Bohr quantisation condition

The starting point for semiclassical inversions is the Bohr quantisation condition... 

It may be helpful to note here that, In the case of only one classically allowed region of motion, the analogue of this equation is simply f Pdq = (n + )h, i.e. the Bohr-Sommerfeld quantisation condition. 

The variables p and q are canonically conjugate, so that the Bohr-Sommerfeld quantisation condition yields ... 

As a practical guide, we may follow the line that any gross breakdown in n characterisation signals difficulties in applying the Bohr-Sommerfeld quantisation condition and should be investigated. It is important (as indicated above) to select groups of levels which interact with each other, since otherwise, their behaviour may be ruled by complexity rather than... 

Here we conclude our account of Bohr s theory. Although it has led to an enormous advance in our knowledge of the atom, and in particular of the laws of line spectra, it involves many difficulties of principle. At the very outset, the fundamental assumption of the validity of Bohr s frequency condition amounts to a. direct and unexplained contradiction of the laws of the classical theory. Again, the purely formal quantisation rule, which stands at the head of the theory, is a foreign element which in the first instance is absolutely unintelligible from the physical point of view. We shall see later how both of these difficulties are removed in a perfectly natural way in wave mechanics. 

This problem was considered by Einstein [529], who realised that the Bohr-Sommerfeld quantisation rule is not generally applicable, to any system because it is not a canonically invariant condition, i.e. it is only valid in the specific case where the Hamiltonian is fully integrable, with N constants of the motion which can be separately determined. 

The strongest pieces of evidence complex atoms provide in favour of independent electron modes and simple Bohr-Sommerfeld quantisation are (i) the existence of Rydberg series and (ii) the regularity of the periodic table of the elements. As a corollary, we should look for quantum chaos (if it occurs) in atoms for which there is some breakdown in the quality of the shell structure, combined with prolific and heavily perturbed overlapping series of interacting levels. These conditions are most readily met, as will be shown below, in the spectra of the alkaline-earth elements, as a result of d-orbital collapse. 

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