量子力学中的不变本征算符方法及其应用
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摘要
在量子力学中,求解系统能谱是基础而重要的问题。处理此类问题的时候,人们通常使用的是Schr(?)dinger方程,由于涉及到微分方程,很多时候不容易求解。另一方面,和Schr(?)dinger方程同样重要的Heisenberg方程,却很少被直接用于求解能谱。经过我们研究发现,由Heisenberg的思想出发并结合Schr(?)dinger算子,可以得出一种求解系统能谱的新方法,称为“不变本征算符方法(invarianteigen-operator method,简称IEO方法)”。此方法主要对算符进行操作,无须涉及系统的具体量子态或波函数,从而回避了复杂的微分方程,可以方便的对很多系统进行求解。本文主要内容就是介绍IEO方法的发展及其在分子物理、固体物理、量子光学和量子场论等领域的应用。
一、经过追溯Schr(?)dinger量子化方案的起源,我们对比Schr(?)dinger方程和Heisenberg方程,从而引入关于本征算符的方程。由于本征算符和系统能级差之间的对应关系,我们将得到可用于求解系统能谱的IEO方法。其核心思想就是构造系统哈密顿量的不变本征算符,从而得出对应的本征值,即系统能级差,由能级差即可得到整个能谱。
二、通过求解几个相对简单的少体系统模型,演示IEO方法的基本流程和独特的便利性之后,我们将运用IEO方法来处理固体物理中比较典型的链状哈密顿量系统。由于在固体物理中,晶格振动的频率就对应于系统的能级差,可以发现IEO方法正适合于晶格振动问题的求解,并且由于晶格的周期性,可以有标准化的构造不变本征算符的思路。
三、一些结构比较复杂的哈密顿系统也可以用IEO方法来求解,如半无限原子链和奇异谐振子等模型。由于结构更为复杂,不变本征算符的构造通常需要针对系统的具体结构来进行。
四、非对易空间中的量子力学(NCQM)最近引起了超弦理论领域物理学家们的兴趣。由于不同粒子的坐标算符之间相互不对易,用通常方法求解变得困难。我们把IEO方法运用到非对易空间中,对NCQM的几个模型进行求解,发现非对易因素在这里并不造成困扰。可见IEO方法在此领域中具有相当的优越性,有望推广实用。
五、当然IEO方法远非完善,还存在相当的局限性。如何针对含时系统应用IEO方法还没有得到解决,而且和传统的Schr(?)dinger方程求解一样,对要处理的哈密顿量的形式也有一定限制,很多问题无法用IEO方法直接解决。基于对标准IEO方法的补充,最后我们介绍一些扩展方法,如赝不变本征算符和算符微扰论等,来扩大IEO方法的适用范围。
How to obtain the energy spectrum of quantum systems is a basic and important question in quantum mechanics. People usually make use of Schrodinger equation to dealing energy spectrum problems. Whereas Schrodinger equation is a differential equation, it is hard to solve in most situations. On the other hand, Heisenberg equation is seldom employed to solve energy spectrum problems, although it is important in quantum mechanics as well as Schrodinger equation.
By researching these circumstances, we combine Heisenberg's idea with Schrodinger operator, and then find a new method to obtain the energy spectrum of quantum systems, which we name as "invariant eigen-operator method", or "IEO method" for short. Applying this method, we mostly deal with the quantum operators, and don't have to refer the quantum states and wave functions, consequently obviate the difficult fluxional equations. Thus IEO method is convenient in solving many quantum systems. The main contents of this paper are to introduce the development of IEO method and its applications in molecule physics, solid state physics, quantum optics and non-commute quantum mechanics.
1. First we cast back the origin of Schrodinger quantization scheme, and compare Schrodinger equation with Heisenberg equation, thereby establish the eigen-equation of quantum operators. In virtue of the corresponding relation between the eigen-operators and energy level gaps, we can educe the IEO method to solve the energy spectrum of quantum systems. The kernel of IEO method is to construct eigen-operators of the Hamiltonian of quantum systems, then accordingly obtain the corresponding eigenvalues, namely the energy level gaps of quantum systems. The full energy spectrum can be known by the energy level gaps.
