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There exists an electron vortex solution with orbital angular momentum quantum in a non-spin-orbit coupling system which has nonconservative orbital angular momentum. We discuss the system with spin-orbit coupling and nonconservative orbital angular momentum, and we can find that the electrons with the total angular momentum numbers also have vortex beam solutions. And the vortex beam is expressed as an entangled wave function of the spin wave function and the vortex wave function. Taking the electrons in the central force field for example, in this paper constructed is a spinor vortex structure which is caused by the propagation of electrons carrying a fixed quantum number of total angular momentum along the z-axis. The spinor vortex structure is under the condition that the orbital angular momentum caused by spin-orbit coupling is non-conserved but the total angular momentum is conserved. The corresponding electron vortex beams in spin-vortex entanglement are solved by perturbation method, and the Foldy-Wouthuysen transformation is utilized to show that the vortex solution of the four-component spinor does exist in the case of relativity, when the electron with a fixed total angular momentum quantum number propagates along the z-axis in the central force field. The spinor provides theoretical support for the existence of the vortex structure for the system where the orbital angular momentum is not conserved but the total angular momentum is conserved due to spin-orbit coupling.
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Keywords:
- relativistic electron vortices /
- central potential fields /
- Foldy-Wouthuysen transformation /
- spin-orbit coupling
[1] Uchida M, Tonomura A 2010 Nature 464 737
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[2] Verbeeck J, Tian H, Schattschneider P 2010 Nature 467 301
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[6] Saitoh K, Hasegawa Y, Hirakawa K 2013 Phys. Rev. Lett. 111 074801
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[7] Nye J F, Berry M V 1974 Proc. R. Soc. London, Ser. A 336 165
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[8] Bliokh K Y, Bliokh Y P, Savel’Ev S 2007 Phys. Rev. Lett. 99 190404
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[12] Karlovets D V 2012 Phys. Rev. A 86 062102
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[13] Van Boxem R, Verbeeck J, Partoens B 2013 Europhys. Lett. 102 40010
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[14] Bliokh K Y, Schattschneider P, Verbeeck J 2012 Phys. Rev. X 2 041011
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[17] Bjorken J D, Drell S D 1964 Relativistic Quantum Mechanics, Relativistic Quanum Fields (Mcgraw: Mcgraw-Hill College) pp47–54
[18] Foldy L L, Wouthuysen S A 1950 Phys. Rev. 78 29
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[19] Silenko A J 2008 Phys. Rev. A 77 012116
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[20] Silenko A J 2008 Eur. Phys. J. Spec. Top. 162 53
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[21] Siegman A E 1986 Lasers (Oxford: Oxford University Press) pp276–279
[22] Barnett S M 2014 New J. Phys. 16 093008
Google Scholar
[23] Allen L, Beijersbergen M W, Spreeuw R J C 1992 Phys. Rev. A 45 8185
Google Scholar
[24] Allen L, Barnett S M, Padgett M J 2016 Optical angular momentum (Boca Raton: CRC Press)
[25] J D 杰克逊 (朱培豫 译) 1978 物理学 (北京: 人民教育出版社) 第129–131页
Jackson J D (translated by Zhu P Y) 1978 Physics (Beijing: People's Education Press) pp129–131 (in Chinese)
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图 1
$l = 1$ 时, 常$\rho $ 曲面与$\phi $ 的等相面上相交出的螺旋线(其中$\rho = r/W(z)$ 为无量纲径向坐标参量,$X = {x}/{{W(0)}}, $ $ Y = {y}/{{W(0)}}, Z = {z}/{{W\left( 0 \right)}}$ ,$x = \rho \sin ({\varphi }/{2})$ ,$y = \rho \cos ({\varphi }/{2})$ , 波形每旋转一周转动波函数${{\rm{e}}^{{\rm{i}}l\varphi }}$ 相位变化$2\pi $ ) (a)$\rho = 1$ 曲面与$\phi $ 的等相面交线; (b)$\rho = 2$ 曲面与$\phi $ 的等相面交线Figure 1. Spiral line intersected by the equiphase
$\phi = \rm constant$ surface and$\rho = \rm constant$ surface in case of$l = 1$ , where$\rho = r/W(z)$ is the dimensionless radial coordinate parameter and$X = {x}/{{W(0)}}, Y = {y}/{{W(0)}}, Z = {z}/{{W\left( 0 \right)}}$ ,$x = $ $ \rho \sin ({\varphi }/{2})$ ,$y = \rho \cos ({\varphi }/{2})$ . The phase increase of the rotation wave function${{\rm{e}}^{{\rm{i}}l\varphi }}$ is$2\pi $ for every periodic rotation of the helix in space. (a) The spiral line intersected by the equiphase$\phi = \rm constant$ surface and$\rho = 1$ surface; (b) the spiral line intersected by the equiphase$\phi = \rm constant$ surface and$\rho = 2$ surface.
