N.6 Is the variational approximation best?

Clearly, “best” is a subjective term. If you are looking for the wave function within a definite set that has the most accurate expec­tation value of energy, then minimizing the expec­tation value of energy will do it. This function will also approximate the true eigen­function shape the best, in some technical sense {A.7}. (There are many ways the best approxi­mation of a function can be defined; you can demand that the maximum error is as small as possible, or that the average magnitude of the error is as small as possible, or that a root-mean-square error is, etcetera. In each case, the “best” answer will be different, though there may not be much of a practical difference.)

But given a set of approximate wave functions like those used in finite element methods, it may well be possible to get much better results using additional mathematical techniques like Richardson extra­polation. In effect you are then deducing what happens for wave functions that are beyond the approximate ones you are using.