Editorial: Non-Hermitian quantum mechanics
Editorial: Non-Hermitian quantum mechanics
复制标题
社论：非厄米量子力学
DOI：
10.1093/ptep/ptaa182
复制
发表时间：
2020
影响因子：
3.5
通讯作者：
Hatano Naomichi
中科院分区：
文献类型：
--
作者：
Kobayashi Koji;Wada Miku;Ohtsuki Tomi;Hatano Naomichi
据推测，第一次尝试将哈密顿量扩展到非厄米算符是为了解释核衰变。实验装置中放射性核素发射的α或β粒子穿过核素外部的自由空间，理论上会丢失到无限空间，或者实验上会被自由空间周围的探测器吸收。在这两种情况下，发射的粒子永远不会回到核素，因此核素的能量不守恒。在目前的术语中，放射性核素是一个开放的量子系统;它不是封闭的，而是对α和β粒子或宏观探测器的无限自由空间环境开放。同样的情况也可以出现在介观尺度上;在一个典型的实验装置中，量子点对量子线开放，量子线由电极终止于外部。根据冯·诺依曼，整个宇宙的哈密顿量可能是厄米的，但它的一部分，例如放射性核素，量子点或任何与宏观宇宙其余部分相连的东西，不守恒能量，因此可以在消除环境自由度后用有效的非厄米哈密顿量来描述[1]。以这种方式获得的非厄米哈密顿量产生谐振态的复本征值。共振态的本征函数在空间上发散，因此是不可归一化的，这有点臭名昭著，但它实际上意味着环境是宏观的。另一个反驳冯·诺依曼教条的论点则更加雄心勃勃。著名的宇称-时间（PT）对称性理论[2]至少最初假设整个宇宙的哈密顿量是非厄米的，但在真实的本征值的参数范围内。在时间反演和奇偶运算的组合下的对称性，或者更一般地说，在反线性运算与线性运算的组合下的对称性，产生复本征值的真实的或共轭对，如果参数被如此调整，则可以使复本征值的实数对或共轭对排他地成为真实的。最简单的例子是两点紧束缚模型的哈密顿量，
Presumably the first attempt to extend the Hamiltonian to a non-Hermitian operator was made to explain nuclear decay. Alpha or beta particles emitted from a radioactive nuclide in an experimental apparatus travel through free space outside the nuclide and, theoretically, are lost to infinite space, or experimentally, are absorbed by detectors that surround the free space. In either case, the emitted particles never come back to the nuclide, and therefore the energy of the nuclide is not conserved. In the present terminology, the radioactive nuclide is an open quantum system; it is not closed but open to the environment of infinite free space for alpha and beta particles or macroscopic detectors. The same situation can emerge at a mesoscopic scale; in a typical experimental setup, a quantum dot is open to quantum wires, which are terminated outside by electrodes. The Hamiltonian of the whole universe may be Hermitian according to von Neumann, but a part of it, for example, a radioactive nuclide, a quantum dot, or whatever is connected to the rest of the macroscopic universe, does not conserve energy, and hence can be described by an effective non-Hermitian Hamiltonian after eliminating the environmental degrees of freedom [1]. The non-Hermitian Hamiltonian obtained in this way produces complex eigenvalues of resonant states. It is somewhat infamous that the eigenfunction of a resonant state diverges spatially, and hence is unnormalizable, but it in fact means that the environment is macroscopic. The other counterargument to von Neumann’s dogma is more ambitious. The celebrated theory of parity–time (PT) symmetry [2] assumed, at least originally, that the Hamiltonian of the whole universe is non-Hermitian but in a parameter regime of real eigenvalues. The symmetry under the combination of time-reversal and parity operations, or more generally an antilinear operation combined with linear operations, produces either real or conjugate pairs of complex eigenvalues, which may be made exclusively real if the parameters are so tuned. The simplest example would be the Hamiltonian for a two-site tight-binding model,