(2+1)维Dirac振子的Foldy-Wouthuysen变换

Foldy-Wouthuysen transformation of the(2+1)-dimensional Dirac oscillator

时空(2+1)维的Dirac振子是相对论量子力学中的一个常用模型,经常被用于研究量子效应和原理的相对论性拓展.由于非等距且负的激发能谱以及自旋-轨道耦合等相对论效应的存在,其在标准表象下的本征态为自旋态矢和角动量态矢组成的复杂缀饰态,这给针对它的研究造成了困难.利用FoldyWouthuysen(F-W)变换将其自旋-轨道耦合完全分离,并使两个自旋态各自完全封闭于正负能态中,极大地简化了其哈密顿量和对应本征态的形式.但这一新表象下自旋和轨道角动量算符是对标准表象中彼此的复杂组合.这些结果对推进相对论量子力学和自旋-轨道耦合的研究具有重要意义.

The(2+1)-dimensional Dirac oscillator is a fundamental model used to study the relativistic extensions of quantum effects and principles.Due to the influence of relativistic effects,including the nonequidistant and negative excitation spectrum and the spin-orbit coupling,the eigenstates are complicated dressed states composed of spin and angular momentum state vectors;in turn,this renders theoretical research difficult.In this work,we decouple the spin and angular momentum state vectors and separate the spin-up and-down components into positive-and negative-energy states,respectively,using the FoldyWouthuysen(F-W)transformation.The Hamiltonian and eigenstates of the Dirac oscillator are then largely simplified in the F-W representation;nevertheless,we find the forms of the operators for spin and angular momentum in the same representation with complex combinations of each other.The results are useful in advancing research in relativistic quantum mechanics and spin-orbit coupling.