量子系统哈密顿量的必要条件

在此前的量子理论中，哈密顿量被要求是厄米算符，这既保证了其本征值谱为实又保证了动力学演化过程几率守恒。近年来，一些非厄米哈密顿量被发现同样满足这两条要求。然而，这两条要求都是哈密顿量可描述量子系统动力学的充分条件而非必要条件。文章中我们从量子化条件和动力学演化方程出发，考察一般形式的哈密顿量，表述为产生算符和湮灭算符之正规积的形式，欲为恰当的哈密顿量所应满足的必要条件。对于形如$hat{H}=hat{p}^2+mathrm{i} hat{x}^3$和$hat{H}=hat{p}^2-hat{x}^4$这样的近年来得到广泛关注的非厄米哈密顿量，容易验证它们满足必要性条件。必要性条件可以用来及时排除那些不恰当的非厄米哈密顿量形式。

Necessary condition for a Hamiltonian to be proper in quantum mechanics

In quantum mechanics, the Hamiltonians are required to be hermitian, since hermiticity guarantees that the energy spectrum is real and the time evolution is unitary. However, some non-hermitian Hamiltonians are also found meeting these requirements. The hermiticity is essentially a sufficient condition. In the current article, we formulate the necessary condition for a Hamiltonian to be proper in quantum mechanics, regarding the quantization condition it follows and the role it plays in the governing equation of dynamic evolution. It can be confirmed that the Hamiltonians adopted in quantum mechanics, even the non-hermitian ones such as$hat{H}=hat{p}^2+mathrm{i} hat{x}^3$ and $hat{H}=hat{p}^2-hat{x}^4$, meet such a necessary condition. The necessary condition provides the first criterium for the candidate Hamiltonians to be introduced.