Non-Hermitian momentum operator for the particle in a box

Seyong Kim, Alexander Rothkopf
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Abstract

We construct a discrete non-Hermitian momentum operator, which implements faithfully the non-self-adjoint nature of momentum for a particle in a box. Its eigenfunctions are strictly limited to the interior of the box in the continuum limit, with the quarter wave as first nontrivial eigenstate. We show how to construct the corresponding Hermitian Hamiltonian for the infinite well as a concrete example to realize unitary dynamics. The resulting Hilbert space can be decomposed into a physical and unphysical subspace, which are mutually orthogonal. The physical subspace in the continuum limit reproduces that of the continuum theory and we give numerical evidence that the correct probability distributions for momentum and energy are recovered.

Abstract Image

箱中粒子的非ermitian动量算子
我们构建了一个离散的非赫米提动量算子,它忠实地实现了粒子在盒中动量的非自交性质。在连续极限中,它的特征函数严格限制在盒子内部,四分之一波是第一个非难特征状态。我们以实现单元动力学为例,展示了如何为无限井构建相应的赫米特哈密顿。由此得到的希尔伯特空间可以分解为物理子空间和非物理子空间,它们是相互正交的。连续极限的物理子空间再现了连续理论的物理子空间,我们给出了数值证据,证明动量和能量的正确概率分布得以恢复。
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CiteScore
8.60
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