wilkinson P B, Fromhold T M, Eaves L, Sheard F W, Miura N, Takamasu T 1996 Nature 380 606
[11] Marcus C M, Rimberg A J, Westervelt R M, Hopkins P F, Gossard A C 1992 Phys. Rev. Lett. 69 506
[12] Shudo A, Shimizu Y 1993 Phys. Rev. E 47 54
[13] Šeba P, Zyczkowski 1991 Phys. Rev. A 44 3457
[14] Kudroli A, Kidambi V, Sridhar S 1995 Phys. Rev. Lett. 75 822
[15] Stöckmann H J 1999 Quantum Chaos (Cambridge: Cambridge Vniversity Press) pp86-92
[16] Stöckmann H J, Stein J 1990 Phys. Rev. Lett. 64 2215
[17] Kaufman D L, Kosztin I, Schulten K 1999 Am. J. Phys. 67 133
[18] Heller E J 1984 Phys. Rev. Lett. 53 1515
[19] Bogomolny E, Schmit C 2004 Phys. Rev. Lett. 92 244102
[20] Bogomolny E, Dietz B, Friedrich T 2006 Phys. Rev. Lett. 97 254102
[21] Du M L, Delos J B 1988 Phys. Rev. A 38 1896
[22] Zhang Y H, Xu X Y, Lin S L 2009 Chin. Phys. B 18 35
[23] Du M L, Zhang Y H, Xu X Y 2005 Acta Phys. Sin. 54 4538 (in Chinese) [杜孟利, 张延惠, 徐学友 2005 物理学报 54 4538]
[24] Balian R, Bloch C 1974 Ann. Phys. 85 514
[25] Haak F 1991 Quantum Signature of Chaos (Berlin, Heidelberg: Springer) pp52-54
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