On the quantum dynamics of degenerate systems

§1. It is well known that if Fi =n i h ,i ═ 1, 2, ... (1) be a set of quantum conditions applicable to a class of dynamical systems, then Fi must satisfy the definite condition: ∂Fi /∂a ═ 0, (2) wherea is a parameter, such as an external field, etc., which is followed to undergo a slow non-systematic variation. In other words, Fi must be an “adiabatic invariant” of the class of systems. Burgers has shown, on the basis of Newtonian dynamics, that Ii ═ ∫0 P i dq i fulfils this condition in the case of a conditionally periodic system of several degrees of freedom whereq i p i are separable Hamiltonian co-ordinates,provided the system he non-degenerate, i. e , provided no relation of the form ∑i s i j ν i = 0 (3) exist between the frequenciesν i , wheres i j is an integer, positive or negative, In the case of a system of charged particles, W. Wilson has shown that on the basis of the general theory of relativity,p i should be replaced by πi where πi =p i +e Ai , (4)