306 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA
21.1 Review of Vectors and Matrices
AD-dimensionalcolumnvectorisasetofDcomplexnumbersarrangedinacolumn
v 1
v 2
v 3
.
.
.
vD
(21.2)
andaD-dimensionalrowvectorisasetofDcomplexnumbersarrangedinarow
[w 1 ,w 2 ,w 3 ,...,wD] (21.3)
Tosavetyping,IwilluseD= 2 whendisplayingrowandcolumnvectorsexplicitly.
Foreachcolumnvector,andoneeachrowvector,thereisdefinedaninnerprod-
uct
[w 1 ,w 2 ]·
[
v 1
v 2
]
= w 1 v 1 +w 2 v 2
=
∑D
i=1
wivi (21.4)
Toeachcolumnvector,thereisacorrespondingrowvector(andvice-versa)
[
v 1
v 2
]
=⇒[v 1 ∗,v 2 ∗] (21.5)
Thenorm|v|ofavectorvisdefinedasthesquare-rootoftheinnerproductofthe
columnvectorwithitscorrespondingrowvector
|v|^2 =[v 1 ∗,v∗ 2 ]·
[
v 1
v 2
]
=
∑D
i=1
vi∗vi (21.6)
AmatrixM isaD×Dsquarearrayofcomplexnumbers
M=
[
m 11 m 12
m 21 m 22
]
(21.7)
whichtransformscolumnvectorsvintocolumnvectorsv′accordingtotherule
[
v′ 1
v′ 2
]
=
[
m 11 m 12
m 21 m 22
][
v 1
v 2
]
=
[
m 11 v 1 +m 12 v 2
m 21 v 1 +m 22 v 2
]
(21.8)