2. We shall show the basic process and unique advantage of the IEO method by solving some few-particle systems. Afterward we apply the IEO method to deal with linear chain Hamiltonians which are typically in solid state physics. Where in solid state physics, lattice vibrating modes correspond to the energy level gaps, we can see that the IEO method is quite suitable for this situation. Due to the lattice periodicity, there is actually a standard technique to construct the eigen-operators.
3. Some complicated Hamiltonians can also be solved by IEO method, such as semi-infinite chain model and singular oscillator model. Owing to the complication forms, the constructing of eigen-operators have to be determined basing on particular of the Hamiltonians.
4. Recently physicists working on superstring theory paid much attention to the quantum mechanics on non-commutative spaces (NCQM). Systems in NCQM are hard to deal with by ordinary methods, since the non-commutation between the coordinate operators of different particles. We apply the IEO method to some models in NCQM, and notice that the non-commutation does not cause any trouble in our way. Thus we can say the IEO method has superiority in NCQM, and we are hopeful that it can be extend to study more models in NCQM.
5. Of course the IEO method is not consummate now, there are still many problems to face at. Just like the Schrodinger equation solution, the Hamiltonians which can be figured out by IEO method are also restricted, and we hope to generalize the IEO method to the time-dependent case in the future. In the final of this paper, we introduce some generalization of the standard IEO method, such as pseudo invariant eigen-operator method and quantum operator perturbation method.
一、经过追溯Schr(?)dinger量子化方案的起源,我们对比Schr(?)dinger方程和Heisenberg方程,从而引入关于本征算符的方程。由于本征算符和系统能级差之间的对应关系,我们将得到可用于求解系统能谱的IEO方法。其核心思想就是构造系统哈密顿量的不变本征算符,从而得出对应的本征值,即系统能级差,由能级差即可得到整个能谱。
二、通过求解几个相对简单的少体系统模型,演示IEO方法的基本流程和独特的便利性之后,我们将运用IEO方法来处理固体物理中比较典型的链状哈密顿量系统。由于在固体物理中,晶格振动的频率就对应于系统的能级差,可以发现IEO方法正适合于晶格振动问题的求解,并且由于晶格的周期性,可以有标准化的构造不变本征算符的思路。
三、一些结构比较复杂的哈密顿系统也可以用IEO方法来求解,如半无限原子链和奇异谐振子等模型。由于结构更为复杂,不变本征算符的构造通常需要针对系统的具体结构来进行。
四、非对易空间中的量子力学(NCQM)最近引起了超弦理论领域物理学家们的兴趣。由于不同粒子的坐标算符之间相互不对易,用通常方法求解变得困难。我们把IEO方法运用到非对易空间中,对NCQM的几个模型进行求解,发现非对易因素在这里并不造成困扰。可见IEO方法在此领域中具有相当的优越性,有望推广实用。
五、当然IEO方法远非完善,还存在相当的局限性。如何针对含时系统应用IEO方法还没有得到解决,而且和传统的Schr(?)dinger方程求解一样,对要处理的哈密顿量的形式也有一定限制,很多问题无法用IEO方法直接解决。基于对标准IEO方法的补充,最后我们介绍一些扩展方法,如赝不变本征算符和算符微扰论等,来扩大IEO方法的适用范围。
How to obtain the energy spectrum of quantum systems is a basic and important question in quantum mechanics. People usually make use of Schrodinger equation to dealing energy spectrum problems. Whereas Schrodinger equation is a differential equation, it is hard to solve in most situations. On the other hand, Heisenberg equation is seldom employed to solve energy spectrum problems, although it is important in quantum mechanics as well as Schrodinger equation.
By researching these circumstances, we combine Heisenberg's idea with Schrodinger operator, and then find a new method to obtain the energy spectrum of quantum systems, which we name as "invariant eigen-operator method", or "IEO method" for short. Applying this method, we mostly deal with the quantum operators, and don't have to refer the quantum states and wave functions, consequently obviate the difficult fluxional equations. Thus IEO method is convenient in solving many quantum systems. The main contents of this paper are to introduce the development of IEO method and its applications in molecule physics, solid state physics, quantum optics and non-commute quantum mechanics.
