Ch4

4 . T w o - l e v e l s y s te m s

4 . 1 G e n e r al i t i e s

4 . 2 R o t at i o n s an d an g u l ar m o m e n t u m

4 . 2 . 1 C l a s s i c a l r o t a t i o n s

4 . 2 . 2 Q M a n g u l a r mo m e n t u m a s g e n e r a t o r o f r o t a t i o n s

4 . 2 . 3 E x a m p l e o f T w o - Le v e l S y s t e m: N e u t r o n I n t e r f e r o me t r y

4 . 2 . 4 S p i n o r b e h a v i o r

4 . 2 . 5 T h e S U ( 2 ) a n d S O ( 3 ) g r o u p s

4 . 1 Ge n e r a l i t i e s

W e h a v e a lr e a d y s e e n s o m e e x a m p le s o f s y s t e m s d e s c r ib e d b y t w o p o s s ib le s t a t e s . A n e u t r o n in a n in t e r f e r o m e t e r , t a k in g e it h e r t h e u p p e r o r lo w e r p a t h . A p h o t o n lin e a r ly p o la r iz e d e it h e r h o r iz o n t a lly o r v e r t ic a lly . A t w o le v e l s y s t e m ( T L S ) is t h e s im p le s t s y s t e m in q u a n t u m m e c h a n ic s , b u t it a lr e a d y illu s t r a t e s m a n y c h a r a c t e r is t ic s o f Q M a n d it d e s c r ib e s a s w e ll m a n y p h y s ic a l s y s t e m s . It is c o m m o n t o r e d u c e o r m a p q u a n t u m p r o b le m s o n t o a T L S . W e p ic k t h e m o s t im p o r t a n t s t a t e s - t h e o n e s w e c a r e a b o u t – a n d t h e n d is c a r d t h e r e m a in in g d e g r e e s o f f r e e d o m , o r in c o r p o r a t e t h e m a s a c o lle c t io n o r c o n t in u u m o f o t h e r d e g r e e s o f f r e e d o m t e r m e d a b a t h .

In a m o r e a b s t r a c t w a y , w e c a n t h in k o f a T L S a s c a r r y in g a b in a r y in f o r m a t io n ( t h e a b s e n c e o r p r e s e n c e o f s o m e t h in g , t h e in f o r m a t io n a b o u t a p o s it io n , s u c h a s le f t o r r ig h t , o r u p o r d o w n , e t c .) . T h u s a T L S c a n b e t h o u g h t a s c o n t a in in g a b it o f in f o r m a t io n . B y a n a lo g y w it h c la s s ic a l c o m p u t e r s a n d in f o r m a t io n t h e o r y , T L S a r e t h u s c a lle d q u b i t s . T h e ir b a s is s t a t e s a r e u s u a lly d e fi n e d a s | 0 ) a n d | 1 ) w it h a v e c t o r r e p r e s e n t a t io n :

0 1

| 0 ) = [ 1 ] , | 1 ) = [ 0 ]

| ) | ) | ) | | | |

A g e n e r a l s t a t e is t h e n ψ = α 0 + β 1 . If it is n o r m a liz e d w e h a v e α 2 + β 2 = 1 . T h e n , t h is s t a t e c a n a ls o b e w r it t e n q u it e g e n e r a lly in t e r m s o f t h e t w o a n g le s ϑ a n d ϕ :

i ϕ

| ψ ) = c o s ( ϑ / 2 ) | 0 ) + e s in ( ϑ / 2 ) | 1 )

F o r t h is s t a t e , t h e p r o b a b ilit y o f fi n d in g t h e s y s t e m in t h e 0 [1 ] s t a t e is c o s 2 ( θ / 2 ) [s in 2 ( θ / 2 ) ]. N o t ic e t h a t I c o u ld h a v e w r it t e n t h e s t a t e a ls o a s

| ψ ) ≡ | φ ) = e − i φ / 2 c o s ( ϑ / 2 ) | 0 ) + e i φ / 2 s in ( θ / 2 ) | 1 )

T h e t w o s t a t e s a r e in f a c t e q u iv a le n t u p t o a g l o b a l p h a s e f a c t o r . W h ile r e la t iv e p h a s e f a c t o r s ( in a s u p e r p o s it io n ) a r e v e r y im p o r t a n t , g lo b a l p h a s e s a r e ir r e le v a n t , s in c e t h e y y ie ld t h e s a m e r e s u lt s in a m e a s u r e m e n t o u t c o m e .

? Q u e s t i o n : S h o w t h a t a g l o b a l p h a s e f a c t o r d o e s n o t c h a n g e m e a s u r e me n t o u t c o me s a n d m e a s u r e me n t s t a t i s t i c s . 1 . ( M e a s u r e me n t o u t c o m e ) A s t h e p o s s i b l e m e a s u r e me n t o u t c o me s a r e t h e e i g e n v a l u e s o f t h e m e a s u r e me n t o p e r a t o r s , t h e fi r s t i s t r i v i a l l y t r u e .

