Default Normal Template - faculty.kfupm.edu.safaculty.kfupm.edu.sa/PHYS/imnasser/papers/all-lectures.pdf · لﻭﻷﺍ: ﻰﺴﺍﺭﺩﻟﺍ لﺼﻔﻟﺍ ﻡﻭﻠﻌﻟﺍ:ﻪﻴﻠﻜﻟﺍ

: : 1+ 2: : : :

) . ( . . : )9 ( 2:00 12:20 :

- -1 - _

. - : -2

. - - -3

: . . :

1- Group Theory for Atoms, Molecules and Solids, B. S. Wherrett, 1986, Prentic Hall International.

2- Finite Group and Quantum Mechanics D. B. Chesnut, 1974, John Wiley. 3- Elements of Group Theory for Physics, A. W. Joshi, 1973. 4- "Molecular Symmetry and Group Theory, A. Vincent, 2001, John Wiley. 5- Physical Chemistry P. W. Atkins, 5th edition, Oxford University, 1994. 6- Chemical Application of Group Theory, F. A. Cotton, third edition , 1998,

John Wiley. 7- Group Theory and Its Application to Physical Problems, M. Hamermesh,

Dover, 1989. 8- Atoms and Molecules, M. Weissbluth, Academic Press, 1978.

, " " "-1 .1994 -

- . . . " "-2 .) . (2003,

: /imnasser/PHYS/sa.edu.kfupm.users://http

: . .... , , : : : ( -1 ....) , : ( -2 . , : -3

Prof. Dr. I. Nasser ([email protected]) 1

http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/http://users.kfupm.edu.sa/PHYS/imnasser/

: ) (

. . .

. ) ( : ) (

.) ( -1

.)( ) ( -2

. -3 . ) ( ).

. .

. .

. . ,

) : - nC

n

)

. 2 . ) (

n

90 x :) : , , ,P x y z )

z x

P (x,z,-y)

-y P(x,y,z)

z

yx


:

'

' 1 0 0' 0 0 1 '' 0 1 0

r rR

x xy yz z

=

r Rr=

| |R

1 : ( .1= = ).

) nC (

2( ), 180nC C n = =

C E , . nn =

3

2

3601 ( ) (120),3

2 22 ( ) (2

23 ( ) (2 ) ,

n n n n

nn

C C C

C C C C C C Cn n

C C n C En

4) ( ),n

= =

= = = =

= = =

( ) 1 2 2 ( )(2 ) ( )m nn nm n mC C C Cn nm

= = =

( )( )

134 4

156 6

1 ,

2 ,

C C

C C

=

=

) ( - ,v h ( , ,P x y z ( , :

( ,)z( :)

xyz


y x

P (x,y,-z)

-z

zz

r B r=

yx

P(x,y,z)

:

'

' 1 0 0' 0 1 0 '' 0 0 1

r rB

x xy yz z

=

1 | |B = | |R =h x

. 1= 1 y =

. . ( ) (

(.E ) ).

xy

2 = =

v

) .h n - )n nS C =n

( )n hS C=

( ) ( )

hh . ) ( n nC C ).(

2n

h =nC

2 2 2( )h hI S C C

. = = =

P(x,y,z)

z

yxP (-x,-y,-z)


:

'

' 1 0 0' 0 1 0 '' 0 0 1

r rB

x xy yz z

= =

r B r

B

1 | = . 3 - = |

2 2( ,h hI C IC

:

-1

) = =

C E2h 22 = E =

C 1 -2 .

.

2h

) (:

-12 .

'2 2

second rotation first rotation

(2 )U U C =

" C" -22 .

'

second reflection first reflection

(2 )v v C =

'

second reflection first reflection

( 2 )v v C =

' ' ' '(2 ) (2 )v v v v v vC C = =

.


.

) - )

E

nC2( )n

i

nS 2n

) ( :


:

G ( I ( ) G( .

: G

{ }, , , ,n nE C S=,

G a - b G . ,a b G) ) ( - ) ( )c a b ca b =, ,a b c G

E G .

E . - a a E=a Ga G

1a - G E 1 . 1a a a a = =a G

b

a ) ( b a,= . a b G

{ }2,G E S= ={

: :

I }, hG E =

h

: E x I E x

E E I E E E E I I

h

h h

: . :

( ) ( , . ).

