无穷深势阱中相对论粒子的矩阵元及其经典近似

对于无穷深势阱中自旋为０(满足Klein-Gordon方程)和自旋为1/2(满足Dirac方程)的相对论粒子, 分别计算了坐标、动量以及速度算符的矩阵元. 在大量子数极限下, 这些矩阵元给出相应的经典物理量(这里是狭义相对论中的有关量), 并且满足正确的经典关系. 从而表明, Heisenberg对应原理对这样的相对论体系也适用. 关键词： 无穷深势阱 Klein-Gordon方程 Dirac方程 Heisenberg对应原理     

Wel对应、Wigner算符及其相干态推广

经典力学量与量子算符之间通常由 Ｗｅｙｌ对应规则联系，本文从 Ｗｅｙｌ对应的定义出发，讨论了它的实质，论述了它在求量子平均与转换矩阵元方面的应用，导出了Ｗｅｙｌ对应中的积分核——Ｗｉｇｎｅｒ算符的相干态形式和正规乘积形式，在这种新形式下，对于导出一些量子算符公式和量子括号的经典极限，以及求产生和消灭算符函数的Ｗｅｙｌ经典对应函数都是非常方便的．     

角动量平方算符的矢量分析计算

量子物理是从经典物理中发展而来的,在其教学中有意识地挖掘现有教材以便与经典进行对比,指出两者的差异,并说明在什么条件下量子描述退化为经典描述,具有十分重要的教学价值,而角动量的平方算符的推导刚好提供了这样的契机.本文利用矢量算符分析的方法来推导出在球坐标系下角动量平方算符的表达式,同时与经典的角动量平方进行了比较,得到量子角动量平方算符比其经典对应量多出含普朗克常数的项,在经典极限下,前者退化为后者.作为拓展,最后用Bohm规则计算了角动量平方算符.     

量子扩散通道中Wigner算符的演化规律

众所周知,量子态的演化可用与其相应的Wigner函数演化来代替.因为量子态的Wigner函数和量子态的密度矩阵一样,都包含了概率分布和相位等信息,因此对量子态的Wigner函数进行研究,可以更加快速有效地获取量子态在演化过程的重要信息.本文从经典扩散方程出发,利用密度算符的P表示,导出了量子态密度算符的扩散方程.进一步通过引入量子算符的Weyl编序记号,给出了其对应的Weyl量子化方案.另外,借助于密度算符的另一相空间表示-Wigner函数,建立了Wigner算符在扩散通道中演化方程,并给出了其Wigner算符解的形式.本文推导出了Wigner算符在量子扩散通道中的演化规律,即演化过程中任意时刻Wigner算符的形式.在此结论的基础上,讨论了相干态经过量子扩散通道的演化情况.     

关于光子相算符的Weyl对应理论

本文用湮灭算符的极分解的思想来研究单模相算符,指出相算符可以作为量子谐振子海森堡方程的解而被引入。我们还审视量子光学的相算符的经典对应,即用算符的Weyl编序公式和Wigner算符的Weyl排序形式我们直接导出光子相算符的经典Weyl对应,发现它确实对应一个经典相。     

用Weyl-Winger对应求证相干态的一个重要性质

一般而言,一个量子算符只有其在某个表象中所有的矩阵元都知道了才能被确定.可是当一个量子算符的相干态平均值(对角表示)知道了,这个算符本身就确定了,这是一个值得注记的性质.本文用Weyl-Winger对应的唯一性证明这一性质.     

Wigner算符的正规乘积形式和相干态形式的应用

本文导出了Wigner算符的正规乘积形式和相干态形式及其若干应用, 其中包括若干新量子算符公式的导出, Moyal定理的相干态推广, 计算以前文献未曾得到的若干与经典函数对应的量子Weyl算符以及若干与量子算符对应的Weyl经典函数.     

Heisenberg对应原理下氢原子1/r矩阵元的量子-经典对应

通过详细计算表明,在准经典情况下,氢原子1r的矩阵元的量子力学结果与它的Heisenberg矩阵元近似相等,在经典极限下,它们相同. 关键词： 量子力学 对应原理     

介子交换流的球面波展开

本文利用球面波展开在非相对论近似下推导了介子交换流公式, 它们将便于计算介子交换流算符对核电磁矩阵元的修正.     

基于压缩光的量子精密测量

对任何物理量的测量都有一定的噪声, 经典测量所能达到的最小噪声一般称为散粒噪声, 对应着测量的标准量子极限. 利用压缩光可以突破标准量子极限, 从而提高测量精度. 本文介绍了压缩态光场用于突破标准量子极限的基本原理, 以及压缩态光场在相位测量、光学横向小位移及倾斜测量、磁场测量以及时钟同步等精密测量领域的应用和最新进展.     

