Relativity quantum mechanics with an application to compton scattering

Abstract

The new quantum mechanics, introduced by Heisenberg and since developed from different points of view by various authors, takes its simplest form if one assumes merely that the dynamical variables are numbers of a special type (called q-numbers to distinguish them from ordinary or c-numbers) that obey all the ordinary algebraic laws except the commutative law of multiplication, and satisfy instead of this the relations q r q s – q s q r =0, p r p s – p s p r = 0 } q r q s – p s q r = 0 ( r ≠ s ) or ih ( r = s ) where the p' s and q' s are a set of canonical variables and h is a c-number euqal to (2π) -1 times the usual Planck’s constant. Equations (1) may be regarded as replacing the commutative law of the classical theory, as one can, with their help, build up a complete algebraic theory of quantities that are analytic functions of a set of canonical variables. Further, it may easily be seen that the quantity [ x, y ] defined by xy – yz = ih [ x, y ] is completely analogous to the Poisson bracket of the classical theory. By means of this analogy the whole of the classical dynamical theory, in so far as it can be expressed in terms of P. B.’s instead of differential coefficients, may be taken over immediately into the quantum theory.