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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 i a j ø j a i ø i a a j hø j j ø i i 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 i 1 and
X 2
ja i j 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) 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) 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) 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