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

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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.