中心力场中的电子涡旋

doi:10.3969/j.issn.1000-5641.2023.04.016

华东师范大学学报（自然科学版） ›› 2023, Vol. 2023 ›› Issue (4): 151-163.doi: 10.3969/j.issn.1000-5641.2023.04.016

中心力场中的电子涡旋

周永香¹, 薛迅^1,^2,^3,*()

1. 华东师范大学物理与电子科学学院, 上海　200241
2. 新疆大学理论物理中心, 乌鲁木齐　830046
3. 华东师范大学重庆研究院, 重庆　401120

收稿日期:2022-04-08 出版日期:2023-07-25 发布日期:2023-07-25
通讯作者: 薛迅 E-mail:xxue@phy.ecnu.edu
基金资助:
国家自然科学基金(11775080, 11865016); 重庆市自然科学基金(CSTB2022NSCQ-MSX0351)

Electron vortices in the central field

Yongxiang ZHOU¹, Xun XUE^1,^2,^3,*()

1. School of Physics and Electronic Science, East China Normal University, Shanghai　200241, China
2. Center for Theoretical Physics, Xinjiang University, Urumqi　830046, China
3. Chongqing Institute of East China Normal University, Chongqing　401120, China

Received:2022-04-08 Online:2023-07-25 Published:2023-07-25
Contact: Xun XUE E-mail:xxue@phy.ecnu.edu

摘要/Abstract

摘要：

电子涡旋波最早是在轨道角动量守恒的系统中被发现. 对于轨道角动量不守恒的系统, 涡旋波的存在与否尚不清楚. 以相对论情况下的中心力场中的电子为例, 构建了在轨道角动量不守恒但总角动量守恒的情况下, 当携带固定总角动量的电子沿 $z$ 轴传播时, 对此时系统所对应的电子涡旋波进行微扰求解; 并结合Foldy-Wouthuysen (F-W)变换, 说明了在相对论情况下, 中心力场下携带轨道角动量的电子沿 $z$ 轴传播时确实存在涡旋解, 同时展示了相对应的涡旋波解和螺旋等相位面.

关键词: 相对论电子涡旋波, 中心力场, Foldy-Wouthuysen (F-W) 变换, 轨道角动量

Abstract:

Electron vortex beams were first discovered in systems that have a conservable orbital angular momentum; for systems where orbital angular momentum is not conserved, the existence of the electron vortices is uncertain. This article takes the electrons in the central field as examples and, in case of relativity, constructs a case where the orbital angular momentum is not conserved while the total angular momentum is conserved. When the electrons that carry a fixed total angular momentum propagate along the z-axis, the perturbation solution of the electron vortex beams corresponding to the system at this time is calculated and combined with the Foldy-Wouthuysen (F-W) transformation. Accordingly, we can prove, in the case of relativity central field, that the vortex solution does exist when the electrons with orbital angular momentum propagate along the z-axis. Consequently, the corresponding vortex wave solution and spiral isophase surface are shown in this article.

Key words: relativistic electron vortices, central field, Foldy-Wouthuysen (F-W) transformation, orbital angular momentum

中图分类号:

O412

周永香, 薛迅. 中心力场中的电子涡旋[J]. 华东师范大学学报（自然科学版）, 2023, 2023(4): 151-163.

Yongxiang ZHOU, Xun XUE. Electron vortices in the central field[J]. Journal of East China Normal University(Natural Science), 2023, 2023(4): 151-163.

