Dirac Equation For Electrons Research Articles

Quantum equations of motion and the geometrical imperative: relativistic

We extract the square root of the Minkowski metric using Dirac/Clifford matrices. The resulting 4 × 4 operator dS that represents the square root can be used to transform four vectors between relatively moving observers. This effects the usual Lorentz transformation. In addition, it acts on a Dirac bi-spinor. The operator is essentially a Hamiltonian that can be used to write an equation of motion for a relativistic spinor. This turns out to be the Dirac equation for electrons in standard form. We believe that it is a new approach to familiar results.

Spin eigen-states of Dirac equation for quasi-two-dimensional electrons

Dirac equation for electrons in a potential created by quantum well is solved and the three sets of the eigen-functions are obtained. In each set the wavefunction is at the same time the eigen-function of one of the three spin operators, which do not commute with each other, but do commute with the Dirac Hamiltonian. This means that the eigen-functions of Dirac equation describe three independent spin eigen-states. The energy spectrum of electrons confined by the rectangular quantum well is calculated for each of these spin states at the values of energies relevant for solid state physics. It is shown that the standard Rashba spin splitting takes place in one of such states only. In another one, 2D electron subbands remain spin degenerate, and for the third one the spin splitting is anisotropic for different directions of 2D wave vector.