n维耦合谐振子的能量谱条件数理论研究
Theoretical Study of Energy Spectrum Condition Number of n-Dimension Coupled Harmonic Oscillator

作者: 胡 奥 * , 钟志成 * , 丁世学 :湖北文理学院,物理与电子工程学院;

关键词: n维耦合谐振子谱条件数算子范数n-Dimension Coupling Harmonic Oscillator Spectrum Condition Number Operator Norm

摘要: 本文根据n维耦合谐振子矩阵形式的薛定谔(Schrödinger)方程,在表象理论的基础上,得到了哈密顿(Hamiltonian)算子ˆH 的矩阵元Hnn。通过构建一类完备的赋范线性空间,由泛函理论,证明了Hnn在此空间中是有界算子,同时求得哈密顿算子的本征值E;进而利用矩阵理论得到E的谱条件数公式。从这个表达式出发,得到了能量E的算子范数与谐振子的状态之间的关系式;研究了E的谱条件数与算子范数之间的关系,并估算E的算子范数上、下界的值,给出了能量谱条件数值大小的原因。结果表明:求出谱条件数与算子范数的大致范围,就可以根据谱条件数的准确值,在表象理论框架内,估测谐振子两个状态之间的差异程度,分析谐振子的状态特征。

Abstract: This paper according to the matrix form of Schrödinger equation of n-dimension coupled harmonic oscillator, on the basis of representation theory, the element of matrix Hnn of Hamiltonian operator ˆH is derived. Through con-structing one of complete normed linear space, the Hnn is proved to be boundedness in this space by using functional theory, and eigenvalue E of Hamiltonian operator is obtained; and then the author get the spectrum condition number formula of E that is made use of matrix theory. From this expressions, the formula of operator norm of energy E and harmonic oscillator’s state is acquired. Researching the relationship between spectrum condition number of E and operator norm, the supremum and infimum of E operator norm is estimated, the reason of numerical size of energy spectrum condition number is presented. It turned out that: when the approximately range of spectrum condition number and operator norm is achieved, under the representation theory frame, the difference degree between two states of har-monic oscillator are estimated, which in terms of the exactly value of spectrum condition number, and analysis the fea-ture states of harmonic oscillator.

文章引用: 胡 奥 , 钟志成 , 丁世学 (2012) n维耦合谐振子的能量谱条件数理论研究。 现代物理, 2, 77-81. doi: 10.12677/MP.2012.24013

参考文献

[1] 郁渭铭. N维谐振子和N维氢原子的联系[J]. 南京师大学报(自然科学版), 1990, 13(8): 28-34.

[2] 王秀利, 张运海. 利用二次型求解n模耦合谐振子能量本征值精确解[J]. 大学物理, 2009, 28(6): 53-56.

[3] 张仲, 卢纪材, 吴献等. 二次型方法求解坐标动量耦合的n维谐振子能量本征值[J]. 大学物理, 2011, 30(3): 11-13.

[4] 凌瑞良, 冯进, 冯金福. 三维各向异性耦合谐振子体系的量子化能谱与精确波函数[J]. 物理学报, 2010, 59(12): 8348- 8358.

[5] 张恭庆, 郭懋正. 泛函分析讲义(上册)[M]. 北京: 北京大学出版社, 1990: 20-67.

[6] W. Pauli. General principles of quantum mechanics. Berlin: Springer- Verlag, 1980: 13-61.

[7] 程其襄, 张奠宙, 魏国强等. 实变函数与泛函分析基础[M].北京: 高等教育出版社, 2003: 124-202.

[8] B. K. Driver. Topology and functional analysis. San Diego: Department of Mathematics, University of California, 2001: 0112.

[9] 张恭庆, 郭懋正. 泛函分析讲义(下册)[M]. 北京: 北京大学出版社, 1990: 45-130.

[10] 方保镕, 周继东, 李医民. 矩阵论[M]. 北京: 清华大学出版社, 2004: 28-120.

[11] J. J. Qi, G. S. Xu. On the spectrum of singular Hamiltonian differential systems. Annual of Differential Equations, 2005, 21(3): 389-396.

[12] 王忠, 孙炯. J-自共轭微分算子谱的定性分析[J]. 数学进展, 2001, 30(5): 405-413.

[13] 门少平, 封建湖. 应用泛函分析[M]. 北京: 科学出版社, 2005: 23-94.

[14] 胡克. 解析不等式的若干问题[M]. 武汉: 武汉大学出版社, 2003: 1-37.

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