Exponential unitary transformations of the elementary operators

Having expanded the transformed creation and annihilation operators (3.2.16) and (3.2.17) in the original ones, let us now develop a more convenient representation of the transformed operators. Consider the operators 

We shall now demonstrate that the operators Cp in (3.2.18) are identical to the transformed creation operators a], in (3.2.16), thereby establishing the relationship (3.2.3). The corresponding expression for the annihilation operators (3.2.4) then follows by taking the Hermitian conjugate of (3.2.3). Expanding Op using the BCH expression (3.1.7), we obtain 

---