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76                   General principles of quantum theory

                             Completeness
                             We now evaluate hf j f i in which f and f     are expanded as in equation (3.27),
                             with the two independent summations given different dummy indices
                                           *                 +
                                            X         X           X X                   X
                                                                                               2
                                  hf j f iˆ     a j ø j    a i ø i  ˆ    a a j hø j j ø i iˆ  ja i j
                                                                           j

                                              j        i            j  i                  i
                             Without loss of generality we may assume that the function f is normalized, so
                             that hf j f iˆ 1 and
                                                           X      2
                                                               ja i j ˆ 1                      (3:30)
                                                            i
                               Equation (3.30) may be used as a criterion for completeness. If an eigenfunc-
                             tion ø n with a non-vanishing coef®cient a n were missing from the summation
                             in equation (3.27), then the series would still converge, but it would be
                             incomplete and would therefore not converge to f . The corresponding coef®-
                             cient a n would be missing from the left-hand side of equation (3.30). Since
                             each term in the summation in equation (3.30) is positive, the sum without a n
                             would be less than unity. Only if the expansion set ø i in equation (3.27) is
                             complete will (3.30) be satis®ed.
                               The completeness criterion can also be expressed in another form. For this
                             purpose we need to introduce the variables explicitly. For simplicity we assume
                             ®rst that f is a function of only one variable x. In this case, equation (3.29) is
                                                            …
                                                       X

                                                 f (x) ˆ     ø (x9) f (x9)dx9 ø i (x)
                                                               i
                                                         i
                             where x9 is the dummy variable of integration. Interchanging the order of
                             summation and integration gives
                                                       … "              #
                                                          X
                                                f (x) ˆ      ø (x9)ø i (x) f (x9)dx9
                                                               i
                                                           i
                             Thus, the summation is equal to the Dirac delta function (see Appendix C)
                                                    X
                                                       ø (x9)ø i (x) ˆ ä(x ÿ x9)               (3:31)
                                                         i
                                                     i
                             This expression, known as the completeness relation and sometimes as the
                             closure relation, is valid only if the set of eigenfunctions is complete, and may
                             be used as a mathematical test for completeness. Notice that the completeness
                             relation (3.31) is not related to the choice of the arbitrary function f , whereas
                             the criterion (3.30) is related.
                               The completeness relation for the multi-variable case is slightly more
                             complex. When expressed explicitly in terms of its variables, equation (3.29) is
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