. : ) (

. G . .

HG

Hhh g

z

.| A | r >

x

| r ' >

y

= + = = = + = + = +

cos ; sinx r y r . = =

' cos sin 0' sin cos 0 | ' ( ) |' 0 0 1

=

x xy y rz z

> = >R r

: : ) ( -1

' cos sin 0' sin cos 0 | ' ( ) |' 0 0 1

= > =

x xy y rz z

cos sin 0sin cos 00 0

>R r

-2 ,

:

1


0 00 00 0

i

i

ee

z

1

: z c

11 12 13

21 22 23

31 32 33

cos sin 0sin cos 00 0 1

z

a a ac a

a a a

= =

2 2 211 12 13 1a+ + =

11 21 12 22 13 23 0a a a a

a a

( )

a -1 a. -2

a a. + + + =

: z . y x -1

11 22 21 12det( ) 1zc a a a a = . -2 =

11 22 33trace( )zc a a = a -3 + +

r >2 2( , ) (=

. , ( |

( . ) 45 : . = o

f x y a x ya

' cos45 sin 45 0' sin 45 cos45 0' 0 0 1

=

x xy yz z

( ', ') 2 ' . ' ( , )f x y3, 2= =x y(3,2) 5= . .

' / 5 2, '= =x y 1/ 2(5 / 2) =g / 2, 1 5 .

2 2( , ) (= + )f x y x y ) .

2 2( '= +x y( 'g x

= g x y a x yf a

, ') 'y


c : C 2 2 ( )2

C = = =

2

1 0 00 10 0 1

c =

( )

0

z : xy 1 0 0

( ) 0 1 00 0 1

xy =

1 0 00 1 00 0 1

i =

xy

: i

zzz

c s z : xy

cos sin 0 1 0 0 cos sin( ) sin cos 0 0 1 0 sin cos 0

0 0 1 0 0 1 0 0z zs xy c

= = =

0

1

: E 1 0 00 1 00 0 1

E =

h

: : i2c2h E

i2c E E

E 2c2c 2ch E ii

E 2cih

h

ih

h


, ,E : 1 00 1

=

11 00 1

A = =

212

B =

, 1 33 1

=3C ,1 31

2 3 1

= =

1 3

3 1

,

.

312

D c

= =

23

1 312 3 1

F c

= = : -1

F D C B A E F D C B A E E C B F D E A A A C D E F B B B A E F D C C E F B A C D D D E A C B F F

-2

. ( ) ( ) ( ); , , ; ,E A B C D F

2v

2

1 0 0 1 0 0 1 0 0 1 0 00 1 0 , 0 1 0 , ( ) 0 1 0 , ( ) 0 1 00 0 1 0 0 1 0 0 1 0 0 1

E C xz yz

= = = =

( )

: :

yz 2c2v E 2c E E E 2c2c

E E

2c

( )xz

( )yz ( )xz( )xz ( )yz

2c( )yz ( )xz ( )xz( )xz ( )yz ( )yz

2Trace 3, Trace 1, Trace ( ) 1, Trace ( ) 1.E c yz xz = = = =

z( .

( ) ( ) ( )2; ; ( ), (E c xz y

: :


2

1 0 0 1 0 0 1 0 0( ) 0 1 0 0 1 0 0 1 0 (

0 0 1 0 0 1 0 0 1) xz C yz

= = =


. .

: : 1 cos sin

Rsin cos

=

. :

: , -1

1 2

cos sinR I 0 ,

sin cosi ie e

= = = =

1ie

. : . -2 =

11 2

2

0cos sin0

0sin cos

i

i

xeix x

xe

= +

1 2,

=

x x1 1x =2

1X 1i

.

x i=

1

2

1xx i

= = 112

X =

2 . 11

2X

i

=

2ie =

, 1 11

2X

i i

=

1 0R0

i

i

eX X

e

:

=

cos sin 0R sin cos

0 0

=

: :

01

.


