Quantum Mechanics and Complex Numbers Part II

In part I of this note, we argued that exp(ipx) written as a 2-vector is solution of a 2x2 rotation matrix with cos(p dx) like terms operating on (a(px),b(px)) to yield d/dx operating on this same vector i.e. the translation operator. We did not give a reason why this should be the case. In this note, we focus on developing bound state quantum mechanics as a statistical theory in parallel with classical statistical theory. As argued in previous notes, the free particle “probability” becomes the central object because there is no acceleration in the statistical picture. It must possess an invariance related to all x points because none is preferential. Ideas of periodicity and invariance, linked to a statistical picture, seem to arise and lead to the complex periodic/spatially invariant form exp(ipx). A number of the ideas in this note have been touched upon in previous notes, but here we try to develop the ideas strictly by comparing with the classical statistical scenario and stressing spatial invariance