Commutator Relations for Normal Coordinates

As we have discussed in Sect.2.2.1, the classical Hamilton function of an oscillator with mass m, eigenfrequency w, displacement coordinate u and momentum p is given by 

The relation (D.3a) means that both sides are to be applied on an arbitrary wave function ijj and thereby using the explicit expression (D.2) for p  

Equation (D.4) is verified at once by differentiation of uf according to the product rule. The relations (D.3b) are satisfied trivially. Since F is an arbitrary (differentiable) function, we can consider the relations (D.3) as identities. We have thus shown that the representation (D.2) is equivalent to the commutator relations (D.3). 

For two particles ( ) and (, ) of our chain ( , number the unit cells, K, k number the atoms in these unit cells), the generalization of (D.3) is 

we first prove the commutator relations (2.112). From (2.53 and 56) we obtain 

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