: : 2 5 4

A1 2

=

. :

: , -32

1 2

5 4A I 0 7 6

1 21, 6

= = + =

= =

1 1

0

. : . -4 =

11 2

2

5 1 4 00

1 2 1 0x

x xx

= +

1 2,

=

x x1 1x =2 1x = 1

2

11

xx

= =

1X 11

. 112

X =

2 . 41117

X =

2

1 41 1

=

X

2 6 =

,

1 1 411 15

X

=

: 1 1 1 4 5 4 1 4 1 01A

1 1 1 2 1 1 0 65X X X

= = =

A

.X


GL(2) -1

1 2

3 4

' ;' ,

x a x a yy a x a= += +

y

:

1 2 1 2

3 4 3 4

x' x xR ,

y' y ya a a aa a a a

= =

1 0A

0 1

=

(2)SL

)

0

:

. 4

-2

. 2) GL

. 3

1 2

3 4

1a aa a

=

O(2) -3(2)2 2+ =

( )x GL y :

( )2 22 2 21 2 3 4' 'x y a x a y a x a y x y+ = + + + = +2 2 2 21 3 2 4 1 2 3 41; 1; 0;a a a a a a a a+ = + = + =

x' cos sin x; 0

y' sin cos y

2;

: : .

2

=

. z

.

O(3) -42 2 2 3

. x y z+ + =


, A BA B

11 12 11 12

22 22 22 22

,

= =

a a b bA B

a a b b

11 12

22 2211 12 11 12

22 22 21 22 11 12

21 22

0 00 00

0 0 00 0

a aa aa a b b A

a a b b

: : 1

A B B b b

b b

= = =

ace( ) Trace( ) Trace( ) :

Tr B = +A B A

11 12 11 12

22 22 22 22

,

= =

a a b bA B

a a b b

11 12 11 1211 12

21 22 21 2211 12 11 12

22 22 21 22 11 12 11 1222 22

21 22 21 22

11 11 11 12 12 11 12 12

11 21 11 22 12 21 12 22

22 11 22 1

= =

=

b b b ba a

b b b ba a b bA B

a a b b b b b ba a

b b b b

a b a b a b a ba b a b a b a ba b a b 2 22 11 22 12

22 21 22 22 22 21 22 22

a b a ba b a b a b a b

Trace( ) Trace( )Trace( )

: : 2

: =A B A

1 2 1 0,

1 0 0 1

= =

A B

1 2 0 01 2 1 0 1 0 0 01 0 0 1 0 0 1 0

0 0 0 1

= =

A B

Trace( ) 3race( ) Trace( ) 1 2 3

A BA B =+ = +

B

: : 3

T

=


11 12 11 12

22 22 22 22

1 0 1 0 1 0 2 01 20 1 0 1 0 1 0 2

1 0 0 01 0 1 01 0

0 1 0 00 1 0 1

= = =

a a b bA B

a a b b

Trace( ) 2Trace( )Trace( ) 1 2 2

A BA B =

= =


A : X= ,2B X= ,C X,,F X, 3456E X= = D X= =

: -1F D C B A E F D C B A E E E F D C B A A B B C C D D F F

) -2 ) ( ) ( ); , , ; ,E A B C D F

23 : 12b C

= = , ,

1c = = ,1 0

0 1

212

d = = 1 33 1

312

1 3

3 1f = = ,

1 3

3 1

3

1 312 3 1

a C

= =

1 00 1

E =

-1) -2 ) ( ) ( ); , ; , ,E a b c d f

2

1 2 3;

1 3 2p =

3

;

6p =

6

:

:

4p = 1 2 3

;3 1 2

1

1 2 3;

1 2 3p =

5p = 1 2 3

;3 2 1

3

1 22 1 3

p = , 1 2 3 ;2 3 1

p

1p 2p 3p 4p 5p

1p 1p 2p 3p 4p 5p 6p

2p2p

3p 3p

4p 4p

5p 5p

6p 6p



00yi

i

=

1 00 1

1 00 1

I =

z

, :

.

0: 11 0x

=

,

z = ,

I

I I I

I I

x y

x y z

x yizix

yy zi xi

yi xiz z

!

a : cAb d

=

ad 1 = 00 1

A A =

1

1 1 d cAb a

=

0bc

4

34c

24c4c4

:

E 34c

24c4c E E

E 34c

24c4c4c

4c34c

24c

24c E

24c4c

34c

34c E

2h :

h

i2c2h E i2c E E

E 2c2c 2ch E ii

E 2cih

h

ih

h

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