图 2 当携带轨道角动量的电子在中心场中沿
$z$ 轴运动时旋量上分量解$\phi $ 的螺旋等相位面, 轨道量子数$l = 1$ , 波形每旋转一周转动波函数${{\rm{e}}^{{\rm{i}}l\varphi }}$ 相位变化$2\text{π}$ Figure 2. Helical equiphase surface of the spinor upper component solution
$\phi $ when the electrons with orbital angular momentum propagate along z-axis and the orbital quantum number is$l = 1$ . The phase increase of the rotation wave function${{\rm{e}}^{{\rm{i}}l\varphi }}$ is$2\text{π}$ for every periodic rotation of the helix in space.
图 3
$z$ 取值从$ - 6$ 到$6$ 时, 旋量下分量等相面与$\rho = 1.2$ 面所交出的涡旋线, 其中$X={x}/{W(0)}, Y={y}/{W(0)},Z= $ $ {z}/{W\left(0\right)}$ ,$x = \rho \sin ({\varphi }/{2})$ ,$y = \rho \cos ({\varphi }/{2})$ ,$\rho = r/W(z)$ , 波形每旋转一周转动波函数${{\rm{e}}^{{\rm{i}}\left( {l + 1} \right)\varphi }}$ 相位变化$4\text{π}$ Figure 3. Spiral line intersected by the spinor lower equiphase surface and the
$\rho = 1.2$ surface in case of the value of Z ranges from$ - 6$ to$6$ , where$X={x}/{W(0)}, Y={y}/{W(0)}, $ $ Z={z}/{W\left(0\right)}$ ,$x = \rho \sin ({\varphi }/{2})$ ,$y = \rho \cos ({\varphi }/{2})$ ,$\rho = r/W(z)$ . The phase increase of the rotation wave function${{\rm{e}}^{{\rm{i}}\left( {l + 1} \right)\varphi }}$ is$4\text{π}$ for every periodic rotation of the helix in space.
图 4
$z$ 取值从$ - 6$ 到$6$ 时, 旋量下分量等相面与$\rho = 1.5$ 面所交出的涡旋线, 其中$X={x}/{W(0)}, Y={y}/{W(0)}, Z= $ $ {z}/{W\left(0\right)}$ ,$x = \rho \sin ({\varphi }/{2})$ ,$y = \rho \cos ({\varphi }/{2})$ ,$\rho = r/W(z)$ , 波形每旋转一周转动波函数${{\rm{e}}^{{\rm{i}}\left( {l + 1} \right)\varphi }}$ 相位变化$4\pi $ Figure 4. Spiral line intersected by the spinor lower equiphase surface and the
$\rho = 1.5$ surface in case of the value of Z ranges from$ - 6$ to$6$ , where$X={x}/{W(0)}, Y={y}/{W(0)}, $ $ Z={z}/{W\left(0\right)}$ ,$x = \rho \sin ({\varphi }/{2})$ ,$y = \rho \cos ({\varphi }/{2})$ ,$\rho = r/W(z)$ . The phase increase of the rotation wave function${{\rm{e}}^{{\rm{i}}\left( {l + 1} \right)\varphi }}$ is$4\pi $ for every periodic rotation of the helix in space.
