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

1.我们在通常的狄拉克表示和Foldy-Wouthuysen （FW）表示中构建了“狄拉克振荡器”的相干波包。在非相对论极限下，这些波包都被简化为谐振子（HO）的施罗林格相干波包。我们发现，在FW表示中的波包，虽然它随时间消散，但比在狄拉克表示中表现得更像一个经典粒子。狄拉克方程的Ehrenfest定理可以在FW表示中以自然的形式得到。提出了一种构造狄拉克方程洛伦兹伪标量（PS）型无反射势的方法。这是Kay和Moses的逆散射方法的相对论推广，该方法处理了薛定谔方程的类似问题。我们展示了如何用逆散射法构造狄拉克方程的相对论HO势。我们揭示了HO势对于洛伦兹标量(S)和PS类型是可能的，但对于矢量类型则不可能。研究了非线性狄拉克孤子在S型和PS型外场中的行为。我们发现，对于S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，这一显著特征可以通过所谓的集体变量Ansatz （CVA）的一个版本来理解。然而，对于PS型的外部势，我们证明了CVA不能解释孤子的行为，并且PS势在经典力学中没有明显的对应。这与PS势下孤子质心的齐特比态有关。我们通过“计算量子动力学”（CQD）研究了各种时间相关情况下的Aharonov-Bohm效应。在我们的一个研究计划中，我们将利用CQD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.