电子的自旋极化描述及其相关物理问题的研究
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摘要
自从电子的自旋被人们发现以来,对电子的自旋极化描述问题的研究一直是量子力学与自旋电子学研究的重要内容之一。目前人们对电子自旋极化的认识在理论上还存在许多值得探讨和深思的问题,或者说还不是充分的,而这些问题的解决与否或许会直接关系到我们对目前许多新奇的与自旋极化密切相关的物理现象的理解和阐释,如自旋霍尔效应、Imbert-Fedorov效应等。
现在人们广泛认识到光束的极化具有局域性,自然界不存在严格意义上的均匀极化光束。同样地,除了特殊情况下的均匀极化电子束,在一般情况下电子束自旋极化也是一个局域的物理量,其在空间具有非均匀分布特征。由此,我们可以预测出许多具有新奇极化分布的电子束,比如自旋极化分布具有如同柱矢量光束那样的柱对称结构的电子束,或称之为“柱极化电子束”。
由于电子相对论效应的存在,相对论电子将表现出非相对论电子所不具有的更丰富的物理行为。为了解释这些现象,必须从相对论量子力学的角度对电子诸多物理性质进行阐述,其中首当其冲的便是相对论电子的自旋极化描述问题。为此,我们分别在非相对论和相对论量子理论框架下进一步发展了单电子的自旋极化描述理论,并且将其应用于自由电子的自旋极化描述中;然后,对所涉及到的体系空间对称性、空间位移特性等物理性质做了进一步的研究工作;考虑到高能极化电子束在粒子物理、凝聚态物理等领域中的应用以及将来“相对论自旋电子学”的研究与发展,在最后一章中我们特别地对相对论电子束穿透非对称方势垒结构的物理行为做了详细的分析。本论文的主要工作和创新点如下:
(1)第一,在非相对论量子框架下发展了电子自旋极化的描述理论。我们发现,在仅考虑电子的自旋的情况下,由于泡利矢量三个笛卡尔分量之间不相互对易,在描述电子的自旋极化时,除了要确定自旋量子化方向外,我们还需要另外一个自由度——单位矢量((?))关于自旋量子化方向的方位角Φ。将此自旋极化的描述理论应用到自由电子情形,并且令构成自由电子波函数的每一个平面波分量的自旋量子化方向为波矢kr方向,我们可以得到自由电子自旋极化的一类表示,在给定权重分布函数和广义琼斯矢量时,单位矢量((?))完全决定了两自旋极化基矢的形式。由于此时两自旋极化基矢为螺旋度算符σr (?)pr /pr的本征值分别为“+1”与“-1”的本征态,而此算符本身便描述了自旋与动量之间的某种耦合,可以说单位矢量I(?)扮演着自旋与动量之间耦合的自由度的角色。
(2)第二,研究了表征自由电子自旋极化基矢的自由度——单位矢量I(?)在描述体系空间旋转对称性中所扮演的重要角色;利用Matlab数值模拟,我们展现了由不同单位矢量I(?)描述的自由电子束的对称性质:令电子束沿z轴传播,极角ΘI决定了电子束整体的物理性质,而方位角ΦI则与体系沿z轴的空间旋转变换之间存在着紧密的联系。
(3)第三,首次提出了柱对称自旋极化电子束的概念,它们是z方向总角动量J z的本征态,但它们即不是自旋角动量也不是轨道角动量在z方向上分量的本征态。
(4)第四,简单地回顾了几个重要的描述自由空间相对论电子自旋角动量或自旋极化性质的力学量算符。然后利用极化算符和相应的阶梯算符通过代数方法构造了自由空间相对论电子的自旋极化基矢,与第二章中构造的非相对论电子自旋极化基矢类似,我们发现除了自旋量子化方向外,还需要另外一个单位矢量才能表征这些基矢。关于单位矢量的物理含义的分析讨论,基本上可以完全借鉴在第二章中有关单位矢量物理含义的讨论,此时只需要将泡利矢量算符替换成极化算符。
(5)第五,相对论电子穿透非对称方形单势垒结构过程中,透射电子的自旋极化将发生改变而不再与入射电子的自旋极化相同,由此,我们重点研究了透射电子的与极化相关的空间位移特性,结果表明它不仅依赖于入射电子能量和入射角度,而且依赖于入射电子的极化。这些现象是非相对论电子所不具有的行为,是电子相对论效应的一种体现。这些结果或许可以为高能极化电子束在粒子物理、凝聚态物理等领域中的应用以及将来“相对论自旋电子学”的研究与发展提供必要的理论指导而具有重要的参考价值。
Since the spin of electron was discovered, investigations on the description of spin polarization have been becoming an important subject in both fundamental quantum theory and spintronics theory. Whereas, the knowledge on the spin polarization is not sufficient, there exist some questions to be considered deeply, which may help us interpret some subtle spin-dependent phenomena including the well-known spin Hall effect and Imbert-Fedorov effect.
