Open AccessOn the quantum dynamics of degenerate systems
Publication year - 1925
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1925.0018
Subject(s) - degenerate energy levels , adiabatic process , mathematics , quantum , hamiltonian (control theory) , mathematical physics , basis (linear algebra) , physics , invariant (physics) , hamiltonian system , pure mathematics , quantum system , quantum mechanics , classical mechanics , geometry , mathematical optimization
§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)