Bohr-Sommerfeld 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... 

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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