1. First we cast back the origin of Schrodinger quantization scheme, and compare Schrodinger equation with Heisenberg equation, thereby establish the eigen-equation of quantum operators. In virtue of the corresponding relation between the eigen-operators and energy level gaps, we can educe the IEO method to solve the energy spectrum of quantum systems. The kernel of IEO method is to construct eigen-operators of the Hamiltonian of quantum systems, then accordingly obtain the corresponding eigenvalues, namely the energy level gaps of quantum systems. The full energy spectrum can be known by the energy level gaps.
2. We shall show the basic process and unique advantage of the IEO method by solving some few-particle systems. Afterward we apply the IEO method to deal with linear chain Hamiltonians which are typically in solid state physics. Where in solid state physics, lattice vibrating modes correspond to the energy level gaps, we can see that the IEO method is quite suitable for this situation. Due to the lattice periodicity, there is actually a standard technique to construct the eigen-operators.
3. Some complicated Hamiltonians can also be solved by IEO method, such as semi-infinite chain model and singular oscillator model. Owing to the complication forms, the constructing of eigen-operators have to be determined basing on particular of the Hamiltonians.
4. Recently physicists working on superstring theory paid much attention to the quantum mechanics on non-commutative spaces (NCQM). Systems in NCQM are hard to deal with by ordinary methods, since the non-commutation between the coordinate operators of different particles. We apply the IEO method to some models in NCQM, and notice that the non-commutation does not cause any trouble in our way. Thus we can say the IEO method has superiority in NCQM, and we are hopeful that it can be extend to study more models in NCQM.
5. Of course the IEO method is not consummate now, there are still many problems to face at. Just like the Schrodinger equation solution, the Hamiltonians which can be figured out by IEO method are also restricted, and we hope to generalize the IEO method to the time-dependent case in the future. In the final of this paper, we introduce some generalization of the standard IEO method, such as pseudo invariant eigen-operator method and quantum operator perturbation method.
引文
[1]M.Planck,Verh.D.Phys.Ges.2(1900)202
[2]A.Einstein,Ann.Der Physik,17(1905)132
[3]N.Bohr,Phil.Mag.26(1913)pp.1-25,471-502,857-875
[4]L.de Broglie,Compus Rendus,177(1923)507;Nature,112(1923)540
[5]W.Heisenberg,Zeit.Physik,33(1925)879M.Born,W.Heisenberg,P.Jordan,Zeit.Physik,35(1926)557
[6]E.Schrodinger,Ann.der Physik,79(1926)36,489;80(1926)437;81(1926)109
[7]M.Born,Zeit Physik,38(1926)803
[8]P.A.M.Dirac,Proc,Roy.Soc.A114(1927)243,710
[9]E.Schrodinger,Four Lectures on Wave Mechanics,1928
[10]W.Heisenberg,The Physical Principles of the Quantum Theory,1930
[11]Hong-Yi Fan and Chao Li,Phys.Lett.A321(2004)75
[1]Hong-Yi Fan and Chao Li,Physics Letters A 321(2004)75-78
[2]Hong-yi Fan and Hao Wu,Il Nuovo Cimento B,119(2004)1105
[3]Fan Hong-yi and Fan Yue,Commu.Theor.Phys.V44,No.2(2005)252
[4]Hong-yi Fan,Hai-peng Hu and Xu-bing Tang,J.Phys.A:Math.Gen.38(2005)1
[5]Hong-yi Fan and Hao Wu,Modern Physics Letters B,Vol.19,No.26(2005)1361
[6]Hong-yi Fan,Hao Wu and Xue-Fen Xu,Inter.J.of Mod.Phys.B,Vol.19,No.27(2005)4073
[7]Fan Hong-yi and Tang Xu-Bing,Commu.Theor.Phys.V45,No.6(2006)1003
[8]Fan Hong-yi and Tang Xu-Bing,Commu.Theor.Phys.V46,No.2(2006)209
[9]Fan Hong-yi and Tang Xu-bing,Commu.Theor.Phys.V46,No.4(2006)603
[10]H.Y.Fan,T.T.Wang and Y.L.Yang,Inter.J.of Mod Phys.B,Vol.20,No.32(2006)5417
[11]GUI Wei-Jun,GUI Jia-Zhang,and TANG Gang,Commu.Theor.Phys.V45,No.4(2006)614
[12]Fang Hong-yi and Da Cheng,Commu.Theor.Phys.V45,No.2(2006)255
[13]Hong-yi Fan and Hao Wu,Il Nuovo Cimento B,Vol.122(2007),257
[14]Hao Wu and Hong-yi Fan,Modern Physics Letters B,Vol.21,No.26(2007)1751
[15]Fan Hong-yi and Tang Xu-bing,Commu.Theor.Phys.V47,No.5(2007)865
[16]Fan Hong-yi,Tang Xu-bing and Wang Tong-tong,Commu.Theor.Phys.V48,No.4(2007)633
[17]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49 No.3(2008)759
[18]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49,No.1(2008)50
[19]Gerald D Mahan,Many Particle Physics,Plenum Press,New York(1981)348
[20]S.Ghoshal and A.Chatterjee,Phys.Rev.B 52(1995)982;Phys.Lett.A 233(1996)195
[1]Elliott R.J.and Gibson A.F.,An Introduction to SolidState Physics and its Applications,(The Macmillan Press LTD,London)1974.
