Chapter 1 - The Basic Framework

7/30/2019 Chapter 1 - The Basic Framework

1/18

C h a p t e r 1T h e B a s i c F r a m e w o r k

1 T h e c o n c e p t o f a g r o u pT h e a i m o f t h i s c h a p t e r is t o i n t r o d u c e t h e i d e a o f a g ro u p , t o g i ve s o m ep h y s ic a ll y i m p o r t a n t e x a m p l e s , a n d t h e n t o i n d ic a t e i m m e d i a t e l y h o w th i sn o t i o n a r is e s n a t u r a l l y i n p h y s ic a l p r o b l e m s , a n d h o w t h e r e l a t e d c o n c e p t o f ag r o u p r e p r e s e n t a t i o n l ie s a t t h e h e a r t of t h e q u a n t u m m e c h a n i c a l f o r m u l a t i o n . 9W i t h t h e b a s i c f r a m e w o r k e s t a b l i s h e d , t h e n e x t f o u r c h a p t e r s w i l l e x p l o r e i nm o r e d e t a i l t h e r e l e v a n t p r o p e r t i e s o f g r o u p s a n d t h e i r r e p r e s e n t a t i o n s b e f o r et h e a p p l i c a t i o n t o p h y s i c a l p r o b l e m s i s t a k e n u p i n e a r n e s t i n C h a p t e r 6 .

T o m a t h e m a t i c i a n s a g r o u p is a n o b j e c t w i t h a v e r y p re c i se m e a n i n g . I tis a s e t o f e l e m e n t s t h a t m u s t o b e y f o u r g r o u p a x i o m s . O n t h e s e is b a s e da mo s t e l a b o ra t e an d f a s c i n a t i n g t h eo ry , n o t a l l o f w h i ch i s co v e red i n t h i sb o o k . T h e d e v e l o p m e n t o f t h e t h e o r y d o e s n o t d e p e n d o n t h e n a t u r e o f t h ee l e m e n t s t h e m s e l v e s , b u t i n m o s t p h y s i c a l a p p l i c a t i o n s t h e s e e l e m e n t s a r et r a n s f o r m a t i o n s o f o n e k i n d o r a n o t h e r , w h i c h i s w h y T w i ll b e u s e d t o d e n o t ea t y p i c a l g r o u p m e m b e r .D e f i n i t i o n Group gA se t g o f e l e m ent s i s ca l l ed aare sa t i s f i ed :

"g ro u p " i f t h e fo l lo w i n g fo u r "g ro u p a x i o m s "

( a) T h e r e e x i s t s a n o p e r a t i o n w h i c h a s s o c i a t e s w i t h e v e r y p a i r o f e l e m e n t s Ta n d T ~ o f g a n o t h e r e l e m e n t T " o f g . T h i s o p e r a t i o n is c a l le d m u l t i p l i -c a t i o n a n d i s w r i t t e n a s T " = T T ~, T " b e i n g d e s c r i b e d a s t h e " p r o d u c tof T w i th T t '' .

( b) F o r a n y t h r e e e l e m e n t s T , T ~ a n d T " o f g( T T ' ) T " = T ( T ' T " ) . (1 .1)

T h i s is k n o w n a s t h e " a s s o c i a ti v e l a w " f or g r o u p m u l t i p l i c a t io n . ( T h ei n t e r p r e t a t i o n o f t h e l e f t - h a n d s i d e o f E q u a t i o n ( 1. 1) is t h a t t h e p r o d u c t


2/18

2 G R O U P T H E O R Y I N P H Y S I C S

T T ~ is t o b e e v a l u a t e d f ir st , a n d t h e n m u l t i p l ie d b y T " w h e r e a s o n t h er i g h t - h a n d s id e T is m u l t i p l ie d b y t h e p r o d u c t T ' T ' . )

( c) T h e r e e x i s t s a n i d e n t i t y e l e m e n t E w h i c h i s c o n t a i n e d i n ~ s u c h t h a tT E = E T = T

f o r e v e r y e l e m e n t T o f G .( d ) F o r e a c h e l e m e n t T o f G t h e r e e x i s t s a n i n v e r s e e l e m e n t T - 1 w h i c h i s

a l so c o n t a i n e d i n G s u c h t h a tT T - 1 = T - 1 T = E .

T h i s d e f i n i t i o n co v e r s a d iv e r s e r an g e o f p o s s ib i l i ti e s , a s t h e fo l l o win g ex -a m p l e s i n d i c a t e .E x a m p l e I T h e m u l t i p l i c a t i v e g r o u p o f r ea l n u m b e r sT h e s i m p l e s t e x a m p l e ( f r om w h i c h t h e c o n c e p t o f a g r o u p w a s g e n e r a l i z e d )i s t h e s e t o f a l l r e a l n u m b e r s ( e x c l u d i n g z e r o ) w i t h o r d i n a r y m u l t i p l i c a t i o na s t h e g r o u p m u l t i p l i c a t i o n o p e r a t i o n . T h e a x i o m s ( a ) a n d ( b ) a re o b v i o u s l ys a t is f i e d , t h e i d e n t i t y i s t h e n u m b e r 1 , a n d e a c h r e a l n u m b e r t ( ~ 0 ) h a s i t sr e c i p r o c a l 1 / t as i t s i n v e r s e .E x a m p l e I I T h e a d d i t i v e g r o u p o f r ea l n u m b e r sT o d e m o n s t r a t e t h a t t h e g r o u p m u l t i p l i c a t i o n o p e r a t i o n n e e d n o t h a v e a n yc o n n e c t i o n w i t h o r d i n a r y m u l t i p l ic a t i o n , t a k e G t o b e t h e s e t of a ll r e a l n u m -b e r s w i t h o r d i n a r y a d d i t i o n a s t h e g r o u p m u l t i p li c a t i o n o p e r a t i o n . A g a i na x i o m s ( a ) a n d ( b ) a r e o b v i o u s l y s a t i s f i e d , b u t i n t h i s c a s e t h e i d e n t i t y i s 0( a s a + 0 - 0 + a = a ) a n d t h e i n v e rs e o f a r e a l n u m b e r a is i t s n e g a t i v e - a( a s a + ( - a ) = ( - a ) + a - - 0 ).E x a m p l e I I I A f i n i t e m a t r i x g r ou pM a n y o f t h e g r o u p s a p p e a r i n g i n p h y s ic a l p r o b l e m s c o n s is t o f m a t r i c e s w i t hm a t r i x m u l t i p l i c a t io n a s t h e g r o u p m u l t i p l i c a ti o n o p e r a t i o n . ( A b r i ef a c c o u n to f t h e t e r m i n o l o g y a n d p r o p e r t i e s o f m a t r i c e s is g i v en in A p p e n d i x A . ) A s a ne x a m p l e o f s u c h a g r o u p l e t G b e t h e s e t o f e i g h t m a t r i c e s[ I [ 1 0 ]0 M 2 = ,M 1 = 0 1 ' 0 - 1[ ] [ o _ 1 ]4 = - 1 0 M 5 = ,0 1 ' 1 0

[ ] [ 17 = 0 1 M s = 0 - 11 0 ' - 1 0 "- 1 0 1M 3 - 0 - 1 '[ ~6 = - 1 0 '

B y e x p l i c it c a l c u l a t i o n i t c a n b e v e ri fi ed t h a t t h e p r o d u c t o f a n y t w o m e m b e r so f G i s a l s o co n t a in e d i n G , s o t h a t ax io m (a ) is s a t i s f ied . A x io m (b ) i s


3/18

T H E B A S I C F R A M E W O R K 3a u t o m a t i c a l l y t r u e f or m a t r i x m u l t i p li c a t i o n , M 1 i s t h e i d e n t i t y o f a x i o m ( c)a s i t i s a u n i t m a t r i x , an d f i n a l l y ax io m (d ) is s a t i s f i ed a s

M ~ -1 = M 1 , M 2 1 = M 2 , M 3 1 = M 3 , M ~ -1 = M4,M 5 1 = M 6 , M 6 1 = M 5 , M ~ - i = M T , M s 1 = M s .

