First-order Perturbation Theory for a Non-degenerate Level

In discussing many problems which cannot be directly solved, a solution can be obtained of a wave equation which differs from the true one only in the omission of certain terms whose effect on the system is small. Perturbation theory provides a method of treating such problems, whereby the approximate equation is first solved and then the small additional terms are introduced as corrections  

We assume that it is possible to expand H in terms of some parameter X, yielding the expression 

The functions j/k form a complete orthogonal set as discussed in Section 22, and, if we assume that they have also been normalized, they satisfy the equation (Appendix III) 

If the perturbation is really a small one, the terms of these series will become rapidly smaller as we consider the coefficients of larger powers of X i.e., the series will converge. 

We now substitute these expansions for H, pk, and Wk into the wave equation 23-1, obtaining the result, after collecting coefficients of like powers of X, 

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FIRST-ORDER PERTURBATION THEORY FOR A NON-DEGENERATE LEVEL... 

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