图 5 中心力场中携带轨道角动量的电子沿
$z$ 轴传播时其旋量下分量$\eta $ 的涡旋解等相面, 所对应的轨道量子数$l + 1 = 2$ , 其中$X={x}/{W(0)}, Y={y}/{W(0)}, Z={z}/{W\left(0\right)}$ ,$x = \rho \sin ({\varphi }/{2})$ ,$y = \rho \cos ({\varphi }/{2})$ ,$\rho = r/W(z)$ , 波形每旋转一周转动波函数${{\rm{e}}^{{\rm{i}}\left( {l + 1} \right)\varphi }}$ 相位变化$4\pi $ Figure 5. Helical equiphase surface of the spinor lower component solution
$\eta $ when the electrons with orbital angular momentum propagate along z-axis in the central field and corresponding orbital quantum number is$l + 1 = 2$ , in which$X={x}/{W(0)}, Y={y}/{W(0)}, Z={z}/{W\left(0\right)}$ ,$x = \rho\; \times $ $ \sin ({\varphi }/{2})$ ,$y = \rho \cos ({\varphi }/{2})$ ,$\rho = r/W(z)$ . The phase increase of the rotation wave function${{\rm{e}}^{{\rm{i}}\left( {l + 1} \right)\varphi }}$ is$4\pi $ for every periodic rotation of the helix in space. -
[1] Uchida M, Tonomura A 2010 Nature 464 737
Google Scholar
[2] Verbeeck J, Tian H, Schattschneider P 2010 Nature 467 301
Google Scholar
[3] McMorran B J, Agrawal A, Anderson I M, et al. 2011 Science 331 192
Google Scholar
[4] Schattschneider P, Stoeger-Pollach M, Verbeeck J 2012 arXiv: 1205.2329
[5] Guzzinati G, Schattschneider P, Bliokh K Y 2013 Phys. Rev. Lett. 110 093601
Google Scholar
[6] Saitoh K, Hasegawa Y, Hirakawa K 2013 Phys. Rev. Lett. 111 074801
Google Scholar
[7] Nye J F, Berry M V 1974 Proc. R. Soc. London, Ser. A 336 165
Google Scholar
[8] Bliokh K Y, Bliokh Y P, Savel’Ev S 2007 Phys. Rev. Lett. 99 190404
Google Scholar
[9] Bliokh K Y, Dennis M R, Nori F 2011 Phys. Rev. Lett. 107 174802
Google Scholar
[10] Schattschneider P, Verbeeck J 2011 Ultramicroscopy 111 1461
Google Scholar
[11] Bliokh K Y, Nori F 2012 Phys. Rev. Lett. 108 120403
Google Scholar
[12] Karlovets D V 2012 Phys. Rev. A 86 062102
Google Scholar
[13] Van Boxem R, Verbeeck J, Partoens B 2013 Europhys. Lett. 102 40010
Google Scholar
[14] Bliokh K Y, Schattschneider P, Verbeeck J 2012 Phys. Rev. X 2 041011
[15] Barnett S M 2017 Phys. Rev. Lett. 118 114802
Google Scholar
[16] Zou L, Zhang P, Silenko A J 2020 J. Phys. G: Nucl. Part. Phys. 47 055003
Google Scholar
[17] Bjorken J D, Drell S D 1964 Relativistic Quantum Mechanics, Relativistic Quanum Fields (Mcgraw: Mcgraw-Hill College) pp47–54
[18] Foldy L L, Wouthuysen S A 1950 Phys. Rev. 78 29
Google Scholar
[19] Silenko A J 2008 Phys. Rev. A 77 012116
Google Scholar
[20] Silenko A J 2008 Eur. Phys. J. Spec. Top. 162 53
Google Scholar
[21] Siegman A E 1986 Lasers (Oxford: Oxford University Press) pp276–279
[22] Barnett S M 2014 New J. Phys. 16 093008
Google Scholar
[23] Allen L, Beijersbergen M W, Spreeuw R J C 1992 Phys. Rev. A 45 8185
Google Scholar
[24] Allen L, Barnett S M, Padgett M J 2016 Optical angular momentum (Boca Raton: CRC Press)
[25] J D 杰克逊 (朱培豫 译) 1978 物理学 (北京: 人民教育出版社) 第129–131页
Jackson J D (translated by Zhu P Y) 1978 Physics (Beijing: People's Education Press) pp129–131 (in Chinese)
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