An Approximate Second-order Perturbation Treatment

The sum may be rearranged in such a manner as to permit an approximate value to be easily found. On multiplying by 

1 To prove this, we note that if = (as is easily verified by 

As an example let us take the now familiar problem of the polarizability of the normal hydrogen atom, with H = eFz. We know that H u u vanishes. The integral (7T2)u,lt is equal to e2F2(z2) n, i and, inasmuch as r2 = x2 -f- y + z2 and the wave function for the normal state is spherically symmetrical, the value of (z2)i,i, is just one-third that of (r2) Utu, given in Section 21c as 3ag. Thus we obtain 

If we use the value — e2/2a0 for W°, (taking the ionized atom at zero energy), we obtain 

It is interesting to note that if, in discussing the normal state of a system, we take as the zero of energy the first unperturbed excited level, then the sum is necessarily positive and the approximate treatment gives a lower limit to WIn the problem of the normal hydrogen atom this leads to 

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There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

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