E x a m p l e I V T h e g ro u ps U ( N ) a n d S U ( N )U ( N ) f o r N > 1 is d e f i n e d t o b e t h e s e t o f a ll N N u n i t a r y m a t r i c e s uw i t h m a t r i x m u l t i p l i c a t i o n a s t h e g r o u p m u l t i p l ic a t i o n o p e r a t io n . S U ( N ) f o rN >_ 2 i s d e f i n e d t o b e t h e s u b s e t o f s u ch m a t r i c e s u fo r wh ich d e t u = 1 ,w i t h t h e s a m e g r o u p m u l t i p l i c a t i o n o p e r a t i o n . ( A s n o t e d in A p p e n d i x A , i fu is u n i t a r y t h e n d e t u = e x p ( i a ) , w h e r e c~ is s o m e r e a l n u m b e r . T h e " S " o fS U ( N ) i n d i c a t e s t h a t S U ( N ) is t h e " s p e ci a l" s u b s e t o f U ( N ) f or w h i c h t h i s ais zero . )

I t is e a s i l y e s t a b l i s h e d t h a t t h e s e s e t s d o f o r m g r o u p s . C o n s i d e r f ir s t t h es e t U ( N ) . A s ( u l u 2 ) t t t - 1 , if U l a n d u 2 a r e b o t hu 2 u 1 a n d ( u l u 2 ) = u 2 1 u l 1u n i t a r y t h e n s o is U lU 2 . A g a i n a x i o m ( b ) is a u t o m a t i c a l l y v a l i d f or m a t r i xm u l t i p l i c a t i o n a n d , a s t h e u n i t m a t r i x 1 N i s a m e m b e r o f U ( N ) , i t p r o v i d e st h e i d e n t i t y E o f a x i o m ( c ). F i n a l l y , a x i o m ( d ) is s a t is f ie d , a s if u i s a m e m b e ro f U ( N ) t h e n s o is u - 1.

F o r S U ( N ) t h e s a m e c o n s i d e r a t i o n s a p p ly , b u t i n a d d i t i o n if u l a n d u 2b o t h h a v e d e t e r m i n a n t 1, E q u a t i o n ( A . 4 ) sh o w s t h a t t h e s a m e is t r u e o f u l u 2 .M o r e o v e r , 1 N is a m e m b e r o f S U ( N ) , s o i t i s i ts i d e n t it y , a n d u - 1 is a m e m b e ro f S U ( N ) i f t h a t is t h e c a s e f or u .

T h e s e t of g r o u p s S U ( N ) is p a r t i c u l a r l y i m p o r t a n t i n t h e o r e t i c a l p h y s ic s .S U ( 2 ) i s i n t i m a t e l y r e l a t e d t o a n g u l a r m o m e n t u m a n d i s o to p ic s p in , a s w i llb e s h o w n i n C h a p t e r s 1 0 a n d 1 3 , w h i l e S U ( 3 ) i s n o w f a m o u s f o r i t s r o l e i n t h ec l a s s i fi c a t i o n o f e l e m e n t a r y p a r t i c l e s , w h i c h w i l l a l so b e s t u d i e d i n C h a p t e r13.E x a m p l e V T h e g ro u ps O ( N ) a n d S O ( N )T h e s e t o f a l l N N rea l o r t h o g o n a l m a t r i ce s R ( fo r N >_ 2 ) i s d e n o t eda l m o s t u n i v e r s a l l y b y O ( N ) , a l t h o u g h O ( N , I R) w o u l d h a v e b e e n p re f e r a b l e a si t i n d i c a t e s t h a t o n l y r e a l m a t r i c e s a r e i n cl u d e d . T h e s u b s e t o f s u c h m a t r i c e sR w i t h d e t R - 1 is d e n o t e d b y S O ( N ) . A s w i l l b e d e s c r i b e d in S e c t i o n 2 ,O ( 3 ) a n d S O ( 3 ) a r e i n t i m a t e l y r e l a t e d t o r o t a t i o n s i n a r e a l t h r e e - d i m e n s i o n a lE u c l i d e a n s p a c e , a n d s o o c c u r t i m e a n d t i m e a g a i n i n p h y s ic a l a p p li c a t io n s .

O ( N ) a n d S O ( N ) a r e b o t h g r o u p s w i t h m a t r i x m u l t ip l ic a t io n a s t h e g ro u pm u l t i p l i c a t i o n o p e r a t i o n , a s t h e y c a n b e r e g a r d e d a s b e i n g th e s u b s e t s o f U ( N )a n d S U ( N ) r e s p e c t i v e l y t h a t c o n s i st o n l y o f r e a l m a t r i c e s . ( A ll t h a t h a s t ob e o b s e rv e d t o s u p p l e m e n t t h e a r g u m e n t s g iv e n i n E x a m p l e I V is t h a t t h ep r o d u c t o f a n y t w o r e a l m a t r i c e s is r e a l, t h a t 1 N i s r e al , a n d t h a t t h e i n v e rs eo f a r e a l m a t r i x is a ls o r e a l .)

I f T1 T2 = T2T 1 fo r every p a i r o f e l e m e n t s T 1 a n d T 2 o f a g r o u p G ( t h a t i s, ifa l l T1 and T2 o f ~ c o m m u t e ) , t h e n G i s s a i d t o b e " A b e l i a n " . I t w i ll t r a n s p i r e


4/18

4 GROUP THE ORY IN PHYSICS

M1M2M3MaM5M6M7Ms

M1 M2 M3 Ma M5 M6 M7 MsM1 M2 M3 Ma M5 M6 M7 MsM2 M1 Ma M3 Ms M7 M6 M5M3 Ma M1 M2 M6 M5 Ms M7M4 M3 M2 M~ M7 M8 M5 M6M5 M7 M6 Ms M3 M~ Ma M2M6 Ms M5 M7 M1 M3 M2 MaM7 M5 Ms M6 M2 Ma M~ M3Ms M6 M7 M5 Ma M2 M3 Ma

Table 1.1: Multiplication table for the group of Example III.

that such groups have relatively straightforward properties. However, manyof the groups having physical applications are non-Abelian. Of the casesconsidered above the only Abelian groups are those of Examples I and IIand the groups V(1) and SO(2) of Examples IV and V. (One of the non-commuting pairs of products of Example III which makes that group non-Abelian is MsM7 = M4, MTM5 = M2.)The "order" of G is defined to be the number of elements in G, which maybe finite, countably infinite, or even non-countably infinite. A group withfinite order is called a "finite group". The vast majority of groups that arisein physical situations are either finite groups or are "Lie groups", which are aspecial type of group of non-countably infinite order whose precise definitionwill be given in Chapter 3, Section 1. Example III is a finite group of order8, whereas Examples I, II, IV and V are all Lie groups.

For a finite group the product of every element with every other elementis conveniently displayed in a multiplication table, from which all informationon the structure of the group can subsequently be deduced. The multiplica-tion table of Example III is given in Table 1.1. (By convention the order ofelements in a product is such that the element in the left-hand column pre-cedes the element in the top row, so for example M5Ms = M2.) For groupsof infinite order the construction of a multiplication table is clearly completelyimpractical, but fortunately for a Lie group the structure of the group is verylargely determined by another finite set of relations, namely the commutationrelations between the basis elements of the corresponding real Lie algebra, aswill be explained in detail in Chapter 8.

2 G r o u p s o f c o o r d i n a t e t r a n s f o r m a t i o n sTo proceed beyond an intuitive picture of the effect of symmetry operations,it is necessary to specify the operations in a precise algebraic form so thatthe results of successive operations can be easily deduced. Attention will beconfined here to transformations in a real three-dimensional Euclidean spaceIR3, as most applications in atomic, molecular and solid state physics involveonly transformations of this type.


5/18

T H E B A S IC F R A M E W O R K 5

Z

f

J 2 "J

y

F i g u r e 1 .1 : E f fe c t o f a r o t a t i o n t h r o u g h a n a n g l e 0 i n t h e r i g h t - h a n d s c r e ws e n s e a b o u t O x.( a ) R o t a t i o n sL e t O x, O y , O z b e t h r e e m u t u a l l y o r t h o g o n a l C a r t e s i a n a x e s a n d l e t O x ~,O y ~, O z' b e a n o t h e r s e t o f m u t u a l l y o r t h o g o n a l C a r t e s i a n a x e s w i t h t h e s a m eo r i g in O t h a t i s o b t a i n e d f ro m t h e f i rs t s e t b y a r o t a t i o n T ab o u t a sp ec i f iedax i s t h r o u g h O . L e t ( x , y , z ) an d (x ' , y ' , z ' ) b e t h e co o rd i n a t e s o f a f ix ed p o i n tP i n t h e s p a c e w i t h r e s p e c t t o t h e s e t w o s e t s o f a x e s. T h e n t h e r e e x i s ts a r e a lo r t h o g o n a l 3 x 3 m a t r i x R ( T ) w h i c h d e p e n d s o n t h e r o t a t i o n T , b u t w h i c h isi n d e p e n d e n t o f t h e p o s i t i o n o f P , s u c h t h a t

r ' = R ( T ) r , ( 1 . 2 )w h e r e i x ] i x ]/ = y1 a n d r - y .

