Mathematical Methods for Physicists: A concise introduction - Site Map

INNER PRODUCT SPACES (UNITARY SPACES)
In our earlier discussion, the inner product of two vectors in E n was de®ned by
Eq. (5.6), a generalization of the inner product of two vectors in E 3 . In a general
linear vector space, an inner product is de®ned axiomatically analogously with the
inner product on E n . Thus given two vectors jUi and jWi
jUi ˆ Xn
iˆ1
u i ji; i
jWi ˆ Xn
iˆ1
w i ji; i
…5:9†
where jUi and jWi are expressed in terms of the n base vectors ji, i the inner
product, denoted by the symbol h UW j i, is de®ned to be
hUjWi ˆ Xn X n
iˆ1 jˆ1
u*w i j h ij ji: …5:10†
hUjis often called the pre-factor and jWithe post-factor. The inner product obeys
the following rules (or axioms):
B.1 h UW j i ˆ hWU
j i*
B.2 hUjUi 0; ˆ 0 if and only if jUi ˆ j0i
B.3 h UX j i‡ jWi† ˆ hUX
j i‡ hUW
j i
B.4 h aUjWi ˆ a* hUW
j i; hUbW
j i ˆ bUW h j i
(skew-symmetry);
(positive semide®niteness);
(additivity);
(homogeneity);
where a and b are scalars and the asterisk (*) denotes complex conjugation. Note
that Axiom B.1 is di€erent from the one for the inner product on E 3 : the inner
product on a general linear vector space depends on the order of the two factors
for a complex vector space. In a real vector space E 3 , the complex conjugation in
Axioms B.1 and B.4 adds nothing and may be ignored. In either case, real or
complex, Axiom B.1 implies that hUU
j i is real, so the inequality in Axiom B.2
makes sense.
The inner product is linear with respect to the post-factor:
hU
jaW ‡ bXi ˆ aU h
and anti-linear with respect to the prefactor,
jWi ‡ bU h jXi;
haU ‡ bXjWi ˆ a* hUW
j i‡ b* hXW
j i:
Two vectors are said to be orthogonal if their inner product vanishes. And we
will refer to the quantity hUjUi 1=2 ˆkU k as the norm or length of the vector. A
normalized vector, having unit norm, is a unit vector. Any given non-zero vector
may be normalized by dividing it by its length. An orthonormal basis is a set of
basis vectors that are all of unit norm and pair-wise orthogonal. It is very handy
to have an orthonormal set of vectors as a basis for a vector space, so for hij ji in
Eq. (5.10) we shall assume
(
hjji i ˆ ij ˆ 1 for i ˆ j
;
0 for i 6ˆ j
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