QMGreensite_merged

308 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

where

!i 1 i 2 ...iD=







+1 if i 1 i 2 ...iD anevenpermutationof 123 ...D

− 1 if i 1 i 2 ...iD an oddpermutationof 123 ...D

0 otherwise

(21.17)

Inparticular,fora 2 × 2 matrix

det

[

m 11 m 12

m 21 m 22

]

=m 11 m 22 −m 12 m 21 (21.18)

TwoveryimportantclassesofmatricesaretheHermitianmatrices,whichhave
theproperty
H=H† (21.19)

or,incomponents,
[H]ij=[H]∗ji (21.20)

andtheunitarymatrices,whichhavetheproperty

U†=U−^1 (21.21)

ByU−^1 ,wemeanamatrixwiththepropertythat

UU−^1 =I (21.22)

whereIistheunitmatrix

I=

[

1 0

0 1

]

(21.23)

Intermsofcomponents
[I]ij=δij (21.24)

Hermitian 2 × 2 matriceshavetheform

H=

[

a c+id

c−id b

]

(21.25)

wherea,b,c,darerealnumbers.Aunitary 2 × 2 matrixhastheform

U=eiθ

[

a+ib −c+id

c+id a−ib

]

where a^2 +b^2 +c^2 +d^2 = 1 (21.26)

Ingeneral,foranyD,aunitarymatrixcanbewritten

U = eiH

=

∑∞

n=0

in

n!

Hn (21.27)