Quantum-classical correspondence of the relativistic equations

According to the Heisenberg correspondence principle, in the classical limit, quantum matrix element of a Hermitian operator reduces to the coefficient of the Fourier expansion of the corresponding classical quantity. In this article, such a quantum-classical connection is generalized to the relativistic regime. For the relativistic free particle or the charged particle moving in a constant magnetic field, it is shown that matrix elements of quantum operators go to quantities in Einstein’s special relativity in the classical limit. Especially, matrix element of the standard velocity operator in the Dirac theory reduces to the classical velocity. Meanwhile, it is shown that the classical limit of quantum expectation value is the time average of the classical variable.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

Classical and Quantum Theories of Spin

A great effort has been devoted to formulating a classical relativistic theory of spin compatible with quantum relativistic wave equations. The main difficulty in connecting classical and quantum theories rests in finding a parameter that plays the role of proper time at a purely quantum level. We present a partial review of several proposals of classical and quantum spin theories from the pioneering works of Thomas and Frenkel, revisited in the classical BMT work, to the semiclassical model of Barut and Zanghi. We show that the last model can be obtained from a semiclassical limit of the Feynman proper time parametrization of the Dirac equation. At the quantum level, we derive spin precession equations in the Heisenberg picture. Analogies and differences with respect to classical theories are discussed in detail.     

Semiclassical limit of the Dirac equation and the g minus 2 experiment

A relativistic mechanics for a Dirac particle is derived as the semi-classical limit of the Dirac equation. The theory resembles ordinary mechanics, except that some of the phase space variables are four by four matrices. We are able to derive from QED the spin precession equation of Bargmann, Michel, and Telegdi and find quantum corrections for inhomogeneous fields.     

Classical analog of quantum phase

A modified version of the Feynman relativistic chessboard model (FCM) is investigated in which the paths involved are spirals in space-time. Portions of the paths in which the particle's proper time is reversed are interpreted in terms of antiparticles. With this interpretation the particle-antiparticle field produced by such trajectories provides a classical analog of the phase associated with particle paths in the unmodified FCM. It is shown that in the nonrelativistic limit the resulting kernel is the correct Dirac propagator and that particle-antiparticle symmetry is in this case responsible for quantum interference.     

First quantization of mass and charge

The proper time is introduced as a parameter into the wave functions of relativistic quantum theory by first quantization of the mass. The classical limit is shown to be given by a recently developed canonical formulation of classical relativistic mechanics. The adjoint spinor is redefined with the help of a sign operator to remove a discrepancy between the classical and quantum actions in the behavior under time inversion. This results in positive energy densities for the Dirac theory. The inclusion of this sign operator into the definition of the probability current then removes negative probabilities from the theory. A five-dimensional formulation with first quantized charge is given.     

Quantum simulation of the Klein paradox with trapped ions

We report on quantum simulations of relativistic scattering dynamics using trapped ions. The simulated state of a scattering particle is encoded in both the electronic and vibrational state of an ion, representing the discrete and continuous components of relativistic wave functions. Multiple laser fields and an auxiliary ion simulate the dynamics generated by the Dirac equation in the presence of a scattering potential. Measurement and reconstruction of the particle wave packet enables a frame-by-frame visualization of the scattering processes. By precisely engineering a range of external potentials we are able to simulate text book relativistic scattering experiments and study Klein tunneling in an analogue quantum simulator. We describe extensions to solve problems that are beyond current classical computing capabilities.     

Relativistic effects in quantum walks: Klein's paradox and zitterbewegung

Quantum walks are not only algorithmic tools for quantum computation but also non-trivial models describing various physical processes. The Letter compares one-dimensional version of the free particle Dirac equation with the discrete time quantum walk (DTQW). It is shown that two relativistic effects associated with the Dirac equation, namely zitterbewegung (quivering motion) and Klein's paradox, are manifested in DTQW. A special case of DTQW for Lorentz invariance not satisfied in the corresponding continuous limit is considered. The effects are examined.     

Hidden Variables with Nonlocal Time

To relax the apparent tension between nonlocal hidden variables and relativity, we propose that the observable proper time is not the same quantity as the usual proper-time parameter appearing in local relativistic equations. Instead, the two proper times are related by a nonlocal rescaling parameter proportional to |ψ|², so that they coincide in the classical limit. In this way particle trajectories may obey local relativistic equations of motion in a manner consistent with the appearance of nonlocal quantum correlations. To illustrate the main idea, we first present two simple toy models of local particle trajectories with nonlocal time, which reproduce some nonlocal quantum phenomena. After that, we present a realistic theory with a capacity to reproduce all predictions of quantum theory.