Cartesian coordinates quantization

Where H contains all terms in (16) except (19). In eq. (16) all quantities were defined through the Cartesian coordinates. For further purposes it will be natural to work in normal coordinates Br. The normal coordinate in second quantized formalism is given as... 

Before 1925/26 quantum mechanics was based on ad hoc quantization procedures such as the Bohr-Sommerfeld quantization method. The disadvantage of this method is that it is coordinate dependent, and usually works only in Cartesian coordinates. In 1917 Einstein suggested a coordinate independent quantization procedure, an important improvement over the Bohr-Sommerfeld quantization scheme. Einstein s quantization method was subsequently extended and improved by Brillouin (1926) and Keller (1958). Therefore, this quantization scheme is referred to as EBK... 

The essential problem of the Born-Oppenheimer approximation lies in the fact, that initially the electronic states are quantized whereas the motion of nuclei remains in classical form. Then the transition from the Cartesian to the normal coordinates is carried out on the basis of Newton mechanics, and finally the nuclear motion is quantized as the system of independent harmonic oscillators. This procedure represents the hierarchical type of quantization, which is a complete contradiction of the fundamental requirement of the second quantization procedure of the total Hamiltonian that must be simultaneous. 

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