Z ! Z

( H e r e a f t e r p o s i t i o n v e c t o r s w i ll a l w a y s b e c o n s i d e r e d a s 3 1 c o l u m n m a t r i c e si n m a t r i x e x p r e s s i o n s u n l e s s o t h e r w i s e i n d i c a t e d , a l t h o u g h f o r t y p o g r a p h i c a lr e a s o n s t h e y w i ll o f t e n b e d i s p l a y e d i n t h e t e x t a s 1 x 3 r o w m a t r i c e s . ) F o re x a m p l e , i f T is a r o t a t i o n t h r o u g h a n a n g l e 0 i n t h e r i g h t - h a n d s c r ew s e ns ea b o u t t h e a x i s O x, t h en , a s i n d i ca t ed i n F i g u re s 1 .1 an d 1 .2 ,

X ! " - X ~

y t _ y c o s O + z s i n O ,zt = - y s i n O + z c o s O ,


6/18

6 GROUP THEO RY IN PHYSICS

9 Ys " I X

I y %J

,,

OrY

F i g u r e 1 . 2 : T h e p l a n e c o n t a i n i n g t h e a x e s Oy, Oz, Oy ~a n d Oz ~c o r r e s p o n d i n gt o t h e r o t a t i o n o f F i g u r e 1 .1 .

s o t h a t

[ 1 o o 1( T ) = 0 c o s 0 s i n 0 . ( 1 . 3 )0 - s i n 0 c o s0T h e m a t r i x R ( T ) o b ey s t h e o r th o g o n a li ty c o n d it io n R ( T ) = R ( T ) - ~ b e -

c a u s e r o t a t i o n s l ea v e i n v a r i a n t t h e l e n g t h o f e v e r y p o s it i o n v e c t o r a n d t h ea n g l e b e t w e e n e v e r y p a i r o f p o s i t i o n v e c t o r s , t h a t is , t h e y l e a v e i n v a r i a n tt h e s c a l a r p r o d u c t r l . r2 o f a n y tw o p o s i ti o n v e c to r s . ( I n d e e d t h e n a m e " o r-t h o g o n a l " s t e m s f r o m th e i n v o l v e m e n t o f s u c h m a t r i c e s in t h e t r a n s f o r m a -t io n s b e i n g c o n s i d e r e d h e r e b e t w e e n s e t s o f o r t h o g o n a l a x e s .) T h e p r o o f t h a tR ( T ) is o r t h o g o n a l d e p e n d s o n t h e f a ct t h a t r l . r 2 c a n b e e x p r e s s e d in m a -t r i x f o r m as r l r 2 . T h e n , i f r~ = R ( T ) r l a n d r~ = R ( T ) r 2 , i t f ol lo w s t h a tr ~ . r 2 = r - ~ r ~ = Y ~ R ( T ) R ( T ) r 2 , w h i c h is e q u a l to Y l r2 f o r a ll r l a n d r 2 if a n do n ly if R ( T ) R ( T ) = 1 .

A s n o t e d i n A p p e n d i x A , t h e o r th o g o n a l i ty c o n d i ti o n i m p li es t h a t d e t R ( T )c a n t a k e o n l y t h e v a lu e s + 1 o r - 1 . I f d e t R ( T ) = + 1 t h e r o t a t i o n is s a i d t ob e " p r o p e r " ; o t h e r w i s e i t is s a id t o b e " i m p r o p e r " . T h e o n l y r o t a t i o n s w h i c hc a n b e a p p l i e d t o a r ig i d b o d y a r e p r o p e r r o t a t i o n s . T h e t r a n s f o r m a t i o n o fE q u a t i o n ( 1 .3 ) gi v es a n e x a m p l e .

T h e s i m p l e s t e x a m p l e o f a n i m p r o p e r r o t a t i o n is t h e s p a t i a l i n v e r s i o n o p -e r a t i o n I f o r w h i c h r ' = - r , s o t h a t[ 1 0 0 ]n ( I ) = 0 - ~ 0 .

0 0 - 1A n o t h e r i m p o r t a n t e x a m p l e is t h e o p e r a t i o n o f r e f le c t io n in a p l a n e . F o r

i n s t a n c e , f o r r e f l e c t i o n i n t h e p l a n e Oyz, f or w h i c h x ' = - x , y ' = y , z ' = z ,


7/18

T H E B A S I C F R A M E W O R K 7t h e t r a n s f o r m a t i o n m a t r i x i s 0 0 ]

0 1 00 0 1T h e " p r o d u c t " T 1 T 2 o f t w o r o t a t i o n s T1 a n d T 2 m a y b e def ined t o b e t h e

r o t a t i o n w h o s e t r a n s f o r m a t i o n m a t r i x i s gi ve n b yR( T~ T2) = R( T 1 ) R( T 2) . (1.4)

( T h e v a l i d i ty o f t h i s d e f in i ti o n i s a s s u r e d b y t h e f a c t t h a t t h e p r o d u c t o fa n y t w o re a l o r t h o g o n a l m a t r i c e s i s i ts e l f r e a l a n d o r t h o g o n a l . ) I n g e n e r a lR ( T ~ ) R ( T 2 ) r R ( T 2 ) R ( T ~ ) , in w h i c h c a se T ~ T 2 =/= T2 T1 . I f r ' = R ( T 2 ) ra n d r " = R ( T ~ ) r ' , t h e n E q u a t i o n ( 1 .4 ) i m p li es t h a t r " = R ( T ~ T 2 ) r , s o t h ei n t e r p r e t a t i o n o f E q u a t i o n ( 1.4 ) is t h a t o p e r a t i o n T2 t a k e s p l a c e before 7"1.T h i s i s a n e x a m p l e o f t h e g e n e r a l co n v e n t i o n ( w h i ch w i ll b e a p p l i e d t h r o u g h o u tt h i s b o o k ) t h a t i n a n y p r o d u c t o f o p e r a t o r s t h e o p e r a t o r o n t h e r i g h t a c t s f ir st .

W i t h t h i s d e fi n it io n ( E q u a t i o n ( 1 .4 ) ) e v e ry i m p r o p e r r o t a t i o n c a n b e c on -s i d e re d t o b e th e p r o d u c t o f t h e s p a t i a l i n v e r si o n o p e r a t o r I w i t h a p r o p e rr o t a t ion . For example , f o r t he r e f l ec t ion in the O y z p l a n e

[ 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 - 1a n d , a s t h e s e c o n d m a t r i x o n t h e r i g h t - h a n d s i d e i s t h e t r a n s f o r m a t i o n o fE q u a t i o n (1 .3 ) w i t h 0 = ~ , i t c o r r e s p o n d s t o a r o t a t i o n t h r o u g h ~ a b o u t O x .

I f a s e t o f m a t r i c e s R ( T ) f o rm s a g r o u p , t h e n t h e c o r r e s p o n d i n g s e t ofr o t a t ions T a l so f o r ms a g r oup in which Equa t ion ( 1 .4 ) de f ine s the g r oup mul -t i p l ic a t i o n o p e r a t o r a n d f o r w h i c h t h e i n v e rs e T - 1 o f T is gi v e n b y R ( T - 1 ) =R ( T ) - 1 . A s t h e s e tw o g r o u p s h a v e t h e s a m e s t r u c t u r e , t h e y a r e sa i d t o b e" is o m o r p h ic " ( a c o n c e p t w h i c h w ill b e e x a m i n e d i n m o r e d e t a i l in C h a p t e r 2 ,Sec t ion 6) .E x a m p l e I The group o f a l l ro ta t ionsT h e s e t o f a ll r o t a t i o n s , b o t h p r o p e r a n d i m p r o p e r , f o r m s a L ie g r o u p t h a t isi s o m o r p h i c t o th e g r o u p 0 ( 3 ) t h a t w a s i n t r o d u c e d i n E x a m p l e V o f S e c t io n1.

