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In Griffith's Introduction to Quantum Mechanics p. 256, after figuring out $$E_n^2=\sum_{m\neq n} \frac{|\langle\psi_m^0|H'|\psi_n^0\rangle|^2}{E_n^0-E_m^0}$$

he says

We could go on to calculate the second-order correction to the wave function $\psi_n^2$, the third-order correction to the energy $E_n^3$, and so on, but in practice this equation is ordinarily as far as it is useful to pursue this method.

Does it mean that calculating higher-order corrections will only improve the result slightly? Or it will make the result even worse than before? Are there any simple examples (one-body, non-relativisitic) about this?

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    $\begingroup$ I think that it is all about the biggest problem with PT: the fact that it is not guaranteed to converge. For instance, in the domain of quantum chemistry, MPPT approach is known to often have slow, oscillatory, or even non-existent convergence. And yes, in practice MP3 and MP4 provides a little or no improvements over MP2 in molecular properties, or they can be even worse in the case of oscillatory or non-existent convergence. $\endgroup$ – Wildcat Dec 5 '14 at 15:14

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