PT-对称算子的表示及非正交量子态的区分

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.

Representation of PT-symmetric Operator and Discrimination of Nonorthogonal Quantum States

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.