Wouthuysen Representation Research Articles

Dimensions of relativistic quantum mechanical paths

The notion of the dimension of a relativistic quantum mechanical path is illustrated. An elementary introduction to the concept of a fractal curve allows the discussion of the evolution of the quantum mechanical wave packet for free particles in terms of Hausdorff dimensions of quantum ‘‘trajectories.’’ The results of a previous work by Abbott and Wise can be generalized to relativistic quantum mechanics of spin-1/2 particles in the Foldy–Wouthuysen representation. A further extension to spin-zero particles is briefly outlined. By simple considerations, an understanding of the notion of a Hausdorff dimension and its values can be introduced.

On the equations of motion for particles with arbitrary spin in nonrelativistic mechanics

It is well known that the electron motion in the external electromagnetic field is described by the relativistic Dirac equation. In this case, in the Foldy–Wouthuysen representation, the Hamiltonian includes the terms corresponding to the interaction of the particle magnetic moment with a magnetic field (∼ (1/m)(σH)) and the terms which are interpreted as a spin-orbit coupling (∼ (σ/m2){(p−eA)×E)). Apart from these constituents the Hamiltonian includes the Darwin term (∼ (1/m) divE) [1]. Such a description is in good accordance with the experimental data. It was shown by Bargman [2] that it is possible to introduce the particle spin in the nonrelativistic quantum mechanics by performing the central extension of the Galilei group. In connection with this Bargman result the problem of finding the motion equations, which are invariant with respect to the extended Galilei group G, arises naturally. Such a problem has been considered in [3–5]. The equations obtained in [5] have redundant components and, besides, these equations do not describe the spin-orbit and the Darwin couplings if one makes the replacement pμ → πμ = pμ − eAμ in them. The aim of this note is to find such motion equations for a particle with spin which are invariant relative to the group G, have no redundant components and describe the spin-orbit and the Darwin couplings of the particle with the field. It is reached by the supposition that the free nonrelativistic particle Hamiltonian has two energy signs just as the Dirac Hamiltonian. This is equivalent to the requirement that the theory (equations) be invariant under such a transformation: