matrix representation of angular momentum

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Matrix representation of

Angular momentum

dAA kiik =

*

! aA : angular momentum A' =

dAYYA lmmllmml =

*

'',''

And for particular value of l which 'll = thus m and 'm has the 1.2 +l value , because lml

So the matrix of the angular momentum has the dimension )12).(12( ++ ll , this is called squar matrix

( 1 ) - Matrix representation of

2L at 1=l

dYLYL lmmlmm =

2*

''

2

'

+= dYLYllh lmml2*

''

2 )1(

'2

)1( mmllh += Where

'..........0

'...........1' {mm

mmmm

==

For 1=l the matrix has dimension of 33 , and we can write it as the following :

2

2 2

, '

2

2 0 0

0 2 0

0 0 2

m m

h

L h

h

=



It is a diagonal matrix where the diagonal elements equal to the eigen

value of 2

L

( 2 ) - Matrix representation of zL

at 1=l By the same way , we can write,

'' ' mmmm mhLz = So the matrix can be written as

h' 0 0

0 0 0

0 0 -h'

zL

=

The diagonal elements equal to the eigen value of zL

(3)-Matrix representation of xL

and yL

at 1=l lader operator/ . -

yx iLLL +=+ )(2

1+ += LLLx

yx iLLL =+ 1

( )2

yL L Li

+ =

+2 0 = LL*

3 : :/ 7 / L L+ 45

1,, +++ = mlml YcYL

1,, = mlml YcYL

And for particular value of of l which 'll =

dYLYL lmmlmm +=+

*

'''

dYcY lmml +=

*

''

dYYc lmml ++= 1

*

''

1,'' ++=+ mmmm cL

(1)

Where



1'............0

1'............11,' {++=+ =

mm

mmmm

By the same treatment we can deduce that :

' ', 1mm m mL c = (2)

Where

1'............0

1'............11,' {= =

mm

mmmm

+ L L+c

c

: dYcYcdYLYL mlmlmlml ,

*

1,,

*

, )()( +++++ =

dYYc lmml ++= 1

*

''

2

2

+= c

Then

dYLLYdYLYLc mlmlmlml ++++ == ,

**

,,

*

,

2 )(

dYYLL mlml+= ,

*

,

Then

)).((2 yxyx iLLiLLLLc +== ++

yxyxxxLiLLiLLL ..)( 22 ++=

],.[22 yxz LLiLL +=

mhmhllh2222 '')1( +=

])1([' 22 mmllh +=

So that

)1)((' ++=+ mlmlhc



And by the same treatment we can easy to deduce that

)1)((' ++= mlmlhc

) 2( ) 1(; 27 / c c+

1,'',.)1)((' +++=+ mmmm mlmlhL

5

1,'',.)1)((' ++= mmmm mlmlhL

@? : : +L L : 0 ' 2 0

0 0 ' 2

0 0 0

h

L h+

=

0 0 0

' 2 0 0

0 ' 2 0

L h

h

=

A C- ;:

)(2

1+ += LLLx

0 ' 2 01

' 2 0 ' 22

0 ' 2 0

x

h

L h h

h

=

A C- 5

1( )

2yL L L

i+ =

0 ' 2 0 0 0 01

0 0 ' 2 ' 2 0 02

0 0 0 0 ' 2 0

x

h

L h h

h

= +



0 ' 2 0 0 0 01

0 0 ' 2 ' 2 0 02

0 0 0 0 ' 2 0

y

h

L h hi

h

=

0 ' 2 01

' 2 0 ' 22

0 ' 2 0

y

h

L h hi

h

=

Derivation of Pauli matrices

For spin angular momentum 2

1=s

, put 2

1=l

then the projection are

2

1,

2

1=sm

because sms s , so that the matrices

zyx ssss ,,,2

of the dimension 22

( 1 ) - 2

s matrix :

2

2 2

, ' , '

2

3' 0

4( ) ( 1) '

30 '

4

m m m m

h

s s s h

h

= + =

( 2 ) - zs

matrix :

, ' . '

01 02

( ) '0 12

02

z m m m m

h

hs mh

h

= = =



( 3 ) - yxss ,

matrices :

)(2

1+ += sss x

1,'', .)1)((' +++=+ mmmm msmshs

)(2

1+ = ss

is y

1,'', .)1)((' ++= mmmm msmshs Then

. '

0 ' 0 1'

0 0 0 0m m

hs h

+ = =

. '

0 0 0 0'

' 0 1 0m m

s hh

= =

0 0 0 0 0 11 1 '( ) '

0 0 1 0 1 02 2 2x

hs s s h+

= + = + =

0 1 0 0 01 1 '( )

0 0 1 0 02 2 2y

ihs s s

ii i+

= = =

We can write the spin matrices in the term of Pauli matrices :



2

hs =

@? :/ :A ./ DE /

1 0 0 1 0, ,

0 1 1 0 0z x z

i

i

= = =

matrix representation of angular momentum

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