哈密顿量宇称-时间对称性的刻画

Depiction of Hamiltonian PT-symmetry

宇称-时间(PT)对称性理论描述了具有实能级的非厄密特哈密顿量,在量子物理学和量子信息科学中起着重要作用,是量子力学中活跃且重要的主题.研究者们对如何描述哈密顿量的PT对称性的问题给予了高度关注.本文基于PT对称理论和哈密顿量归一化特征函数,提出了算子F的定义.然后,在找到算子CPT和算子F的对易子和反对易子的特性后,给出了刻画了无量纲情况下哈密顿量的PT对称性的第一种方法.进一步研究发现,该方法还可以量化哈密顿量在无量纲情况下的PT对称性.此外,提出了另一种基于哈密顿量特征值实部和虚部来描述哈密顿量PT对称性的方法,该方法仅用于判断哈密顿量是否具有PT对称性.

The theory of PT-symmetry describes the non-hermitian Hamiltonian with real energy levels,which means that the Hamiltonian H is invariant neither under parity operator P,nor under time reversal operator T,PTH=H.Whether the Hamiltonian is real and symmetric is not a necessary condition for ensuring the fundamental axioms of quantum mechanics:real energy levels and unitary time evolution.The theory of PT-symmetry plays a significant role in studying quantum physics and quantum information science,Researchers have paid much attention to how to describe PT-symmetry of Hamiltonian.In the paper,we define operator F according to the PT-symmetry theory and the normalized eigenfunction of Hamiltonian.Then we first describe the PT-symmetry of Hamiltonian in dimensionless cases after finding the features of commutator and anti-commutator of operator CPT and operator F.Furthermore,we find that this method can also quantify the PT-symmetry of Hamiltonian in dimensionless case.I(CPT,F)=||[CPT,F]||CPT represents the part of PT-symmetry broken,and J(CPT,F)=||[CPT,F]||CPT represents the part of PT-symmetry.If I(CPT,F)=||[CPT,F]||CPT=0,Hamiltonian H is globally PT-symmetric.Once I(CPT,F)=||[CPT,F]||CPT≠0,Hamiltonian H is PT-symmetrically broken.In addition,we propose another method to describe PT-symmetry of Hamiltonian based on real and imaginary parts of eigenvalues of Hamiltonian,to judge whether the Hamiltonian is PT symmetric.ReF=1/4||(CPTF+F)||CPT represents the sum of squares of real part of the eigenvalue En of Hamiltonian H,ImF=1/4||(CPTF–F)||CPT is the sum of imaginary part of the eigenvalue En of a Hamiltonian H.If ImF=0,Hamiltonian H is globally PT-symmetric.Once ImF≠0,Hamiltonian H is PT-symmetrically broken.ReF=0 implies that Hamiltonian H is PT-asymmetric,but it is a sufficient condition,not necessary condition.The later is easier to realize in the experiment,but the studying conditions are tighter,and it further requires that CPTϕn(x)=ϕn(x).If we only pay attention to whether PT-symmetry is broken,it is simpler to use the latter method.The former method is perhaps better to quantify the PT-symmetrically broken part and the part of local PT-symmetry.