The Van Vleck and contact transformations

We now describe two other methods of deriving an effective Hamiltonian, both of which are widely used. Although we shall not go into details, the mathematical development will show that the two methods are exactly equivalent and, in addition, that they are very nearly equivalent to the method based on projection operators given in the previous section. The equivalence ofthe three methods is not really very surprising since they are all solutions of the problem by perturbation theory, differing only in the mathematical techniques employed. 

We first imagine that we have constructed a matrix representation of X in the 

Since the matrix T is unitary (7 t = T l), Xwill have the same eigenvalues as Xbut different eigenfunctions. We then set 

We now show how the matrix elements of Si can be selected to remove the effects on Xof matrix elements off-diagonal in tj. Since it can easily be verified that °(1h [3C , Si] i , j)° is always zero, irrespective of the form of 6), we see from (7.54) that the diagonal matrix elements of X) are just the diagonal matrix elements of X  

Without any loss in generality therefore, we can set the matrix elements of Si diagonal 

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