标题:
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
关键字:
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
期刊名称:
《Modern Physics》, Vol.2 No.4, 2012-11-22
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。
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.