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

References 16

1	BERMUDEZ A, MARTIN-DELGADO M A, SOLANO E. Mesoscopic superposition states in relativistic landau levels [J]. Physical Review Letters, 2007, 99(12): 123602.
2	BENZAIR H, BOUDJEDAA T, MERAD M. Path integral for Dirac oscillator with generalized uncertainty principle [J]. Journal of Mathematical Physics, 2012, 53(12): 123516.
3	MUÑOZ E, PEÑA F J. Quantum heat engine in the relativistic limit: The case of a Dirac particle [J]. Physical Review E, 2012, 86(6): 061108.
4	MORENO M, ZENTELLA A. Covariance, CPT and the Foldy-Wouthuysen transformation for the Dirac oscillator. Journal of Physics A, 1989, 22 (17): L821- L825.
5	CHARGUI Y. Extended Dirac oscillator in (2 + 1)-dimensional space-time [J]. Physics Letters A, 2020, 384(25): 126484.
6	MOSHINSKY M, SZCZEPANIAK A. The Dirac oscillator. Journal of Physics A, 1989, 22 (17): L817- L819.
7	BERMUDEZ A, MARTIN-DELGADO M A, SOLANO E. Exact mapping of the 2 + 1 Dirac oscillator onto the Jaynes-Cummings model: Ion-trap experimental proposal. Physical Review A, 2007, 76(4), 041801.
8	BERMUDEZ A, MARTIN-DELGADO M A, LUIS A. Nonrelativistic limit in the 2 + 1 Dirac oscillator: A Ramsey-interferometry effect [J]. Physical Review A, 2008, 77(3): 033832.
9	ZHANG K, ZHOU L, MEYSTRE P, et al. Relativistic measurement backaction in the quantum Dirac oscillator [J]. Physical Review Letters, 2018, 121(11): 110401.
10	STRANGE P. Relativistic Quantum Mechanics [M]. 1st ed. Cambridge: Cambridge University Press, 1998.
11	SETARE M R, MAJARI P. (2 + 1)-dimensional f-deformed Dirac oscillator as f-deformed AJC model [J]. European Physical Journal Plus, 2017, 132(11): 458. DOI: 10.1140/epjp/i2017-11745-8.
12	VERLAN E M, RAZUMOVA M A. Excitation of coherent correlated states in the Jaynes-Cummings model by external deterministic forces: II. Optics and Spectroscopy, 2001, 91 (5): 735- 740.
13	BLIOKH K Y, DENNIS M R, NORI F. Position, spin, and orbital angular momentum of a relativistic electron [J]. Physical Review A, 2017, 96(2): 023622.
14	MAROLF D, ROVELLI C. Relativistic quantum measurement [J]. Physical Review D, 2002, 66(2): 023510.
15	SARGOLZAEIPOR S, HASSANABADI H, CHUNG W S. Dirac oscillator under the new generalized uncertainty principle from the concept doubly special relativity [J]. Communications in Theoretical Physics, 2019, 71(11): 1301-1303.
16	AL-HASHIMI M H, WIESE U J. Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics. Annals of Physics, 2009, 324 (12): 2599- 2621.

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