Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

8.1 VECTOR SPACES


In the above basis we may express any two vectorsaandbas


a=

∑N

i=1

aiˆei and b=

∑N

i=1

bieˆi.

Furthermore,in such an orthonormal basiswe have, for anya,


〈ˆej|a〉=

∑N

i=1

〈ˆej|aieˆi〉=

∑N

i=1

ai〈ˆej|eˆi〉=aj. (8.16)

Thus the components ofaare given byai=〈ˆei|a〉. Note that this isnottrue


unless the basis is orthonormal. We can write the inner product ofaandbin


terms of their components in an orthonormal basis as


〈a|b〉=〈a 1 ˆe 1 +a 2 eˆ 2 +···+aNˆeN|b 1 eˆ 1 +b 2 ˆe 2 +···+bNeˆN〉

=

∑N

i=1

a∗ibi〈ˆei|eˆi〉+

∑N

i=1

∑N

j=i

a∗ibj〈eˆi|ˆej〉

=

∑N

i=1

a∗ibi,

where the second equality follows from (8.14) and the third from (8.15). This is


clearly a generalisation of the expression (7.21) for the dot product of vectors in


three-dimensional space.


We may generalise the above to the case where the base vectorse 1 ,e 2 ,...,eN

arenotorthonormal (or orthogonal). In general we can define theN^2 numbers


Gij=〈ei|ej〉. (8.17)

Then, ifa=


∑N
i=1aieiandb=

∑N
i=1biei, the inner product ofaandbis given by

〈a|b〉=

〈N

i=1

aiei







∑N

j=1

bjej


=

∑N

i=1

∑N

j=1

a∗ibj〈ei|ej〉

=

∑N

i=1

∑N

j=1

a∗iGijbj. (8.18)

We further note that from (8.17) and the properties of the inner product we


requireGij=G∗ji. This in turn ensures that‖a‖=〈a|a〉is real, since then


〈a|a〉∗=

∑N

i=1

∑N

j=1

aiG∗ija∗j=

∑N

j=1

∑N

i=1

a∗jGjiai=〈a|a〉.
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