Quantum mechanics and a preliminary investigation of the hydrogen atom

Abstract

Although the classical electrodynamic theory meets with a considerable amount of success in the description of many atomic phenomena, it fails completely on certain fundamental points. It has long been thought that the way out of this difficulty lies in the fact that there is one basic assumption of the classical theory which is false, and that if this assumption were removed and replaced by something more general, the whole of atomic theory would follow quite naturally. Until quite recently, however, one has had no idea of what this assumption could be. A recent paper by Heisenberg* provides the clue to the solution of this question, and forms the basis of a new quantum theory. According to Heisen­berg, if x and y are two functions of the co-ordinates and momenta of a dyna­mical system, then in general xy is not equal to yx . Instead of the commutative law of multiplication, the canonical variables q r p r ( r = 1... u ) of a system of u degrees of freedom satisfy the quantum conditions, which were given by the author in the form q r q s ― q s q r = 0 p r p s ― p s p r = 0 q r p s ― p s q r = 0 q r p r ― p r q r = ih ( r ≠ s ) } (1) where i is a root of — 1 and h is a real universal constant, equal to (2 π ) -1 times the usual Planck’s constant. These equations are just sufficient to enable one to calculate xy — yx when x and y are given functions of the p’ s and q’ s, and are therefore capable of replacing the classical commutative law of multi­plication. They appear to be the simplest assumptions one could make which would give a workable theory.