E x a m p l e I I The group o f a l l proper ro ta t ionsT h e s e t of a ll p r o p e r r o t a t i o n s f o rm s a L ie g r o u p t h a t is i s o m o r p h i c t o t h eg r o u p S O ( 3 ) .E x a m p l e I I I The crys ta l lograph ic po in t g roup D4A gr oup o f r o t a t io ns th a t l e ave inva r i an t a c r ys t a l l a t t i c e i s c a l l ed a " c r ys-t a l l o g r a p h i c p o i n t g r o u p " , t h e " p o i n t " i n d i c a t i n g t h a t o n e p o i n t , t h e o r i g i nO , i s l ef t u n m o v e d b y t h e o p e r a t i o n s o f t h e g r o u p . T h e r e a r e o n l y 3 2 s u c h


8/18

8 G R O U P T H E O R Y I N P H Y S I CS

%

"%%

%%

\

\

' l / "T xx 0

/I , , /i r /

,,rp

e,

J/f//

P/J

J o/

F i g u r e 1 . 3 : T h e r o t a t i o n a x e s Ox, Oz, Oc a n d Od o f t h e c r y s t a l l o g r a p h i c p o i n tg r o u p D 4 .

g r o u p s , a ll of w h i c h a r e f in i te . A c o m p l e t e d e s c r i p t i o n i s g i v e n i n A p p e n d i x C .T h e o n l y p o s s ib l e a n g l e s o f r o t a t i o n a r e 27r/n, w here n - 2 , 3 , 4 , o r 6 . (T hisr e s t r i c t i o n o n t h e v a l u e o f n i s a c o n s e q u e n c e o f t h e t r a n s l a t i o n a l s y m m e t r yo f a pe r fec t c ry s t a l ( cf . C ha p t e r 7 , S ec t i on 6 ) . F o r a "qu as i c ry s t a l " , w h ichh a s n o s u c h t r a n s l a t i o n a l s y m m e t r y , t h i s r e s t r i c t i o n n o l o n g e r a p p l i e s , a n d s oi t is pos s ib l e t o hav e o th e r va lues o f n a s we ll , inc l ud i ng , i n pa r t i cu l a r , t hev a l u e n - 5 .) I t is c o n v e n i e n t t o d e n o t e a p r o p e r r o t a t i o n t h r o u g h 27r/n a b o u ta n a x i s O j b y Cnj. T h e i d e n t i ty tr a n s f o r m a t i o n m a y b e d e n o t e d b y E , s ot h a t R ( E ) - 1 , a n d i m p r o p e r r o t a ti o n s ca n b e w r i t t e n i n t h e f o rm ICn j . A sa n e x a m p l e , c o n s i d e r t h e c r y s t a l l o g r a p h i c p o i n t g r o u p D 4 , t h e n o t a t i o n b e i n gth a t o f S chS nf li e ss (1923) . D4 con s i s ts o f t he e igh t ro t a t i o ns :

E : t h e id e n t i t y ;C 2x, C 2y , C 2~" p rop er ro t a t i on s t h r ou gh 7r ab ou t Ox, Oy, Oz r e s p e c t i v e l y ;

C-1.4y, ay p r o p e r r o t a t i o n s t h r o u g h 7 r / 2 a b o u t O y i n t h e r i g h t - h a n d a n dl e f t -hand s c rew senses r e spec t i ve ly ;

C2c, C2d" p r o p e r r o t a t i o n s t h r o u g h l r a b o u t Oc a n d Od r e spec t i ve ly .H e r e Ox, Oy, Oz a r e m u t u a l l y o r t h o g o n a l C a r t e s i a n a x e s , a n d Oc, Od a rem u t u a l l y o r t h o g o n a l a x e s i n t h e p l a n e O xz w i t h Oc m a k i n g a n a n g l e o f 7r /4w i t h b o t h O x a n d Oz, a s i n d i c a t e d i n F i g u r e 1 .3 . T h e t r a n s f o r m a t i o n m a t r i c e sa re [ 1 0 0 ] [ 1 0 0 ]R ( E ) = 0 1 0 , R ( C 2 ~ ) = 0 - 1 0 ,

0 0 1 0 0 - 1[ _ 1 o o ] [_ 1 oR(C2y) = 0 1 0 , R(C2z) = 0 -I

0 0 -1 0 00 ]0 ,1


9/18

T H E B A S I C F R A M E W O R K 9

R ( C 4 y )

a(c~) [ 1 [ ]0 0 - 1 0 0 10 1 0 R (C 4~ ) = 0 1 01 0 0 - 1 0 0[0 0 1 ] [0 0 1 ]0 - 1 0 , R ( C 2 d ) = 0 - 1 01 0 0 1 0 0

EC2~C~yC2zC4y%1C2~C2d

E C2~ C2y C2z C4~ C ~ ~ C2c C2dE C2~ C2y C2z C4y C{~ C2c C2dC2~ E C2z C2y C2d C2~ C {1 C4yC2y C2z E C2x C4y1 C4y C2d C2cC2z C2y C2x g C2c C2d C4y C4y1C4y C2c C41 C2d C2y g C2z C2xC4~ C2d C4y C2c E C2y C2x C2zC2c C4~ C2d C ~ 1 C2~ C2z E C2yC2d C4y1 C2c C4y C2z C2x C2y E

Tab le 1 .2 : Mul t ip l i ca t ion t ab le fo r the c rys ta l log raph ic po in t g roup D4 .The mul t ip l i ca t ion t ab le i s g iven in Tab le 1 .2 . Th i s example w i l l be used

to i l lu s t r a te a num ber o f concep t s in Cha p te r s 2 , 4 , 5 and 6 .( b ) T r a n s l a t i o n sS u p p o s e n o w t h a t O x , O y , O z is a s e t o f m u t u a l l y o r t h o g o n a l C a r t e s i a n a x e sa n d O~x I , 01 y I, 0 l z t i s ano the r s e t , ob ta ined by f i r s t ro ta t ing the o r ig ina l s e ta b o u t s o m e a x is t h r o u g h 0 b y a r o t a t i o n w h o s e t r a n s f o r m a t i o n m a t r i x i sR ( T ) , a n d t h e n t r a n s l a t i n g 0 t o O / a lo n g a v e c to r - t ( T ) w i t h o u t f u r t h e rro ta t ion . ( In IR 3 any two se t s o f Ca r te s ia n axes can be r e la ted in th i s way .)T h e n E q u a t i o n ( 1 . 2 ) g e n e r a l i z e s t o

r ' = R ( T ) r + t ( T ) . (1 .5 )I t is u s e fu l t o r e g a r d t h e r o t a t i o n a n d t r a n s l a t i o n a s b e i n g tw o p a r t s o f a s inglec o o r d i n a t e t r a n s f o r m a t i o n T , a n d s o it is c o n v e n ie n t t o r e w r i te E q u a t i o n ( 1. 5)as

r ' = { R ( T ) I t( T ) } r,t h e r e b y d e fi n in g t h e c o m p o s i te o p e r a t o r { R ( T ) I t ( T ) } . I n d ee d , in t h e n o n -s y m m o r p h i c s p a c e g r o u p s ( s e e C h a p t e r 7 , S e c t i o n 6 ) , t h e r e e x i s t s y m m e t r yo p e r a t i o n s i n w h i c h t h e c o m b i n e d r o t a t i o n a n d t r a n s l a t i o n l e a v e t h e c r y s t a ll a t t i c e i n v a r i a n t w i t h o u t t h i s b e i n g t r u e f o r t h e r o t a t i o n a l a n d t r a n s l a t i o n a lp a r t s s e p a r a t e l y .

T h e g e n e r a l i z a t i o n o f E q u a t i o n ( 1. 4) c a n b e d e d u c e d b y c o n s i d e ri n g t h et w o s u c c es s iv e t r a n s f o r m a t i o n s r ' = { R ( T 2 ) [ t ( T 2 ) } r =- R ( T 2 ) r + t (T 2 ) a n dr ' = { R ( T 1 ) ] t ( T 1 ) } f f - R ( T 1 ) r ' + t (T 1 ) , w h ic h g iv e

r " - - R ( T 1 ) R ( T 2 ) r + [ R ( T 1 ) t ( T2 ) + t ( T 1 )] . (1.6)


10/18

10 G R O U P T H E O R Y I N P H Y S I C ST h u s t h e n a t u r a l c h oi ce o f t h e d e f in i ti o n o f t h e " p r o d u c t " T 1 T 2 o f t w o g e n e r a ls y m m e t r y o p e r a t i o n s T 1 a n d T 2 i s

{ R ( T ~ ) I t ( T ~ ) } = { R ( T ~ ) R ( T 2 ) I R ( T 1 ) t ( T 2 ) + t ( T 1 ) } . ( 1 . 7 )Th is p ro du c t a lways s a t is f ie s the g roup as soc ia t ive l aw of E qu a t i on (1 .1 ) .