The polarization of light beams is a local quantity, similarly, the polarization of electron beams is generally local except for linear-polarized electron beams, it exhibits inhomogeneous distribution. In view of the above, we may predict kinds of electron beams with novel polarization including“cylindrically symmetric electron beams”.
Furthermore, due to relativistic effect, the physical behavior of electron in non-relativistic and relativistic cases will exhibits great distinctions. Therefore, in this thesis, we develop a theory of electron spin polarization, taking into account of both non-relativistic and relativistic electrons, and apply this theory to describe the spin polarization of free electron. We also make some effort to the investigations on symmetry properties and spatial displacement. The main results and innovations contain:
(1) First, new degrees of freedom having the form of a unit vector are identified for characterizing the spin polarization of free electron. It is shown that when only the spin is considered, the non-commutativity of the Cartesian components of the Pauli vector allows us to use the azimuthal angle of a second direction, denoted by unit vector I, with respect to the quantization direction to characterize the spin polarization. Upon utilizing this approach to a free electron and letting the quantization direction for each plane wave be the wave vector, we arrive at a representation in which the unit vector I functions as an independent index to characterize the spin polarization.
(2) Second, we investigate the important role of unit vector I(?) to describe the spatial symmetry of single free electron systems; Making use of numerical simulation, it is shown that electron systems with different unit vector have different symmetry properties; We have also illustrated the close relation between azimuthal angleΦI of unit vector and spatial rotation of system along z axis.
(3) Third, cylindrically symmetric electron beams in spin polarization are reported for the first time. They are shown to be the eigen-states of total angular momentum in z direction. But they are neither the eigen-states of spin nor the eigen-states of orbital angular momentum in that direction.
(4) Fourth, we briefly review several important mechanical operators which can be served as spin angular momentum of relativistic electron. Making use of polarization operator preferably, we propose a method to construct spin polarization bases of free relativistic electron, and find that, besides the direction of spin quantization, there exists another degree of freedom---unit vector to characterize these bases. Similar discussions can be carried out by referring to those in chapter 2 for non-relativistic electron, just replacing Pauli vector by polarization operator.
(5) Fifth, when relativistic electron transmit a non-symmetric square single potential structure, the spin polarization of transmitting electron will change compared with the spin polarization of incident electron, we also discuss the spin-dependent spatial displacements of transmitting electron beams, which also depend on the energy, incident angle and spin polarization of incident electron.These phenomena can not occur for non-relativistic electron. These results may provide necessary theoretical instructions and references for the application of high-energy electron beam in particle physics, high-energy physics and condensed matter physics, et al., and the investigation and development of“relativistic spin electronics”in future.
