48 CHAPTER4. THEQUANTUMSTATE
vector.Ifthecomponentsof%varerealnumbers,thisisgivenby
%v·%v = [v 1 ,v 2 ,v 3 ...,vN]
v 1
v 2
v 3.
.
vN
= v^21 +v^22 +...+vN^2 (4.8)Ontheotherhand,ifthecomponentsof%varecomplex,thiscanleadtoanegative
innerproduct, andan imaginarynorm. Invectoralgebrathe normof avector is
alwaysreal,so the definitionof arow vector ismodified: its components arethe
complexconjugateofthecorrespondingcolumnvector,i.e
%v·%v = [v 1 ∗,v 2 ∗,v∗ 3 ...,vN∗]
v 1
v 2
v 3
.
.
.
vN
= v 1 ∗v 1 +v∗ 2 v 2 +...+v∗NvN (4.9)Invectoralgebraavectorcomesintwoforms,rowsandcolumns. However,the
notation%vdoesnotdistinguishbetweenrowandcolumnvectors,andsometimesthis
distinctionisuseful.Wethereforeintroducethe”bra-ket”notationinwhicha”ket”
|v>correspondstoacolumnvector
|v>⇐⇒
v 1
v 2
v 3
.
.
.
vN
(4.10)
anda”bra”<v|tothecorrespondingrowvector
<v|⇐⇒[v∗ 1 ,v∗ 2 ,v∗ 3 ,....,vN∗] (4.11)Inthisnotation,theinnerproductis
<v|v>=%v·%v=v∗ 1 v 1 +v 2 ∗v 2 +...+vN∗vN (4.12)