As Equa t ion (1 .5 ) can be inve r ted to g iver = R ( T ) - l r ' - R ( T ) - l t ( T ) ,

t h e i n v e r s e o f { R ( T ) I t ( T ) } m a y b e d e f i n e d b y{ R ( T ) I t ( T ) } - 1 = { R ( T ) - I I - R ( T ) - I t ( T ) } . (1.s)I t i s eas i ly ver i f ied that

{ R ( T 1 T 2 )I t( T 1 T 2 )} - 1 = { R ( T 2 ) I t ( T 2 ) } - I { R (T 1 ) It (T ~ )} - 1 ,the o rde r o f f ac to r s be ing r eve r sed on the r igh t -hand s ide .

I t i s s o m e t i m e s c o n v e n i e n t t o r e f e r t o t r a n s f o r m a t i o n s f o r w h i c h t ( T ) = 0a s " p u r e r o t a t i o n s " a n d t h o s e fo r w h i c h R ( T ) = 1 a s " p u re tr a n s l a t i o n s " .

3 T h e g r o u p o f t h e S c h r S d i n g e r e q u a t i o n(a ) T h e H a m i l t o n i a n o p e r a t o rT h e H a m i l t o n i a n o p e r a t o r H o f a p h y s i c a l s y s t e m p l a y s t w o m a j o r r o l es inqu an tum m echan ics (Schi ff 1968). F i r s t ly , i t s e igenva lues c , a s g iven by thet i m e - i n d e p e n d e n t S c h r S d i n g e r e q u a t i o n

H e = e ra re the on ly a l lowed va lues of the ene rgy o f the sys tem . Second ly , the t i m e de -v e l o p m e n t of t h e s y s t e m is d e t e r m i n e d b y a w a v e f u n c t io n r w h i c h s a ti sf ie st h e t i m e - d e p e n d e n t S c h r S d i n g e r e q u a t i o n

H e = i h o rN o t s u r p r i s i n g l y , a c o n s i d e r a b l e a m o u n t c a n b e l e a r n t a b o u t t h e s y s t e m b ys i m p l y e x a m i n i n g t h e s e t o f t r a n s f o r m a t i o n s w h i c h le av e t h e H a m i l t o n i a ninvar ian t . I nde ed th e m a in fun c t ion o f g ro up theo ry , a s i t i s app l i e d in phys ica lp r o b l e m s , i s t o s y s t e m a t i c a l l y e x t r a c t a s m u c h i n f o r m a t i o n a s p o s s i b l e f r o mt h i s s e t o f t r a n s f o r m a t i o n s .

In o rde r to p resen t the e s sen t i a l f ea tu res a s c lea r ly a s pos s ib le , i t w i l lbe a s sumed in the f i r s t in s tance tha t the p rob lem invo lves so lv ing a " s ing le -p a r ti c l e" S c h r S d i n g e r e q u a t i o n . T h a t is , i t w i ll b e s u p p o s e d t h a t e i t h e r th esys te m con ta ins on ly one pa r t ic l e , o r , i f the re i s mo re than one pa r t i c l e in-v o l v ed , t h e n t h e y d o n o t i n t e r a c t o r t h e i r i n t e r - p a r t i c l e i n t e r a c t i o n s h a v e b e e n


11/18

T H E B A S I C F R A M E W O R K 11t r e a t e d i n a H a r t r e e - F o c k o r s im i l a r a p p r o x i m a t i o n i n s u c h a w a y t h a t e a chp a r t i c l e e x p e r i e n c e s o n l y t h e a v e r a g e fi el d o f a ll o f t h e o t h e r s . M o r e o v e r , i tw i ll b e a s s u m e d t h a t H c o n t a i n s n o s p i n - d e p e n d e n t t e r m s , s o t h a t t h e s ig -n i f ic a n t p a r t o f e v e r y w a v e f u n c t i o n is a s c a la r f u n c t i o n . F o r e x a m p l e , f o r a ne l e c t r o n i n t h i s s i t u a t i o n , e a c h w a v e f u n c t i o n c a n b e t a k e n t o b e t h e p r o d u c to f a n " o r b i t a l " f u n c t i o n , w h i c h i s a s ca l a r , w i t h o n e o f t w o p o s s i b l e s p i n f u n c -t i o n s , s o t h a t t h e o n l y ef fe c t o f t h e e l e c t r o n ' s s p i n is t o d o u b l e t h e " o r b i t a l "d e g e n e r a c y o f e a c h e n e r g y e ig e n v a l u e. ( A d e v e l o p m e n t o f a t h e o r y o f s p i n o r sa l o n g s i m i l a r l i n e s t h a t e n a b l e s s p i n - d e p e n d e n t H a m i l t o n i a n s t o b e s t u d i e d i sg i v en , fo r e x a m p l e , i n C h a p t e r 6 , S e c t i o n 4 , o f C o r n w e l l ( 1 9 8 4 ) . )

W i t h t h e s e a s s u m p t i o n s a t y p i c a l H a m i l t o n i a n o p e r a t o r f o r a p a r t i c l e o fm a s s p h a s t h e f o r m

h 2 02 0 2 c 9 2H ( r ) = - ~ - - ( _~-~o + ~ + ~ ) + V ( r ) , ( 1 .1 0 )O z~ # a x " u y -w h e r e V ( r ) i s t h e p o t e n t i a l f ie ld e x p e r i e n c e d b y th e p a r t i c l e . F o r e x a m p l e ,f or t h e e l e c t r o n o f a h y d r o g e n a t o m w h o s e n u c l eu s i s l o c a t e d a t O ,

h 2 02 0 2 02H ( r ) = - ~ p (0 -~ x + ~ + ~ z 2 ) - e 2 / { x 2 + y 2 + z 2 } 1 / 2 . (1.11)I n E q u a t i o n s ( 1.1 0 ) a n d ( 1. 11 ) th e H a m i l t o n i a n is w r i t t e n a s H ( r ) t o e m p h a -s i z e i t s d e p e n d e n c e o n t h e p a r t i c u l a r c o o r d i n a t e s y s t e m O x y z .( b ) T h e i n v a r i a n c e o f t h e H a m i l t o n i a n o p e r a t o rL e t H ( { R ( T ) I t ( T ) } r ) b e th e o p e r a t o r t h a t is o b t a i n e d f r o m U ( r ) b y s u b s ti -t u t i n g t h e c o m p o n e n t s o f r ' - { R ( T ) [ t ( T ) } r i n p l ac e o f t h e c o r re s p o n d i n gc o m p o n e n t s o f r . F o r e x a m p l e , i f H ( r ) i s g i v en b y E q u a t i o n ( 1 .1 1 ), t h e n

h 2 02 02 02H ( { R ( T ) ] t (T ) } r ) = - ~ ( ~ z , 2 + ~ y ,2 + ~ z ,2 ) - s (1 .12)

/ - / ( { R ( r ) l t ( r ) } r ) c a n t h e n b e r e w r i t te n s o t h a t i t d e p e n d s e x p l i c i t l y o n r .F o r e x a m p l e , i n E q u a t i o n ( 1 .1 2 ) , i f T i s a p u r e t r a n s l a t i o n x p = x + t l , y~ =y + t2 , z ' - z + ta , t h e n

g2 02 02 -~02 )H ( { R ( T ) I t ( T ) } r ) - -2-~(~x2__ + ~5y2 + O z

- - e 2 / { ( x q - t l ) 2 -t- (y + t2) 2 + (z q- t3)2} 1/2.s o t h a t

H ( { R ( T ) I t ( T ) } r ) :/: H ( r ) ,w h e r e a s i f T is a p u r e r o t a t i o n a b o u t O , t h e n a s h o r t a l g e b r a ic c a l c u l a ti o ngives

h 2 02 02 0 2S ( { R ( T ) l t ( T ) } r ) = - ~ ( - 5 ~ x 2 + ~ + - - ~ ) - e : / { x : + y : + z 2} 1 /:O z


12/18

12 G R O U P T H E O R Y I N P H Y S IC Sand hence in th i s case

H ( { R ( T ) I t ( T ) } r ) = H ( r ) .A c o o r d i n a t e t r a n s f o r m a t i o n T f o r w h i c h

H ( { R ( T ) I t ( T ) } r ) = H ( r ) (1.13)is s a id t o le a v e t h e H a m i l t o n i a n " i n v a r ia n t " . F o r t h e h y d r o g e n a t o m t h e a b o v ea n a l y s i s m e r e l y e x p l i c i t l y d e m o n s t r a t e s t h e i n t u i t i v e l y o b v i o u s f a c t t h a t t h es y s t e m i s i n v a r i a n t u n d e r p u r e r o t a t i o n s b u t n o t u n d e r p u r e t r a n s l a t i o n s .

