Angle-action transformations, quantum

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

Concerning the Poincare surface of section, it should be noticed that a sort of quantum surface of section can be constructed by intersection of the Wigner or Husimi transform of the eigenfunctions expressed in the quantum action-angle variables of the effective Hamiltonian, which can provide a comparison with the classical Poincare surfaces of section (e.g., in acetylene). 

Hitherto wc have applied the quantum theory only to mechanical systems whose motion may be calculated by separation of the variables. We proceed now to deal in a general manner with the question of when it is possible to introduce the angle and action variables wk and Jfc so admirably suited to the application of the quantum theory. For this purpose it is necessary, in the first place, to fix the J s by suitable postulates so that only integral linear transformations with the determinant 1 are possible for it is only in such cases that the quantum conditions (1) Jk=nkh... 

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