[2]Myers H.P.,Introductory Solid State Physics,(The Universities Press(Belfast)Ltd,Wiltshire)1990.
[3]Hong-yi Fan and Hao Wu,Il Nuovo Cimento B,Vol.122(2007),257
[4]R.E.Peierls,Quantum Theory of Solids,Oxford Univ.Press,1955
[5]P.A.Lee,T.M.Rice,P.W.Anderson,Solid State Comm.1974,14:703
[6]李正中,《固体物理》,高等教育出版社,北京,2002
[7]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49 No.3(2008)759
[8]L.N.Bylaevskii,Achievement of Physics Science(translated from Russian),1975, 115:263
[9] Hao Wu and Hong-yi Fan, Modern Physics Letters B, Vol. 21, No. 26 (2007) 1751
[10] H. Goldstein, Classical mechanics (2nded), Addison-Wesley, Massachusetts (1980)
[11] E. T. Whittaker, A Treatise on The Analytical Dynamics (4thed), Camb. Univ. Press, Cambridge (1952)
[12] Hong-yi Fan and Hao Wu, Inter. J. of Mod. Phys. B (Submitted)
[1]黄昆,固体物理学,高等教育出版社,北京,1988
[2]Hong-yi Fan and Hao Wu,Commu.Theo.Phys.(Accepted)
[3]D.C.Mattis,The Theory of Magnetism Made Simple,World Sci.Pub.,2004
[4]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49 No.3(2008)759
[5]李正中,《固体物理》,高等教育出版社,北京,2002
[6]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49,No.1(2008)50
[7]Fan Hong-yi and Tang Xu-bing,Commu.Theor.Phys.V47,No.5(2007)865
[8]Hong-yi Fan,Hai-peng Hu and Xu-bing Tang,J.Phys.A:Math.Gen.38(2005)1-8
[9]Hong-yi Fan,Xu-bing Tang and Hai-peng Hu,Commu.Theor.Phys.(Accepted)
[10]F.A.Hopf and G.Ⅰ.Stegeman,Applied Classical Electrodynamics,Volume Ⅱ:Nonlinear Optics,1985
[1]R.J.Szabo,Phys.Rep.378(2003)207
[2]M.R.Douglas and M.A.Nekrasov,Rev.Mod.Phys.73(2001)977
[3]P.M.Ho and H.C.Kao,Phys.Rev.Lett.88(2002)151602
[4]Si-Cong Jing,Qiu-Yu Liu and Hong-Yi Fan,J..Phys.A 38(2005)8409
[5]F.Toseano at el,http://arxiv.org/quant-ph/0508093
[6]D.Leibfried,R.Blatt,C.Monroe and D.Wineland,Rev.Mod Phys.75(2003)281
[7]Fan Hong-yi,Tang Xu-bing and Wang Tong-tong,Commu.Theor.Phys.V48,No.4(2007)633
[8]Hao Wu and Hong-yi Fan,Commu.Theor.Phys.(Submitted)
[9]H.Y.Fan and J.R.Klauder,Phys.Rev.A 49(1994)704
[10]范洪义,量子力学表象与变换论,上海科学技术出版社,1997
[11]A.Einstein,B.Podolsky,N.Rosen,Phys.Rev.47(1935)777
[1]Fan Hong-yi and Tang Xu-Bing,Commu.Theor.Phys.V45,No.6(2006)1003
[2]E.T.Jaynesand and F.W.Cummings,Proc.IEEE 51(1965)89
[3]Hong-yi Fan and Liang-shi Li,Commun.Theor.Phys.25(1996)105
[4]R.A.Zait and N.H.Abd EL-Wahab,J.Phys.B35,3701(2002);Phys.Scr.66,425(2002)
