Cius D, Menon L, Dos Santos M A F, de Castro A S M, Andrade Fabiano M
Programa de Pós-Graduação em Ciências/Física, Universidade Estadual de Ponta Grossa, 84030-900 Ponta Grossa, Paraná, Brazil.
Departamento de Física, Pontifícia Universidade Católica do Rio de Janeiro, 22451-900 Rio de Janeiro, Rio de Janeiro, Brazil.
Phys Rev E. 2022 Nov;106(5-1):054126. doi: 10.1103/PhysRevE.106.054126.
The time-evolution operator obtained from the fractional-time Schrödinger equation (FTSE) is said to be nonunitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the nonunitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.
从分数时间薛定谔方程（FTSE）得到的时间演化算符据说不是幺正的，因为它不能随时间保持矢量态的范数。正如在含时非厄米量子形式体系中所做的那样，对于一个无迹非厄米二能级量子系统，我们证明了有可能将非幺正时间演化算符映射为一个幺正算符。这是通过考虑一个具有含时度规算符的动力学希尔伯特空间来实现的，该度规算符由一个厄米含时戴森映射构造而成，相对于此，系统以幺正方式演化，并且可以正确地进行标准量子力学解释。为了阐明我们的方法，我们考虑了哈密顿算符的三个例子以及从FTSE解中得到的它们相应的幺正动力学，还有各自的戴森映射。
