On the Notion of Velocity in the Dirac Theory of the Electron

Ziro Koba

이현우

Lu Lin

It is shown that, in Dirac theory, there is a spatial velocity of a free electron which commutes with the Hamiltonian, so it is a conserved quantity of the motion. Furthermore, there is a spatial orbital angular momentum which also commutes with the Hamiltonian and is a constant of the motion.

Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechanical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can- not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e2/(4πǫ0m0c 2) (where e is the electron’s charge, m0 is the mass, c is the speed of light in vacuum, and ǫ0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quantum mechanics is required to understand how the spin property comes about. Its origin is in relativistic quantum mechanics and really was first appreciated by Paul Andrew Maurice Dirac when he derived the Dirac equation which is the cornerstone of relativistic quantum mechanics. Richard Feynman, noted for his distaste for mysticism in physics, wrote about the notion of “spin”: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is down deep in relativistic quantum mechanics.

G. Breit

Hisaiti Yamasaki

This paper aims, in the theory of Dirac electron, at the deliberate use of the hyperbolic metric of velocity, the effectiveness of which Smorodinskii has recently emphasized in relativity. The metric of velocity is fully used together with a proper spin direction vector, to represent explicitly spinor ψ and vector, etc., of ψ*(matrix)ψ, of free Dirac electron. The 3-vector polarization operator is confirmed to correspond to the proper spin vectro. For the electron in a square well potential, ψ is a product of hyperbolic function of the metric with trigonometric or the Bessel function, etc., and the metric turns out to be imaginary or complex in the case of the bound state or free negative energy state. A classical model of Dirac electron is established concretely by obtaining simple time derivatives of the metric and spin vector, from the covariant equations of a spinning particle under general fields, suggested by Frenkel, Kramers and others. The so-called Thomas factor 1/2 is brought inevitably in the equation of the spin vector itself, regardless of the time average motion. These explicit representations may facilitate the visualization of Dirac electron.