T h e f o l l o w i n g k e y t h e o r e m s h o w s h o w a n d w h y g r o u p t h e o r y p l a y s s u c h as i g n i f i c a n t p a r t i n q u a n t u m m e c h a n i c s .T h e o r e m I T h e se t o f c o o r d i n a te t r a n s f o r m a t i o n s t h a t l ea v e t h e H a m i l to -n ian inv a r ian t fo rm a g roup . Th i s g rou p i s u sua l ly ca ll ed " the g rou p o f theSchr5d inger equa t ion" , bu t i s somet imes r e f e r r ed to a s " the inva r iance g roupo f t h e H a m i l t o n i a n o p e r a t o r " .P roo f I t has on ly to be ve r i f i ed tha t the fou r g roup ax ioms a re s a t i s f i ed .F i r s tl y , i f t h e H a m i l t o n i a n i s i n v a r i a n t u n d e r t w o s e p a r a t e c o o r d i n a t e t r a n s -fo rmat ions T1 and T2 , then i t i s inva r ian t under the i r p roduc t T1T2 . ( Invar i -a n ce u n d e r T1 i m p li es t h a t H ( r " ) = H ( r ' ) , w h e r e r " = { R ( T 1 ) I t ( T 1 ) } r ' , a n di n v a ri a n c e u n d e r T2 i m p li e s t h a t H ( r ' ) = H ( r ) , w h e r e r ' = { R ( T 2 ) I t ( T 2 ) } r ,s o t h a t H ( r " ) = H ( r ) , w h e re , b y E q u a t i o n (1 .7 ), r " = {R(T~T2)I t (T1T2)}r ) .Secondly , as noted in Sect ion 2(b) , the associa t ive law is va l id for a l l coor-d i n a t e t r a n s f o r m a t i o n s . T h i r d ly , t h e i d e n t i t y t r a n s f o r m a t i o n o b v i o u s l y l ea v esthe Hami l ton ian inva r ian t , and f ina l ly , a s Equa t ion (1 .13 ) can be r ewr i t t ena s H ( r ' ) = H ( { R ( T ) I t ( T ) } - l r ' ) , w h e r e r ' = { R ( T ) I t ( T ) } r , i f T l ea v e s t h eH a m i l t o n i a n i n v a r i a n t t h e n s o d o e s T - 1 .

F o r t h e c a se o f t h e h y d r o g e n a t o m , o r a n y o t h e r s p h e r i c a ll y s y m m e t r i csys te m in wh ich V ( r ) is a func t ion o f Irl a lone , the g rou p o f the SchrSd ingerequ a t ion i s the g roup o f a l l pu re ro ta t io ns in IR3.

( c ) T h e s c a l a r t r a n s f o r m a t i o n o p e r a t o r s P ( T )A "sca la r f i e ld" i s de f ined to be a quan t i ty tha t t akes a va lue a t each po in tin the space ] a 3 ( in genera l tak ing d i f ferent va lues a t d i f ferent poin ts ) , thev a l ue a t a p o i n t b e i n g i n d e p e n d e n t o f t h e c h o ic e o f c o o r d i n a t e s y s t e m t h a tis u sed to des igna te the po in t . One o f the s imp les t examp les to v i sua l ize isthe den s i ty o f pa r t i c le s . The concep t i s r e l evan t to the p resen t con s ide ra t ionbecause the "o rb i t a l " pa r t o f an e lec t ron ' s w ave func t ion i s a s ca la r f ie ld .

Sup pose th a t the s ca la r fi eld i s spec if ied by a func t ion ~p( r ) wh en thecoord ina tes o f po in t s o f IR3 a re de f ined by a coo rd ina te sy s tem O x, O y , O z ,and th a t t he s am e sca la r f ie ld i s spec if ied by a func t ion r p) wh en ano the rc o o r d i n a t e s y s t e m O tx ~, O~y~, O~z~ s u sed in s tead . I f r and r ~ a r e th e pos i t ion


13/18

T H E B A S IC F R A M E W O R K 13v e c t o r s o f t h e s a m e p o i n t r e f e r r e d t o th e t w o c o o r d i n a t e s y s t e m s , t h e n t h ed e f i n i t i o n o f t h e s c a l a r f i el d imp l i e s t h a t

r : % b ( r) . ( 1 . 1 4 )N o w s u p p o s e t h a t O'x ' , O 'y ' , O ' z ' a r e o b t a i n e d f r o m O x, O y , O z b y a co o rd i -n a t e t r a n sf o r m a t i o n T , so t h a t r ' - { R ( T ) [ t ( T ) } r .

T h e n E q u a t i o n ( 1 . 1 4 ) c a n b e w r i t t e n a sr = r ( 1 .1 5 )

w h i ch p ro v i d es a co n c re t e p re s c r i p t i o n fo r d e t e rm i n i n g t h e fu n c t i o n %9' f ro mt h e f u n c t i o n r n a m e l y t h a t r is t h e f u n c t io n o b t a i n e d b y r e p l a c i n g e a c hc o m p o n e n t o f r in ~ ( r ) b y t h e c o r re s po n d in g c o m p o n e n t o f { R ( T ) I t ( T ) } - l r '.F o r e x a m p l e , if r = x2y 3 a n d T is t h e p u r e r o t a t i o n o f E q u a t i o n ( 1. 3) , a s

{ R ( T ) ] t ( T ) } - ~ r ' = R ( T ) - l r ' = R : ( T ) r '= (x ' , y 'cosO - z ' s i n 0 , y ' s i n 0 + z ' c o s 0 ) ,

t h e nr (r') = x'2(y ' cos0- z' sin 0) 3.

It is very convenient in the following analysis to replace the argument r'of ~' by r (without changing the functional form of ~'). Thus in the aboveexample

r ( r ) = x ( y cos 0 - z sin 0) 3,a n d E q u a t i o n ( 1 . 1 5 ) c a n b e r e w r i t t e n a s

r = ~ b ( { R ( T ) l t ( T ) } - ~ r ) . (1 .1 .6 )A s r is u n i q u e l y d e t e r m i n e d f r om r f or t h e c o o r d i n a t e t r a n s f o r -

m a t i o n T , ~ ' c a n b e r e g a r d e d a s b e i n g o b t a i n e d f r o m ~ b y t h e a c t i o n o f a noperator P(T) , w h i ch i s t h e re fo re defined b y ~b'= P(T)~b, or , equ iva len t ly ,f r o m E q u a t i o n ( 1 . 1 6 ) b y( P ( T ) r = r

T h e t y p o g r a p h y c a n b e s i m p l i f i e d w i t h o u t c a u s i n g c o n f u s i o n b y r e m o v i n g o n eo f t h e s e t s o f b r ack e t s o n t h e l e f t -h an d s id e , g i v in g

P(T)~ ; ( r ) = r ( 1 .1 7 )T h e s e s c a l a r t r a n s f o r m a t i o n o p e r a t o r s p e r f o r m a p a r t i c u l a r ly i m p o r t a n t r ol ei n t h e a p p l i c a ti o n o f g r o u p t h e o r y t o q u a n t u m m e c h a n ic s . T h e i r p r o p e r t i e sw i l l n o w b e e s t ab l i s h ed .

C l e a r l y P(T1) = P( T2 ) on ly i f T1 = T2 . (Here P ( T I ) = P ( T 2 ) m e a n s t h a tP (T 1)% b (r ) = P (T 2 ) r fo r every f u n c t io n r M o r e o v e r , e a c h o p e r a t o rP ( T ) is l inear, t h a t i s

P ( T ) { a r + b e ( r ) } = a P ( T ) r + b P ( T ) r (1 .18)


14/18

14 GRO UP TH EO RY IN PHYSICSf or a n y t w o f u n c t io n s r a n d r a n d a n y t w o c o m p l e x n u m b e r s a a n d b,a s c a n b e v e r i f i e d d i r e c t l y f r o m E q u a t i o n ( 1 . 1 7 ) ; ( s e e A p p e n d i x B , S e c t i o n 4 ) .T h e o t h e r m a j o r p r o p e r t i es o f t h e o p e r a t o r s P (T) a r e m o s t s u c c i n c t l y s t a t e di n t h e f o l l o w i n g f o u r t h e o r e m s .T h e o r e m I I E a c h o p e ra to r P ( T ) is a unitary o p e r a t o r i n t h e H i l b e r t s p a c eL 2 w i t h i n n e r p r o d u c t ( r r d e f in e d b y/?/?/o~ r (r)~ b(r ) dx dy dz, 19)r ~b) (1 .w h e r e t h e i n t e g r a l is o v e r th e w h o l e o f th e s p ac e IR 3 , t h a t is ,

