(2 + 1)维Dirac振子的Foldy-Wouthuysen变换

doi:10.3969/j.issn.1000-5641.2022.04.009

华东师范大学学报（自然科学版） ›› 2022, Vol. 2022 ›› Issue (4): 95-102.doi: 10.3969/j.issn.1000-5641.2022.04.009

(2 + 1)维Dirac振子的Foldy-Wouthuysen变换

孙伟, 张可烨*()

华东师范大学物理与电子科学学院, 上海　200241

收稿日期:2021-04-16 出版日期:2022-07-25 发布日期:2022-07-19
通讯作者: 张可烨 E-mail:kyzhang@phy.ecnu.edu.cn
基金资助:
国家自然科学基金 (11974116); 中央高校基本科研业务费专项资金(40500-20101-222091)

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

Wei SUN, Keye ZHANG*()

School of Physics and Electronic Science, East China Normal University, Shanghai　200241, China

Received:2021-04-16 Online:2022-07-25 Published:2022-07-19
Contact: Keye ZHANG E-mail:kyzhang@phy.ecnu.edu.cn

摘要/Abstract

摘要：

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Foldy-Wouthuysen (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.

关键词: Dirac振子, 缀饰态, Foldy-Wouthuysen变换

Abstract:

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 non-equidistant 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 Foldy-Wouthuysen (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.

Key words: Dirac oscillator, dressed states, Foldy-Wouthuysen transformation

中图分类号:

O412

孙伟, 张可烨. (2 + 1)维Dirac振子的Foldy-Wouthuysen变换[J]. 华东师范大学学报（自然科学版）, 2022, 2022(4): 95-102.

Wei SUN, Keye ZHANG. Foldy-Wouthuysen transformation of the (2 + 1)-dimensional Dirac oscillator[J]. Journal of East China Normal University(Natural Science), 2022, 2022(4): 95-102.

图/表 2

图1

表1

自旋算符和角动量算符在2个表象的数学形式"

表象	算符	解析通式	数学形式		总角动量
表象	算符	解析通式	弱相对极限	强相对极限	总角动量
标准表象	自旋算符	${ {\widehat{{S} }_{z}=\tfrac{\hbar}{2}\widehat{{\sigma} }_{z} } }$	$\widehat{{S} }_{z}=\tfrac{\hbar}{2}\widehat{{\sigma} }_{z}$	$\widehat{{S} }_{z}=\tfrac{\hbar}{2}\widehat{{\sigma} }_{z}$	$\widehat{{J} }_{z}=\widehat{{J} }_{z,{\rm{F} }{\text{-} }{\rm{W} } }$
标准表象	轨道角动量算符	$\widehat{{L} }_{z}=\hbar(\widehat{ {{n} } }_{\rm{r} }-\widehat{ {\boldsymbol{n} } }_{\rm{ {l} } })$	$\widehat{{L} }_{z}=\hbar(\widehat{ {{n} } }_{\rm{r} }-\widehat{ {{n} } }_{\rm{ {l} } })$	$\widehat{{L} }_{z}=\hbar(\widehat{ {{n} } }_{\rm{r} }-\widehat{ {{n} } }_{\rm{ {l} } })$
F-W 表象	自旋算符	$\widehat{ {{S} } }_{z,\text{F-W} }=\tfrac{\hbar}{2}\begin{pmatrix}\widehat{ {{A} } }_{n_{ {\rm{l} } } }^{2}-\widehat{ {{B} } }_{n_{ {\rm{l} } } }^{2}&2\frac{\widehat{ {{A} } }_{n_{ {\rm{l} } } }\widehat{ {{B} } }_{n_{ {\rm{l} } } } }{\sqrt{\widehat{ {{n} } }_{ {\rm{l} } } } }\widehat{ {{a} } }_{ {\rm{l} } }^{\dagger}\\2\widehat{ {{a} } }_{ { {\rm{l} } } }\frac{\widehat{ {{A} } }_{n_{ {\rm{l} } } }\widehat{ {{B} } }_{n_{ {\rm{l} } } } }{\sqrt{\widehat{ {{n} } }_{ {\rm{l} } } } }&-(\widehat{ {{A} } }_{n_{ {\rm{l} } }+1}^{2}-\widehat{ {{B} } }_{n_{ {\rm{l} } }+1}^{2})\end{pmatrix}$	$\widehat{ {{S} } }_{z,\text{F-nr} }=\widehat{ {{S} } }_{z}$	$\widehat{ {{S} } }_{z,\text{F-r} }=\frac{1}{\sqrt{\widehat{ {{n} } }_{ {\rm{l} } } } }\widehat{ {{a} } }_{ {\rm{l} } }^{\dagger}\widehat{ {{S} } }_{+}+{\rm{h} }.{\rm{c} }.$	$\widehat{{J} }_{z}=\widehat{{J} }_{z,{\rm{F} }{\text{-} }{\rm{W} } }$
F-W 表象	轨道角动量算符	$\widehat{ {{L} } }_{z,\text{F-W} }=\hbar(\widehat{ {{n} } }_{ {\rm{r} } }-\widehat{ {{n} } }_{ {\rm{l} } })+\hbar\begin{pmatrix}\widehat{ {{B} } }_{n_{ {\rm{l} } } }^{2}&-\frac{\widehat{ {{A} } }_{n_{ {\rm{l} } } }\widehat{ {{B} } }_{n_{ {\rm{l} } } } }{\sqrt{\widehat{ {{n} } }_{ {\rm{l} } } } }\widehat{ {{a} } }_{ {\rm{l} } }^{\dagger}\\-\widehat{ {{a} } }_{ { {\rm{l} } } }\frac{\widehat{ {{A} } }_{n_{ {\rm{l} } } }\widehat{ {{B} } }_{n_{ {\rm{l} } } } }{\sqrt{\widehat{ {{n} } }_{ {\rm{l} } } } }&-\widehat{ {{B} } }_{n_{ {\rm{l} } }+1}^{2}\end{pmatrix}$	$\widehat{ {{L} } }_{z,\text{F-nr} }=\hbar(\widehat{ {{n} } }_{ {\rm{r} } }-\widehat{ {{n} } }_{ {\rm{l} } })$	$\begin{array}{l}{ {\widehat {{L} } }_{z,{\rm{F }{\text{-} } {\rm{r} } } } } = \hbar ({ {\widehat {{n} } }_{\rm{r} } } - { {\widehat {{n} } }_{\rm{l} } }) + { {\widehat {{S} } }_z} - \\\;\;\;\;\;\;\;\;\;\;(\frac{1}{ {\sqrt { { {\widehat {{n} } }_{\rm{l} } } } } }\widehat {{a} }_{\rm{l} }^\dagger { {\widehat {{S} } }_ + } + {\rm{h} }.{\rm{c} }.)\end{array}$

表1

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(2 + 1)维Dirac振子的Foldy-Wouthuysen变换

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

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