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参考文献 34

1	SAFFMAN P G. Vortex Dynamics [M]. Cambridge: Cambridge University Press, 1993.
2	OGAWA A. Book reviews(7R47. Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics). Applied Mechanics Reviews, 2002, 55 (4): B77- B78.
3	TILLEY D R. Superfluidity and Superconductivity [M]. 1st ed. New York: Routledge, 1990.
4	DONNELLY R J. Quantized Vortices in Helium II [M]. Cambridge: Cambridge University Press, 1991.
5	LEGGETT A J. Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems [M]. Oxford: Oxford University Press, 2006.
6	PISMEN L M. Vortices in Nonlinear Fields: From Liquid Crystals to Superfluids, From Non-Equilibrium Patterns to Cosmic Strings [M]. Oxford: Oxford University Press, 1999.
7	NYE J F. Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations [M]. Philadelphi: Institute of Physics Publishing, 1999.
8	YAO A, PADGETT M. Orbital angular momentum: Origins, behavior and applications. Advances in Optics and Photonics, 2011, 3 (2): 161- 204.
9	ANDREWS D L, BABIKER M. The Angular Momentum of Light [M]. Cambridge: Cambridge University Press, 2012.
10	UCHIDA M, TONOMURA A. Generation of electron beams carrying orbital angular momentum. Nature, 2010, 464 (7289): 737- 739.
11	VERBEECK J, TIAN H, SCHATTSCHNEIDER P. Production and application of electron vortex beams. Nature, 2010, 467 (7313): 301- 304.
12	MCMORRAN B J, AGRAWAL A, ANDERSON I M, et al. Electron vortex beams with high quanta of orbital angular momentum. Science, 2011, 331 (6014): 192- 195.
13	SCHATTSCHNEIDER P, STOEGER-POLLACH M, VERBEECK J. A novel vortex generator and mode converter for electrons [EB/OL]. (2012-05-10)[2022-03-11]. https://arxiv.org/abs/1205.2329.
14	GUZZINATI G, SCHATTSCHNEIDER P, BLIOKH K Y, et al. Observation of the Larmor and Gouy rotations with electron vortex beams. Physical Review Letters, 2013, 110 (9): 093601.
15	SAITOH K, HASEGAWA Y, HIRAKAWA K, et al. Measuring the orbital angular momentum of electron vortex beams using a forked grating. Physical Review Letters, 2013, 111 (7): 074801.
16	NYE J F, BERRY M V. Dislocations in wave trains. Proceedings of the Royal Society A, 1974, 336 (1605): 165- 190.
17	BLIOKH K Y, BLIOKH Y P, SAVEL’EV S, et al. Semiclassical dynamics of electron wave packet states with phase vortices. Physical Review Letters, 2007, 99 (19): 190404.
18	BLIOKH K Y, DENNIS M R, NORI F. Relativistic electron vortex beams: Angular momentum and spin-orbit interaction. Physical Review Letters, 2011, 107 (17): 174802.
19	SCHATTSCHNEIDER P, VERBEECK J. Theory of free electron vortices. Ultramicroscopy, 2011, 111 (9/10): 1461- 1468.
20	BLIOKH K Y, NORI F. Relativistic hall effect. Physical Review Letters, 2012, 108 (12): 120403.
21	KARLOVETS D V. Electron with orbital angular momentum in a strong laser wave. Physical Review A, 2012, 86 (6): 062102.
22	VAN BOXEM R, VERBEECK J, PARTOENS B. Spin effects in electron vortex states. Europhysics Letters, 2013, 102 (4): 40010.
23	BLIOKH K Y, SCHATTSCHNEIDER P, VERBEECK J, et al. Electron vortex beams in a magnetic field: A new twist on landau levels and aharonov-bohm states. Physical Review X, 2012, 2 (4): 798- 808.
24	BARNETT S M. Relativistic electron vortices. Physical Review Letters, 2017, 118 (11): 114802.
25	ZOU L P, ZHANG P M, SILENKO A J. Paraxial wave function and Gouy phase for a relativistic electron in a uniform magnetic field. Journal of Physics G, 2020, 47 (5): 055003.
26	BJORKEN J D, DRELL S D. Relativistic Quantum Mechanics [M]. [S. l. ]: Mcgraw-Hill College, 1964: 47-54.
27	FOLDY L L, WOUTHUYSEN S A. On the Dirac theory of spin 1/2 particles and its non-relativistic limit. Physical Review, 1950, 78 (1): 29- 36.
28	SILENKO A J. Foldy-Wouthyusen transformation and semiclassical limit for relativistic particles in strong external fields. Physical Review A, 2008, 77 (1): 012116.
29	SILENKO A J. Foldy-Wouthyusen transformation and exact description of spin dynamics in strong external fields. The European Physical Journal Special Topics, 2008, 162 (1): 53- 58.
30	SIEGMAN A E. Lasers [M]. [S. l. ]: University Science Books, 1986: 276-279.
31	BARNETT S M. Optical Dirac equation. New Journal of Physics, 2014, 16, 093008.
32	ALLEN L, BEIJERSBERGEN M W, SPREEUW R J C, et al. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Physical Review A, 1992, 45 (11): 8185- 8189.
33	ALLEN L, BARNETT S M, PADGETT M J. Optical Angular Momentum [M]. 1st ed. Boca Raton, FL USA: CRC Press, 2003.
34	杰克逊 J D. 经典电动力学 [M]. 朱培豫, 译. 北京: 人民教育出版社, 1978: 129-131.

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中心力场中的电子涡旋

Electron vortices in the central field

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