伪自伴量子系统的酉演化与绝热定理

经典量子系统的哈密尔顿是自伴算子.哈密尔顿算符的自伴性不仅确保了系统遵循酉演化,而且也保证了它自身具有实的能量本征值.但是,确实有一些物理系统,其哈密尔顿是非自伴的,但也具有实的能量本征值,这种具有非自伴哈密尔顿的系统就是非自伴量子系统.具有伪自伴哈密尔顿的系统是一类特殊的非自伴量子系统,其哈密尔顿相似于一个自伴算子.本文研究伪自伴量子系统的酉演化与绝热定理.首先,给出了伪自伴算子定义及其等价刻画;其次,对于伪自伴哈密尔顿系统,通过构造新内积,证明了伪自伴哈密尔顿在新内积下是自伴的,并给出了系统在新内积下为酉演化的充分必要条件.最后,建立了伪自伴量子系统的绝热演化定理及与绝热逼近定理.

Unitary Evolution and Adiabatic Theorem of Pseudo Self-adjoint Quantum Systems

Hamiltonian of a classical quantum system is a self-adjoint operator. The self-adjoint property of a Hamiltonian not only ensures that the system follows unitary evolution, but also ensures that it has real energy eigenvalues. However, there exist indeed some physical systems, their Hamiltonians are nonself-adjoint, but also have real energy eigenvalues. The systems with nonself-adjoint Hamiltonians are called nonself-adjoint quantum systems (NSAQSs). The systems with pseudo self-adjoint Hamiltonians are a special class of NSAQSs, their Hamiltonians are similar to selfadjoint operators. In this paper, we discuss unitary evolution and adiabatic theorem of pseudo self-adjoint quantum systems (PSAQSs). First, pseudo self-adjoint operators are defined and characterized. Second, a given time-dependent pseudo self-adjoint Hamiltonian H(t) is self-adjoint with respect to a new time-dependent inner product, and then a condition for the system to be unitary evolving under the new inner product is derived. Last, adiabatic evolving theorem and adiabatic approximation theorem for a PSAQS are proved.