Spin ½ and Nonrelativistic Quantum Mechanics

Although the idea of spin ½ was known from the Stern Gerlach experiment, it had to be added to the Schrodinger equation by W. Pauli. It was not until P. Dirac linearized the relativistic Klein-Gordon equation that spin ½ seemed to appear naturally in a theory. In a previous note (1), we argued that not only does a photon solution (which emerges from the classical Maxwell electromagnetic equations with no charge sources and currents) have a wavelength and frequency exp(-iEt + ipx) with E=p, but also an intrinsic physical sense of three space because El (the electric field), B (the magnetic field which is perpendicular to El) and p the momentum (which is perpendicular to El and B) form a basis for three-space. This is important because one may always choose the z direction along p (the momentum) so momentum alone does not intrinsically define three-space. De Broglie postulated that an electron has a wavelength like a photon. In (1) we suggested that perhaps the electron should also have an intrinsic physical sense of three-space. The Klein-Gordon equation, however, contains pp which is a scalar with the same value for a fixed value p pointing in any direction. As a result, three-space is not represented even if one has (px,py,pz). We noted in (1) that one may linearize the Klein-Gordon equation using a two vector if one considers p as pz only. If one wishes to linearize with p=(px,py,pz) then one is forced to use four-vectors and ( s dot p) emerges in the equations where s is a 2x2 Pauli matrix such that s dot p s dot p = pp. Thus, linearizing the Klein-Gordon equation does not necessarily bring in spin. It is by insisting that px, py and pz be present i.e. forcing an intrinsic sense of three-space that the si Pauli matrices appear. These are associated with spin, a kind of angular motion which creates a sense of three space within an electron. In this note, we try to extend these ideas to the nonrelativistic Schrodinger equation which is not linearized. We also point out that Eberlein in 1962 (2) already suggested spin is a “geometric property” of nonrelativistic Euclidean 3-space