
PT-对称算子的表示及非正交量子态的区分
Representation of PT-symmetric Operator and Discrimination of Nonorthogonal Quantum States
经典量子系统中的哈密尔顿为自伴算子,这不仅保证了系统能量本征值全部为实数,而且相应的本征态(单位长度的特征向量)构成了状态空间的一组正规正交基.然而存在一类PT-对称的物理系统,哈密尔顿的自伴性(共轭转置)被物理的PT-对称性所代替.一个完整的PT-对称哈密尔顿,其谱全部为实数且能构造一个合理的CPT-内积.本文研究一类PT-对称算子.固定时间反演算子T,得到宇称算子P的矩阵表示,进而给出每一组PT-对称哈密尔顿的具体表示形式.作为应用,选择一组确定的{P,T}算子,及PT-对称的哈密尔顿,给出两个在传统量子力学中不正交的量子态区分的刻画.
Hamiltonian of a classical quantum system is a self-adjoint operator which ensures that the energy eigenvalues are real and the eigenstates (unit eigenvectors) form an orthonormal basis for the state space. However, there exists the parity-time-reversal (PT) symmetric physical system, the Hermiticity (transpose and complex conjugate) of a Hamiltonian is replaced by the physically transparent condition of PT-symmetry. If a Hamiltonian has an unbroken PT-symmetry, then the spectrum is real and further more one can construct a CPT inner product with a positive-definite inner product. In this paper, we discuss the PT-symmetric operator in the system. First, given the fixed time reversal operator T as the complex conjugation, the matrix representations of both the parity operator P and PT-symmetric Hamiltonian H are obtained. Then all possible concrete forms of P and the corresponding forms of H are expressed. Next, as an application, it is established that PT-symmetric quantum theory for realizing the discrimination of two quantum states which are not orthogonal in the conventional quantum mechanics.
PT-对称 / 复共轭 / 区分 / 内积 / 正交 {{custom_keyword}} /
PT-symmetric / complex conjugation / discrimination / inner product / orthogonal {{custom_keyword}} /
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国家自然科学基金资助项目(11971283,11871318,11771009,11601300,61602291,11571213);陕西省科技厅项目(2018KJXX-054)及国家留学基金委资助
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