Mathematical Methods for Physics and Engineering - Matematica.NET

8.1 VECTOR SPACESIn the above basis we may express any two vectors a and b asN∑N∑a = a i ê i and b = b i ê i .i=1Furthermore, in such an orthonormal basis we have, for any a,N∑N∑〈ê j |a〉 = 〈ê j |a i ê i 〉 = a i 〈ê j |ê i 〉 = a j . (8.16)i=1Thus the components of a are given by a i = 〈ê i |a〉. Note that this is not trueunless the basis is orthonormal. We can write the inner product of a and b interms of their components in an orthonormal basis as〈a|b〉 = 〈a 1 ê 1 + a 2 ê 2 + ···+ a N ê N |b 1 ê 1 + b 2 ê 2 + ···+ b N ê N 〉N∑N∑ N∑= a ∗ i b i 〈ê i |ê i 〉 + a ∗ i b j 〈ê i |ê j 〉=i=1N∑a ∗ i b i,i=1i=1j≠iwhere the second equality follows from (8.14) and the third from (8.15). This isclearly a generalisation of the expression (7.21) for the dot product of vectors inthree-dimensional space.We may generalise the above to the case where the base vectors e 1 , e 2 ,...,e Nare not orthonormal (or orthogonal). In general we can define the N 2 numbersi=1i=1G ij = 〈e i |e j 〉. (8.17)Then, if a = ∑ Ni=1 a ie i and b = ∑ Ni=1 b ie i , the inner product of a and b is given by〈 ∣∑ N ∣∣∣∣∣ 〉∑N〈a|b〉 = a i e i b j e j==N∑i=1i=1 j=1N∑i=1 j=1j=1N∑a ∗ i b j 〈e i |e j 〉N∑a ∗ i G ij b j . (8.18)We further note that from (8.17) and the properties of the inner product werequire G ij = G ∗ ji . This in turn ensures that ‖a‖ = 〈a|a〉 is real, since thenN∑ N∑N∑ N∑〈a|a〉 ∗ = a i G ∗ ija ∗ j = a ∗ j G ji a i = 〈a|a〉.i=1 j=1j=1 i=1245