非厄密系统哈密顿量的本征值和本征态

The eigenvalue and eigenvector of a non-Hermitian Hamiltonian

利用扭结哈密顿算子构造了非厄密哈密顿量,探讨了非厄密哈密顿量本征值和本征态的性质.结果表明:非厄密哈密顿量的能量本征值为复数,且随着模型参数和角度的变化而变化,可能会出现奇异点;通过理论推导可确定本征值奇异点的位置和个数.非厄密哈密顿量本征矢量之间的正交归一性呈现与传统量子力学完全不同的特点;考虑基尔霍夫电流定律,用电阻、电感及电容构造电路,可获得非厄密哈密顿量.

The non-Hermitian Hamiltonian is constructed with the knot operators,and then the eigenvalue and corresponding eigenvector are evaluated respectively.It manifests that the eigenvalue of a non-Hermitian Hamiltonian is a complex number,and changes with the angle and the tunable parameter.The number and position of exceptional points are obtained theoretically.Moreover,the biorthogonal normalization of the right and left eigenvectors of the non-Hermitian Hamiltonian is discussed,which is different from the case in the traditional quantum mechanics.Finally,according to the Kirchhoff’s current law,the non-Hermitian Hamiltonian is realized experimentally in an electric circuit with resistor,inductor and capacitor components.