[5]R.A.Zait,Int.J.Mod Phys.B17,5795(2003)
[6]R.R.Puri,R.K.Bullough,J.Opt.Soc.Am.B,5(1988)2021;
[7]T.Sleator and M.Wilkens,Phys.Rev.A48(1993)3286
[8]F.Cooper,A.Khare and U.Sukhatme,Phys.Rep.251(1995)267
[9]曾谨言,《量子力学》,科学出版社.北京,2000
[2]A.Einstein,Ann.Der Physik,17(1905)132
[3]N.Bohr,Phil.Mag.26(1913)pp.1-25,471-502,857-875
[4]L.de Broglie,Compus Rendus,177(1923)507;Nature,112(1923)540
[5]W.Heisenberg,Zeit.Physik,33(1925)879M.Born,W.Heisenberg,P.Jordan,Zeit.Physik,35(1926)557
[6]E.Schrodinger,Ann.der Physik,79(1926)36,489;80(1926)437;81(1926)109
[7]M.Born,Zeit Physik,38(1926)803
[8]P.A.M.Dirac,Proc,Roy.Soc.A114(1927)243,710
[9]E.Schrodinger,Four Lectures on Wave Mechanics,1928
[10]W.Heisenberg,The Physical Principles of the Quantum Theory,1930
[11]Hong-Yi Fan and Chao Li,Phys.Lett.A321(2004)75
[1]Hong-Yi Fan and Chao Li,Physics Letters A 321(2004)75-78
[2]Hong-yi Fan and Hao Wu,Il Nuovo Cimento B,119(2004)1105
[3]Fan Hong-yi and Fan Yue,Commu.Theor.Phys.V44,No.2(2005)252
[4]Hong-yi Fan,Hai-peng Hu and Xu-bing Tang,J.Phys.A:Math.Gen.38(2005)1
[5]Hong-yi Fan and Hao Wu,Modern Physics Letters B,Vol.19,No.26(2005)1361
[6]Hong-yi Fan,Hao Wu and Xue-Fen Xu,Inter.J.of Mod.Phys.B,Vol.19,No.27(2005)4073
[7]Fan Hong-yi and Tang Xu-Bing,Commu.Theor.Phys.V45,No.6(2006)1003
[8]Fan Hong-yi and Tang Xu-Bing,Commu.Theor.Phys.V46,No.2(2006)209
[9]Fan Hong-yi and Tang Xu-bing,Commu.Theor.Phys.V46,No.4(2006)603
[10]H.Y.Fan,T.T.Wang and Y.L.Yang,Inter.J.of Mod Phys.B,Vol.20,No.32(2006)5417
[11]GUI Wei-Jun,GUI Jia-Zhang,and TANG Gang,Commu.Theor.Phys.V45,No.4(2006)614
[12]Fang Hong-yi and Da Cheng,Commu.Theor.Phys.V45,No.2(2006)255
[13]Hong-yi Fan and Hao Wu,Il Nuovo Cimento B,Vol.122(2007),257
[14]Hao Wu and Hong-yi Fan,Modern Physics Letters B,Vol.21,No.26(2007)1751
[15]Fan Hong-yi and Tang Xu-bing,Commu.Theor.Phys.V47,No.5(2007)865
[16]Fan Hong-yi,Tang Xu-bing and Wang Tong-tong,Commu.Theor.Phys.V48,No.4(2007)633
[17]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49 No.3(2008)759
[18]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49,No.1(2008)50
[19]Gerald D Mahan,Many Particle Physics,Plenum Press,New York(1981)348
[20]S.Ghoshal and A.Chatterjee,Phys.Rev.B 52(1995)982;Phys.Lett.A 233(1996)195
[1]Elliott R.J.and Gibson A.F.,An Introduction to SolidState Physics and its Applications,(The Macmillan Press LTD,London)1974.