现在人们广泛认识到光束的极化具有局域性,自然界不存在严格意义上的均匀极化光束。同样地,除了特殊情况下的均匀极化电子束,在一般情况下电子束自旋极化也是一个局域的物理量,其在空间具有非均匀分布特征。由此,我们可以预测出许多具有新奇极化分布的电子束,比如自旋极化分布具有如同柱矢量光束那样的柱对称结构的电子束,或称之为“柱极化电子束”。
由于电子相对论效应的存在,相对论电子将表现出非相对论电子所不具有的更丰富的物理行为。为了解释这些现象,必须从相对论量子力学的角度对电子诸多物理性质进行阐述,其中首当其冲的便是相对论电子的自旋极化描述问题。为此,我们分别在非相对论和相对论量子理论框架下进一步发展了单电子的自旋极化描述理论,并且将其应用于自由电子的自旋极化描述中;然后,对所涉及到的体系空间对称性、空间位移特性等物理性质做了进一步的研究工作;考虑到高能极化电子束在粒子物理、凝聚态物理等领域中的应用以及将来“相对论自旋电子学”的研究与发展,在最后一章中我们特别地对相对论电子束穿透非对称方势垒结构的物理行为做了详细的分析。本论文的主要工作和创新点如下:
(1)第一,在非相对论量子框架下发展了电子自旋极化的描述理论。我们发现,在仅考虑电子的自旋的情况下,由于泡利矢量三个笛卡尔分量之间不相互对易,在描述电子的自旋极化时,除了要确定自旋量子化方向外,我们还需要另外一个自由度——单位矢量((?))关于自旋量子化方向的方位角Φ。将此自旋极化的描述理论应用到自由电子情形,并且令构成自由电子波函数的每一个平面波分量的自旋量子化方向为波矢kr方向,我们可以得到自由电子自旋极化的一类表示,在给定权重分布函数和广义琼斯矢量时,单位矢量((?))完全决定了两自旋极化基矢的形式。由于此时两自旋极化基矢为螺旋度算符σr (?)pr /pr的本征值分别为“+1”与“-1”的本征态,而此算符本身便描述了自旋与动量之间的某种耦合,可以说单位矢量I(?)扮演着自旋与动量之间耦合的自由度的角色。
(2)第二,研究了表征自由电子自旋极化基矢的自由度——单位矢量I(?)在描述体系空间旋转对称性中所扮演的重要角色;利用Matlab数值模拟,我们展现了由不同单位矢量I(?)描述的自由电子束的对称性质:令电子束沿z轴传播,极角ΘI决定了电子束整体的物理性质,而方位角ΦI则与体系沿z轴的空间旋转变换之间存在着紧密的联系。
(3)第三,首次提出了柱对称自旋极化电子束的概念,它们是z方向总角动量J z的本征态,但它们即不是自旋角动量也不是轨道角动量在z方向上分量的本征态。
(4)第四,简单地回顾了几个重要的描述自由空间相对论电子自旋角动量或自旋极化性质的力学量算符。然后利用极化算符和相应的阶梯算符通过代数方法构造了自由空间相对论电子的自旋极化基矢,与第二章中构造的非相对论电子自旋极化基矢类似,我们发现除了自旋量子化方向外,还需要另外一个单位矢量才能表征这些基矢。关于单位矢量的物理含义的分析讨论,基本上可以完全借鉴在第二章中有关单位矢量物理含义的讨论,此时只需要将泡利矢量算符替换成极化算符。
(5)第五,相对论电子穿透非对称方形单势垒结构过程中,透射电子的自旋极化将发生改变而不再与入射电子的自旋极化相同,由此,我们重点研究了透射电子的与极化相关的空间位移特性,结果表明它不仅依赖于入射电子能量和入射角度,而且依赖于入射电子的极化。这些现象是非相对论电子所不具有的行为,是电子相对论效应的一种体现。这些结果或许可以为高能极化电子束在粒子物理、凝聚态物理等领域中的应用以及将来“相对论自旋电子学”的研究与发展提供必要的理论指导而具有重要的参考价值。
Since the spin of electron was discovered, investigations on the description of spin polarization have been becoming an important subject in both fundamental quantum theory and spintronics theory. Whereas, the knowledge on the spin polarization is not sufficient, there exist some questions to be considered deeply, which may help us interpret some subtle spin-dependent phenomena including the well-known spin Hall effect and Imbert-Fedorov effect.
