Link Between the One Dimensional Dirac Equations and Two Velocities in Nonrelativistic Quantum Mechanics?

Francesco R. Ruggeri
2019 Zenodo  
The Dirac equation is often obtained through linearization of the Klein-Gordon equation. This leads to a 2x2 matrix system for one dimension, thus two functions u(x) and v(x) from the two- vector appear naturally. The Schrodinger equation may then be obtained through the nonrelativistic limit, as has shown in the literature. In this note, we consider the idea of two velocities in nonrelativistic quantum mechanics, namely 1/m d/dx W / W and (1/m) sqrt[ - d/dx d/dx W(x)/W(x)] where W(x) is the
more » ... relativistic wavefunction. We note the Einstein's energy momentum relation with a scalar potential is: [E-mo-V(x)][E+mo+V(x)] = p*p. The 2m in the 1/2m factor (in the nonrelativistic equation) may be associated with E+mo+V(x) and E-mo-V(x) with the time-independent Schrodinger equation. In a series of notes, we argued there seem to be two velocities in nonrelativistic quantum mechanics: 1/m d/dx W/W and (1/m)sqrt(- d/dx d/dx W/W). We suggest these two velocities might be the starting point for introducing two functions u(x) and v(x). These might then be linked to the relativistic quantities E+mo+V(x) and E-mo-V(x). From this point, we try to obtain the two Dirac equations in one dimension.
doi:10.5281/zenodo.3376305 fatcat:6n7qdkl35rgmvpy7obu2f2ywra