[2]Myers H.P.,Introductory Solid State Physics,(The Universities Press(Belfast)Ltd,Wiltshire)1990.
[3]Hong-yi Fan and Hao Wu,Il Nuovo Cimento B,Vol.122(2007),257
[4]R.E.Peierls,Quantum Theory of Solids,Oxford Univ.Press,1955
[5]P.A.Lee,T.M.Rice,P.W.Anderson,Solid State Comm.1974,14:703
[6]李正中,《固体物理》,高等教育出版社,北京,2002
[7]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49 No.3(2008)759
[8]L.N.Bylaevskii,Achievement of Physics Science(translated from Russian),1975, 115:263
[9] Hao Wu and Hong-yi Fan, Modern Physics Letters B, Vol. 21, No. 26 (2007) 1751
[10] H. Goldstein, Classical mechanics (2nded), Addison-Wesley, Massachusetts (1980)
[11] E. T. Whittaker, A Treatise on The Analytical Dynamics (4thed), Camb. Univ. Press, Cambridge (1952)
[12] Hong-yi Fan and Hao Wu, Inter. J. of Mod. Phys. B (Submitted)
[1]黄昆,固体物理学,高等教育出版社,北京,1988
[2]Hong-yi Fan and Hao Wu,Commu.Theo.Phys.(Accepted)
[3]D.C.Mattis,The Theory of Magnetism Made Simple,World Sci.Pub.,2004
[4]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49 No.3(2008)759
[5]李正中,《固体物理》,高等教育出版社,北京,2002
[6]Hong-yi Fan and Hao Wu,Commu.Theor.Phys.V49,No.1(2008)50
[7]Fan Hong-yi and Tang Xu-bing,Commu.Theor.Phys.V47,No.5(2007)865
[8]Hong-yi Fan,Hai-peng Hu and Xu-bing Tang,J.Phys.A:Math.Gen.38(2005)1-8
[9]Hong-yi Fan,Xu-bing Tang and Hai-peng Hu,Commu.Theor.Phys.(Accepted)
[10]F.A.Hopf and G.Ⅰ.Stegeman,Applied Classical Electrodynamics,Volume Ⅱ:Nonlinear Optics,1985
[1]R.J.Szabo,Phys.Rep.378(2003)207
[2]M.R.Douglas and M.A.Nekrasov,Rev.Mod.Phys.73(2001)977
[3]P.M.Ho and H.C.Kao,Phys.Rev.Lett.88(2002)151602
[4]Si-Cong Jing,Qiu-Yu Liu and Hong-Yi Fan,J..Phys.A 38(2005)8409
[5]F.Toseano at el,http://arxiv.org/quant-ph/0508093
[6]D.Leibfried,R.Blatt,C.Monroe and D.Wineland,Rev.Mod Phys.75(2003)281
[7]Fan Hong-yi,Tang Xu-bing and Wang Tong-tong,Commu.Theor.Phys.V48,No.4(2007)633
[8]Hao Wu and Hong-yi Fan,Commu.Theor.Phys.(Submitted)
[9]H.Y.Fan and J.R.Klauder,Phys.Rev.A 49(1994)704
[10]范洪义,量子力学表象与变换论,上海科学技术出版社,1997
[11]A.Einstein,B.Podolsky,N.Rosen,Phys.Rev.47(1935)777
[1]Fan Hong-yi and Tang Xu-Bing,Commu.Theor.Phys.V45,No.6(2006)1003
[2]E.T.Jaynesand and F.W.Cummings,Proc.IEEE 51(1965)89
[3]Hong-yi Fan and Liang-shi Li,Commun.Theor.Phys.25(1996)105
[4]R.A.Zait and N.H.Abd EL-Wahab,J.Phys.B35,3701(2002);Phys.Scr.66,425(2002)
[5]R.A.Zait,Int.J.Mod Phys.B17,5795(2003)
[6]R.R.Puri,R.K.Bullough,J.Opt.Soc.Am.B,5(1988)2021;
[7]T.Sleator and M.Wilkens,Phys.Rev.A48(1993)3286
[8]F.Cooper,A.Khare and U.Sukhatme,Phys.Rep.251(1995)267
[9]曾谨言,《量子力学》,科学出版社.北京,2000