An inner product gives a notion of length-squared||·||^2 for vectors, with||v||^2 =〈v,v〉Note that whether to specify antilinearity in the first or second variable is a
matter of convention. The choice we are making is universal among physicists,
with the opposite choice common among mathematicians. Our Hermitian in-
ner products will be positive definite (||v||^2 >0 forv 6 = 0) unless specifically
noted otherwise (i.e., characterized explicitly as an indefinite Hermitian inner
product).
An inner product also provides an isomorphismV'V∗by the map
v∈V→lv∈V∗ (4.3)wherelvis defined by
lv(w) = (v,w)
in the real case, and
lv(w) =〈v,w〉
in the complex case (where this is a complex antilinear rather than linear iso-
morphism).
Physicists have a useful notation due to Dirac for elements of a vector space
and its dual, for the case whenV is a complex vector space with a Hermitian
inner product (such as the state spaceHfor a quantum theory). An element of
such a vector spaceVis written as a “ket vector”
|α〉whereαis a label for a vector inV. Sometimes the vectors in question will be
eigenvectors for some observable operator, with the labelαthe eigenvalue.
An element of the dual vector spaceV∗is written as a “bra vector”
〈α|with the labeling in terms ofαdetermined by the isomorphism 4.3, i.e.,
〈α|=l|α〉Evaluating〈α|∈V∗on|β〉∈Vgives an element ofC, written〈α|(|β〉) =〈α|β〉Note that in the inner product the angle bracket notation means something
different than in the bra-ket notation. The similarity is intentional though since
〈α|β〉is the inner product of a vector labeled byαand a vector labeled byβ
(with “bra-ket” a play on words based on this relation to the inner product
bracket notation). Recalling what happens when one interchanges vectors in a
Hermitian inner product, one has
〈β|α〉=〈α|β〉