( P ( T ) r P ( T ) r = ( r r (1 .20)f o r a n y t w o f u n c t i o n s r a n d r o f L 2 ; (s ee A p p e n d i x B , S e c t i o n s 3 a n d 4 ) .Proo f W i t h r '1 d e fi ne d b y r " - { R ( T ) ] t ( T ) } - l r , f ro m E q u a t i o n s (1 .1 7) a n d(1 .19) /?/??P ( T ) r P ( T ) r = r ( r ' l ) r ' ' ) dx dy dz.

o o o o ( x )

H o w e v e r , dx dy dz - J dx" dy" dz", w h e r e t h e J a c o b i a n J i s d e f i n e d b y[ OO x " O zlO u " Ox/Oz"]J = de t O y / O x " O y / O y " O y / O z "Oz/Oz" Oz/Oy" Oz/Oz"(1.21)

A s r = R ( T ) r " + t ( T ) , i t f ol lo w s t h a t Ox /Ox"= R(T)11 , Ox /Oy"= R ( T ) 1 2e t c . , s o t h a t J - d e t R ( T ) - = k l. I n c o n v e r t i n g t h e r i g h t - h a n d s i d e o f E q u a -t i o n ( 1 .2 1 ) t o a t r i p l e i n t e g r a l w i t h r e s p e c t t o x " , y " , z " , t h e r e a p p e a r s a n o d dn u m b e r o f i n t e r ch a n g e s o f u p p e r a n d l ow e r l im i t s fo r a n i m p r o p e r r o t a t i o n ,w h e r e a s f or a p r o p e r r o t a t i o n t h e r e is a n e v e n n u m b e r o f s u c h i n t e rc h a n g e s .( F o r e x a m p l e , f o r s p a t i a l in v e r s io n I , x " - - x , y " - - - y , z " - - z , s o t h e u p -p e r a n d l ow e r l i m i ts a r e i n t e r c h a n g e d t h r e e t i m e s , w h i l e fo r a r o t a t i o n t h r o u g h7r a bou t O z t h e l i m i t s a r e i n t e r c h a n g e d t w i c e . ) T h u s i n a l l c a s e s E q u a t i o n( 1 . 2 1 ) c a n b e w r i t t e n a s

( P ( T ) r 1 6 2 = / ? f ? / ? r 16 2 dy" d z",o o o o o o

f r o m w h i c h E q u a t i o n ( 1. 20 ) f o ll ow s im m e d i a t e ly .T h e o r e m I I I F o r a n y t w o c o o r d in a t e t r an s f o rm a t i o n s T1 a n d T2,

P(TIT2) = P(T1)P(T2). (1.22)Proof I t i s r e q u i r e d t o s h o w t h a t f o r any f u n c t i o n r P(TIT2)r =P(T1)P(T2)r w h e r e i n t h e r i g h t - h a n d si d e P ( T 2 ) a c ts f ir s t o n r a n d


15/18

T H E B A S IC F R A M E W O R K 15P ( T 1 ) ac t s on t he re su l t i ng expre ss ion . Le t r = P (T 2) r so t h a tr = r T h e nP ( T 1 ) r = r = ~ b ( { R ( T 2 ) I t ( T 2 ) } - I { R ( T ~ ) ] t ( T 1 ) } - l r ) ,

t he l a s t equa l i t y be ing a consequence of t he fac t t h a t r isby de f in it ion t he func t i on ob t a in ed f rom r by s imply rep l ac ing t he com po-n e n t s o f r b y t h e c o m p o n e n t s o f { R ( T 1 ) [ t ( T 1 ) } - l r . T h u s , o n u si n g E q u a t i o n(1.9),

P(T~ )P(T2 ) r - r {R(T1 T2 ) ] t(T~ T2 ) } - ~r) = P(T1T2 ) r

T h e o r e m I V T h e s e t o f o p e r a t o r s P ( T ) t h a t c o r r e s p o n d t o t h e c o o r d i n a t et rans fo rm a t ions T of t he g ro up of t he Sch r5d inge r equ a t ion fo rms a g rouptha t i s i somorphic t o t he g ro up of t he Sch r5d inge r equ a t ion .P r o o f T h e p r o d u c t P ( T 1 ) P ( T 2 ) , as de fined i n t he p roo f o f t he p rev ious t he -orem, may be t aken t o spec i fy t he g roup mul t i p l i ca t i on ope ra t i on , so t ha tt he a s soc ia t ive l aw of ax iom (b) is s a ti sf ied . Th e prev ious t heore m th enimpl ies tha t group axiom (a) i s ful f i l l ed, and wi th P ( E ) be ing t he i den t i t yope ra tor i t a lso impl ie s t ha t t he i nve rse op e ra to r P (T ) -1 m ay be de fined byP ( T ) - 1 - P ( T - I ) . Fina l l y , i t a l so i nd i ca t e s t ha t t he two groups a re i somor-phic .T h e o r e m V F o r e v e ry c o o r d in a t e t r a n s f o r m a t i o n T o f t h e g r o u p o f t h eSchrSdinge r equa t ion

P ( T ) H ( r ) - H ( r ) P ( T ) . (1.23)

P r o o f I t ha s t o be e s t ab l i shed t ha t fo r any r and any T of GP ( T ) { H ( r ) r H ( r ) ( P ( T ) ~ 2 ( r ) } .

L e t r H ( r ) ~ ( r ) . T h e n , b y E q u a t i o n ( 1.1 7 ),P ( T ) r - r

= H ( { R ( T ) I t ( T ) } - l r ) r= H ( { R ( T ) I t ( T ) } - l r ) ( P ( T ) ~ b ( r ) } ,

f rom which E qua t ion (1 .24) follows by Eq ua t io n (1 .13) .

(1.24)

4 T h e r o l e o f m a t r i x r e p r e s e n t a t i o n sHaving shown how groups a r i se na tura l l y i n quan tum mechanic s , i n t h i s p re -l imina ry survey i t r ema ins on ly to i n t roduce t he concep t o f a g roup repre sen-t a t i o n a n d t o d e m o n s t r a t e t h a t i t t o o h a s a f u n d a m e n t a l r o l e t o p l a y .


16/18

16 G R O U P T H E O R Y I N P H Y S I C SD e f i n i t i o n Represen ta t ion o f a groupI f each e lemen t T o f a g roup G can be as s igned a non - s ingu la r d x d m a t r ixF ( T ) c o n t a i n e d i n a g ro u p o f m a t r i c e s h a v i n g m a t r i x m u l t i p l i c a t i o n a s i t sg r o u p m u l t i p l i c a t i o n o p e r a t i o n i n s u c h a w a y t h a t

r(T1T ) = ( 1 . 2 5 )for every pai r of e lements T1 and T2 of G, then th is se t of matr ices i s sa id top rov ide a d -d imens iona l " r ep resen ta t ion" r o f G .E x a m p l e I A rep resenta t ion o f the crys ta l lographic po in t group DaT h e g r o u p D 4 i n t r o d u c e d i n E x a m p l e I I I o f S e c t io n 2 h a s t h e f ol lo w i ng t w o -d i m e n s i o n a l r e p r e s e n ta t i o n :

F(E) = M 1 ,r ( C a y ) = M b ,

r(c2~) = M2,r(c = M 6 , F ( C 2 y ) = M 3 ,F (C2c) = MT,

r ( c ~ ) = g 4 ,F(C2d)--- M s,w h e r e M 1 , M 2 , . . . a r e t h e 2 2 m a t r i c e s de f in e d i n E x a m p l e I I I of S e c t io n 1 .Tha t Equa t ion (1 .25 ) i s s a t i s f i ed can be ve r i f i ed s imply by compar ing Tab les1.1 and 1.2.

I t w i l l b e s h o w n i n C h a p t e r 4 t h a t e v e r y g r o u p h a s a n i n f i n i t e n u m b e ro f d i f fe r en t r ep resen ta t ion s , bu t the y a re de r ivab le from a smal le r nu m be ro f bas ic r ep resen ta t ions , the so -ca ll ed " i r r educ ib le r ep re sen ta t ions" . A f in i teg r o u p h a s o n l y a f in it e n u m b e r o f s u c h i rr e d u c ib l e r e p r e s e n t a t i o n s t h a t a r eessent ia l ly d i f ferent .

