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

doi:10.3969/j.issn.1000-5641.2022.04.009

Abstract

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

CLC Number:

O412

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.

Figures/Tables 2

Fig.1

Table 1

Spin and angular momentum operator in the two representations"

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	$\widehat{ {{L} } }_{z,\text{F-W} }=\hbar(\widehat{ {{n} } }_{ {\rm{r} } }-\widehat{ {{n} } }_{ {\rm{l} } })+\hbar\begin{pmatrix}\widehat{ {{B} } }_{n_{ {\rm{l} } } }^{2}&-\frac{\widehat{ {{A} } }_{n_{ {\rm{l} } } }\widehat{ {{B} } }_{n_{ {\rm{l} } } } }{\sqrt{\widehat{ {{n} } }_{ {\rm{l} } } } }\widehat{ {{a} } }_{ {\rm{l} } }^{\dagger}\\-\widehat{ {{a} } }_{ { {\rm{l} } } }\frac{\widehat{ {{A} } }_{n_{ {\rm{l} } } }\widehat{ {{B} } }_{n_{ {\rm{l} } } } }{\sqrt{\widehat{ {{n} } }_{ {\rm{l} } } } }&-\widehat{ {{B} } }_{n_{ {\rm{l} } }+1}^{2}\end{pmatrix}$	$\widehat{ {{L} } }_{z,\text{F-nr} }=\hbar(\widehat{ {{n} } }_{ {\rm{r} } }-\widehat{ {{n} } }_{ {\rm{l} } })$	$\begin{array}{l}{ {\widehat {{L} } }_{z,{\rm{F }{\text{-} } {\rm{r} } } } } = \hbar ({ {\widehat {{n} } }_{\rm{r} } } - { {\widehat {{n} } }_{\rm{l} } }) + { {\widehat {{S} } }_z} - \\\;\;\;\;\;\;\;\;\;\;(\frac{1}{ {\sqrt { { {\widehat {{n} } }_{\rm{l} } } } } }\widehat {{a} }_{\rm{l} }^\dagger { {\widehat {{S} } }_ + } + {\rm{h} }.{\rm{c} }.)\end{array}$

Table 1

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