The polarization of light beams is a local quantity, similarly, the polarization of electron beams is generally local except for linear-polarized electron beams, it exhibits inhomogeneous distribution. In view of the above, we may predict kinds of electron beams with novel polarization including“cylindrically symmetric electron beams”.
Furthermore, due to relativistic effect, the physical behavior of electron in non-relativistic and relativistic cases will exhibits great distinctions. Therefore, in this thesis, we develop a theory of electron spin polarization, taking into account of both non-relativistic and relativistic electrons, and apply this theory to describe the spin polarization of free electron. We also make some effort to the investigations on symmetry properties and spatial displacement. The main results and innovations contain:
(1) First, new degrees of freedom having the form of a unit vector are identified for characterizing the spin polarization of free electron. It is shown that when only the spin is considered, the non-commutativity of the Cartesian components of the Pauli vector allows us to use the azimuthal angle of a second direction, denoted by unit vector I, with respect to the quantization direction to characterize the spin polarization. Upon utilizing this approach to a free electron and letting the quantization direction for each plane wave be the wave vector, we arrive at a representation in which the unit vector I functions as an independent index to characterize the spin polarization.
(2) Second, we investigate the important role of unit vector I(?) to describe the spatial symmetry of single free electron systems; Making use of numerical simulation, it is shown that electron systems with different unit vector have different symmetry properties; We have also illustrated the close relation between azimuthal angleΦI of unit vector and spatial rotation of system along z axis.
(3) Third, cylindrically symmetric electron beams in spin polarization are reported for the first time. They are shown to be the eigen-states of total angular momentum in z direction. But they are neither the eigen-states of spin nor the eigen-states of orbital angular momentum in that direction.
(4) Fourth, we briefly review several important mechanical operators which can be served as spin angular momentum of relativistic electron. Making use of polarization operator preferably, we propose a method to construct spin polarization bases of free relativistic electron, and find that, besides the direction of spin quantization, there exists another degree of freedom---unit vector to characterize these bases. Similar discussions can be carried out by referring to those in chapter 2 for non-relativistic electron, just replacing Pauli vector by polarization operator.
(5) Fifth, when relativistic electron transmit a non-symmetric square single potential structure, the spin polarization of transmitting electron will change compared with the spin polarization of incident electron, we also discuss the spin-dependent spatial displacements of transmitting electron beams, which also depend on the energy, incident angle and spin polarization of incident electron.These phenomena can not occur for non-relativistic electron. These results may provide necessary theoretical instructions and references for the application of high-energy electron beam in particle physics, high-energy physics and condensed matter physics, et al., and the investigation and development of“relativistic spin electronics”in future.
引文
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[23]在文章《Transverse Angular Momentum and Geometric Spin Hall Effect of Light》(A. Aiello,et. al.,Phys. Rev. Lett.,2009,103,100401),《Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum》(Y. Y. Bekshaev,J. Opt. A,2009,11,094003),《Angular momentum and spin-orbit interaction of nonparaxial light in free space》(K. Y. Bliokh,et. al.,Phys. Rev. A,2010,82,063825)均有提到自由空间光束的自旋-轨道耦合的概念并对由此产生的包括自旋和轨道霍尔效应等一些物理现象做了阐述。
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[23]在文章《Transverse Angular Momentum and Geometric Spin Hall Effect of Light》(A. Aiello,et. al.,Phys. Rev. Lett.,2009,103,100401),《Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum》(Y. Y. Bekshaev,J. Opt. A,2009,11,094003),《Angular momentum and spin-orbit interaction of nonparaxial light in free space》(K. Y. Bliokh,et. al.,Phys. Rev. A,2010,82,063825)均有提到自由空间光束的自旋-轨道耦合的概念并对由此产生的包括自旋和轨道霍尔效应等一些物理现象做了阐述。
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