Curious features of relativistic quantum mechanics in its limit to classical mechanics and application of relativistic coherent states

1.在通常的狄拉克表示和Foldy-Woutheysen(FW)表示下，我们构造了‘Dirac振子’的相干波包。在非相对论极限下，这两个波包都简化为谐振子的薛林格相干波包。我们发现，在FW表示中的波包，尽管它在时间上消散，但它的行为比在狄拉克表示中更像一个经典粒子。Dirac方程的Ehrenfest定理可以在FW表示下以自然的形式得到。2.我们提出了一种构造Dirac方程的洛伦兹伪标量(PS)型无反射势的方法。这是Kay和Mosse的逆散射方法的相对论性推广，它处理了薛定谔方程的类似问题。3.我们展示了如何利用逆散射方法来构造Dirac方程的相对论Ho势。我们发现，对于洛伦兹标量(S)型和PS型，Ho势是可能的，而对于矢量型是不可能的。4.我们研究了非线性狄拉克孤子在S和PS型外场中的行为。我们发现，对于S类型的线性势，孤子质心的运动与非线性相互作用的强度无关，这一显着的特征可以用所谓的集体变量ANSATZ的一个版本来解释。然而，对于PS类型的外势，我们发现CVA不能解释孤子的行为，并且PS势与经典力学没有明显的对应关系。这与PS势情况下孤子质心的Zitterbewegung有关。5.我们用计算量子动力学(CQD)的方法研究了各种含时情况下的Aharonov-Bohm效应。在我们的一个研究计划中，我们将通过CQD的方法广泛地研究介观电子系统。

1.We constructed coherent wavepackets of the 'Dirac oscillator' in the usual Dirac representation and in the Foldy-Wouthuysen (FW) representation. In the nonrelativistic limit, these wavepackets are both reduced to Schrolinger's coherent wavepacket for the harmonic oscillator (HO). We found that the wavepacket in the FW representation, although it dissipates in time, behaves much more like a classical particle than in the Dirac representation. Ehrenfest's theorem for the Dirac equation can be obtained, in a natural form, in the FW representation.2.We proposed a method for constructing reflectionless potentials of the Lorentz pseudoscalar (PS) type for the Dirac equation. This is a relativistic generalization of Kay and Moses' inverse scattering method that deals with a similar problem for the Schrodinger equation.3.We showed how the relativistic HO potentials for the Dirac equation can be constructed by means of the inverse scattering method. We revealed that HO potentials are possible for the Lorentz scalar (S) and PS types but not for the vector type.4.We examined the behavior of the nonlinear Dirac soliton in external fields of the S and PS types. We found that for the linear potential of the S type the motion of the soliton centroid is independent of the strength of the nonlinear interaction and this remarkable feature can be understood by means of a version of the so-called collective variable Ansatz (CVA). For external potentials of the PS type, however, we showed that the CVA fails in explaining the behavior of the soliton and the PS potential has no obvious counterpart m classical mechanics. This is related to Zitterbewegung of the soliton centroid in the case of the PS potential.5.We examined the Aharonov-Bohm effect in various time-dependent situations, by means of 'computational quantum dynamics' (CQD). In one of our research plans we will extensively examine mesoscopic electron systems by means of the CQD.