计算自旋-s算子幺正演化矩阵d^s(t)的新方法及其应用

A new method of calculating the unitary evolution matrix d^s(t) of the spin-s operators and its applications

提出了一种严格求解任意自旋-s算子幺正演化矩阵的方法,该方法不同于群论的方法和直接计算的方法,是一种间接的算法.方法的核心是利用两个系统表示的等价性:即自旋-s算子Hamiltonian量Hs=Sx与Heisenberg XX开链带相互作用Jn=(n(N-n))^(1/2)的Hamiltonian量的等价性,由于存在这种等价性,自旋-s算子幺正演化矩阵的计算可通过Heisenberg XX开链中态的演化来实现.采用该方法计算了s=3/2,s=2和s=5/2时对应的幺正演化矩阵.由于初始态|sm〉在算子e-itSx下的演化实质上相当于对态|sm〉进行一个绕x轴转角为β=t的转动,演化矩阵元dsm’m(t)=〈sm′|e-it|Sxsm〉就是转动后的态e-it|Sxsm〉在|sm′〉态上的投影值,所以在t=π时刻的演化矩阵刚好对应Heisenberg XX开链上量子态的理想传输.

We propose a method for calculating the unitary evolution matrix of the arbitrary spin-s operators rigorously. This is an indirect method, which differs from the method of group theory or the method of direct calculation. The kernel of our method is to use the identity of two systems in their expressions, namely the Hamiltonian Hs=Sx of spin-s particle and the Hamiltonian of the Heisenberg XX open chain with interaction Jn=（n（N-n））. Because of this identity, the calculation of the unitary evolution matrix of the spin-s operators is substituted by the calculation of the state evolution matrix in Heisenberg XX open chain. As examples, the unitary evolution matrix of s=3/2, s=2 and s=5/2 are calculated by using our method. Since the evolution of the state ｜sm〉 under the operator e-itSx corresponds to the rotation of the initial state ｜sm〉 around the x-axis by an angle β=t, and the evolution matrix element dsm′m（t）=〈sm′｜e-it｜Sxsm〉 is just the projection of the finial state e-itSx｜sm〉 on the initial state ｜sm′〉, the evolution matrix at t=π corresponds the perfect transmission of quantum state in the Heisenberg XX open chain.