T h e r e p r e s e n t a t io n s o f t h e g r o u p o f t h e S c h r b d i n g e r e q u a t i o n a r e o f p a r t i c -u l a r i n t e r e s t . T h e i n t i m a t e c o n n e c t i o n b e t w e e n t h e m a n d t h e e i g e n f u n c t i o n so f t h e t i m e - i n d e p e n d e n t S c h r b d i n g e r e q u a t i o n i s p r o v i d e d b y t h e n o t i o n o f"bas i s func t ions" o f the r ep resen ta t ion s .D e f i n i t i o n Bas i s func t ions o f a group o f coord ina te t rans format ions GA se t o f d l inea r ly indep ende n t func t ions r ( r ) , r Cd ( r ) fo rms a bas i sfo r a d -d imens iona l r ep resen ta t ion I ' o f ~ i f , f o r eve ry coord ina te t r ans fo rma-t ion T o f G ,

dP ( T )~ b n (r ) - - E r ( T ) r ~ m ( r ) , n - 1, 2 , . . . , d . ( 1.2 6)

m--1

T h e f u n c t i o n C n ( r ) is t h e n s a i d t o " t r a n s f o r m a s t h e n t h r o w " o f t h e r e p r e -s e n t a t i o n r .

The de f in i t ion imp l ie s tha t no t on ly i s each func t ion P ( T ) ~ b n ( r ) r e q u i r e dto be a l inea r com bina t ion o f r r Cd( r ) , bu t the coe ff ic ien t s a r er e q u i r e d t o b e e q ua l t o sp e ci fi ed m a t r i x e l e m e n t s o f F ( T ) . T h e r a t h e r u n u s u a lo r d e r i n g o f r o w a n d c o l u m n i n d i ce s o n t h e r i g h t - h a n d s id e o f E q u a t i o n ( 1. 26 )


17/18

T H E B A S IC F R A M E W O R K 17e n s u r e s t h e c o n s i s t e n c y o f t h e d e f i n i t i o n f o r e v e r y p r o d u c t T I T 2 , f o r , a c c o r d i n gto Equat ions (1 .18) , (1 .22) , (1 .25) , and (1 .26) ,

P(T~T:)r = P(T1)P(T2)rd= P ( T ~ ) { E F (T 2)m n ~b m (r)}

m--1d= ~ r(T:)mnP(T1)r

m--1d d

= E E F(T2)mnF(T1)pmCp(r)m=l p- -1

d= E F ( T iT 2 ) p n Cp ( r ) .

p--1

E x a m p l e I I Som e basis functio ns of the crystallographic point group D4The func t i ons r ( r ) = x , r = z p rov ide a ba s is fo r t he repre sen t a t i on Fof D4 tha t ha s b een co ns t ruc t ed i n Ex am ple I above , a s can be ve r if ied byinspec ti on . (Thi s se t ha s been deduced by a m e tho d t ha t wi ll be desc r ibed i nde ta i l in Chapter 5 , Sec t ion 1 . )T h e o r e m I Th e e igenfunc t ions o f a d - fo ld degen e ra t e e igenva lue c o f t het ime- independent SchrSdinge r equa t ion

H ( r ) r = e rform a bas is fo r a d -d imens iona l r epre sen t a t ion of t he g roup of t he Schr5d inge requa t ion ~ .Proof Le t r r Cd( r ) be a se t o f l inea r ly i ndepe nden t e igenfunc -t ions o f H( r ) w i th e igenva lue e , so t ha t

H ( r ) r = e r n - 1, 2 , . . . , d ,and any o th e r e igenfunc t ion of H ( r ) w i th e igenva lue e is a l inea r com bina -t ion of r ( r ) , r Cd( r ) . For any t r ans f orm a t ion T of t he g roup of t heSchr5d inge r equa t ion , Equa t ion (1 .23) impl i e s t ha t

H ( r ) { P ( T ) r = P ( T ) { H ( r ) r = e { P ( T ) rd e m o n s t r a t i n g t h a t P ( T ) r is a lso an e igenfunc t ion of H ( r ) wi th e igen-va lue e , so tha t P ( T ) r ( r ) may be wr i t t en i n t he fo rm

dP ( T ) r - - E r ( T ) ~ r n - 1, 2 , . . . , d . ( 1.2 7)m--1


18/18

18 GRO UP THE OR Y IN PHYSICSA t t h i s s t a g e t h e F(T)mn a re m ere ly a s e t o f coe f f ic i en ts wi th t he m , n a nd Td e p e n d e n c e e x p l i c i t l y d i s p l a y e d . F o r e a c h T t h e s e t F ( T ) m n c a n b e a r r a n g e dt o f o r m a d d m a t r i x F ( T ) . I t w ill n o w b e sh o w n t h a t

dF (T IT2)m n = E F (T1 )mpF (T2 )pn (1 .28 )

p = lf o r a n y t w o t r a n s f o r m a t i o n s T1 a n d T 2 o f G , t h e r e b y d e m o n s t r a t i n g t h a t t h em a t r i c e s r(T) do a c t u a l l y f o r m a r e p r e s e n t a t i o n o f 6 . E q u a t i o n ( 1..2 7 ) t h e ni m p l i e s t h a t t h e e i g e n f u n c t i o n s r 1 6 2 C d ( r ) f o r m a b a s is f o r t h i sr e p r e s e n t a t i o n .F r o m E q u a t i o n ( 1 .2 7 ) , w i t h T r e p l a c e d b y T 1 , T 2 a n d TIT2 i n t u r n ,

dP(T 1)r = E F(T~)mpCm(r), (1 .29)m - - 1

dP ( T 2 ) r = E r ( T 2 ) p ~ r

p - - 1dP(TIT2)r ( r ) = ~ F ( T I T 2 ) m ~ r ( r ) .

m = lF r o m E q u a t i o n s ( 1 .2 9 ) a n d ( 1 .3 0 )d dP(T1)P(T2)r = E ~ -~ r (T ~ ) m p r ( T 2 ) p n r

m--1 p----1

(1 .31)

(1.32)

a n d , a s P(T1)P(T2)r ( r ) = P(T1T2)r ( r ) b y E q u a t i o n ( 1 . 2 2 ) , t h e r i g h t h a n ds id e s o f E q u a t i o n s ( 1 .3 1 ) a n d ( 1 .3 2 ) m u s t b e e q u a l . A s t h e f u n c t i o n s rr . . . , C d ( r ) h a v e b e e n a s s u m e d t o b e l in e a r l y i n d e p e n d e n t , E q u a t i o n(1 . 28 ) fo l l ows on equa t i ng coef f i c i en t s o f each Cm(r ) .

T h i s t h e o r e m i m p l i e s t h a t e a c h e n e r g y e i g e n v a l u e c a n b e l a b e l l e d b y ar e p r e s e n t a t i o n o f t h e g r o u p o f t h e S c h r h d i n g e r e q u a t i o n . I n C h a p t e r 1 0 i t w i llb e s h o w n t h a t t h e f a m i l ia r c a t e g o r i z a t i o n o f e l e c t r o n i c s t a t e s o f a n a t o m i n t o" s - s t a t e s " , " p - s t a t e s " , " d - s t a t e s " e t c . i s ac tu a l l y j u s t a spec i a l case o f t h i s t yp eo f desc r i p t i on . M ore p rec i se ly , eve ry s - s t a t e e i ge n fun c t i on i s a bas i s fun c t i ono f p a r t i c u l a r r e p r e s e n t a t i o n o f t h e g r o u p o f r o t a t i o n s i n t h r e e d i m e n s i o n s, t h ep - s t a t e e i g e n f u n c t i o n s a r e b a s is f u n c t io n s o f a n o t h e r r e p r e s e n t a t i o n o f t h a tg r o u p , a n d s o o n .H a v i n g e s ta b l i sh e d a p r i m a f a c ie ca s e t h a t g r o u p s a n d t h e i r r e p r e s e n t a ti o n sp l a y a s i g n if ic a n t ro l e in t h e q u a n t u m m e c h a n i c a l s t u d y o f p h y s i c a l s y s t e m s ,t h e n e x t c h a p t e r s w i ll b e d e v o t e d t o a d e t a il e d e x a m i n a t i o n o f t h e s t r u c t u r e o fg r o u p s a n d t h e t h e o r y o f t h e i r r e p r e s e n t a t i o n s . S o fa r o n ly a b r i e f i n d i c a t i o nh a s b e e n g i v e n o f w h a t c a n b e a c h i e v e d , b u t t h e e n s u i n g c h a p t e r s w i ll s h o wt h a t t h e g r o u p t h e o r e t i c a l a p p r o a c h is c a p a b l e o f d e a l in g w i t h a v e r y w i d er a n g e o f p r o f o u n d a n d d e t a i l e d q u e s t i o n s .

Chapter 1 - The Basic Framework

Documents

Transcript of Chapter 1 - The Basic Framework