− i φ / 2 i φ / 2

2

2 i φ 2 2

2 . ( S t a t i s t i c s ) Le t ’ s c o n s i d e r a n o b s e r v a b l e A w i t h e i g e n v e c t o r s | a ) = a 0 | 0 ) + a 1 | 1 ) c o r r e s p o n d i n g t o t h e e i g e n v a l u e s a , t h e n t h e p r o b a b i l i t y o f o b t a i n i n g a f r o m t h e me a s u r e m e n t i s p ( a ) = ( ψ | a ) = | a 0 c o s ( θ / 2 ) + a 1 e s i n ( θ / 2 ) | = ( ϕ | a ) =

| a 0 e c o s θ / 2 + a 1 e s i n θ / 2 | .

2

C o n s id e r f o r e x a m p le | a ) = ( | 0 ) + | 1 ) ) / √ 2 . T h e n w e o b t a in p ( a ) = 1 ( 1 + c o s ϕ s in ϑ ) . T h u s t h e r e l a t i v e p h a s e o f | 0 )

w .r .t . | 1 ) is im p o r t a n t . B u t a g lo b a l p h a s e m u lt ip ly in g t h e s t a t e is n o t .

z

0





y



x

1

Image by MIT OpenCourseWare.

F i g . 2 : B l o c h s p h e r e r e p r e s e n t a t i o n o f a q u b i t ( T LS ) [ F r o m w i k i p e d i a ]

D e s c r ib in g a T L S v ia t h e t w o a n g le s θ a n d φ le a d s t o a s im p le g e o m e t r ic a l p ic t u r e f o r t h e s p a c e o c c u p ie d b y t h is s y s t e m . T h e a n g le s d e fi n e a p o in t o n a s p h e r e o f r a d iu s 1 , w h ic h is c a lle d B l o c h s phe r e . T h e T L S c a n t h e n a s s u m e a n y o f t h e p o in t s o n t h e s u r f a c e o f t h e s p h e r e v ia a u n it a r y t r a n s f o r m a t io n ( in t h e f o llo w in g , w e w ill a ls o in t e r e s t e d in t h e p o in t s i n s i d e t h e s p h e r e , a s w e ll a s m e a n s t o r e a c h t h e m ) . T h e u n it a r y e v o lu t io n f o r t h is p a r t ic u la r s y s t e m c a n t h e n b e d e s c r ib e d a s r o t a t i o n s o f t h e s t a t e v e c t o r in t h e s p h e r e .

U s in g t h e e x a m p le o f a T L S w e a r e t h u s g o in g t o in t r o d u c e t h e c o n c e p t o f r o t a t io n a n d a n g u la r m o m e n t u m , w h ic h c a n b e g e n e r a liz e d a ls o t o la r g e r s y s t e m s .

4 . 2 R o t a t i o n s a n d a n g u l a r m o m e n t u m

4 . 2 . 1 C l a s s i c a l r o t a t i o n s

L e t ’s r e v ie w r o t a t io n in c la s s ic a l m e c h a n ic s ( g e o m e t r y ) . T h e fi r s t p r o p e r t y t h a t w e w a n t t o a n a ly z e is t h e f a c t t h a t s u c c e s s iv e r o t a t io n s a b o u t d iff e r e n t a x e s d o n o t c o m m u t e . C o n s id e r f o r e x a m p le t o s t a r t w it h a v e c t o r a lig n e d a lo n g t h e z a x is a n d t h e n e ff e c t u a t e t w o r o t a t io n s , o n e a b o u t t h e y a x is a n d o n e a b o u t t h e z a x is . D e p e n d in g o n t h e o r d e r , w e o b t a in a r o t a t io n o r n o r o t a t io n a t a ll.

×

R o t a t io n a r e r e p r e s e n t e d in 3 D b y o r t h o g o n a l 3 3 m a t r ic e s . ( a n o r t h o g o n a l m a t r ix is s u c h t h a t R R T = R T R = 1 1 . In p a r t ic u la r , r o t a t io n s a b o u t t h e 3 a x e s a r e a s f o llo w :

R x ( φ ) =

1 0 0

 

—





0 c o s ( φ ) s in ( φ )

0 s in ( φ ) c o s ( φ )

R y ( φ ) = 0 1 0

— s in ( φ ) 0 c o s ( φ )

  c o s ( φ ) 0 s in ( φ )  

R z ( φ ) =

s in ( φ ) c o s ( φ ) 0

0 0 1

  c o s ( φ ) − s in ( φ ) 0  

It is e a s y t h e n t o s h o w t h a t R α ( ϑ ) R β ( ϕ ) / = R β ( ϕ ) R α ( ϑ ) u n le s s α = β . W h a t a b o u t if t h e r o t a t io n a n g le s a r e v e r y s m a ll? W e m ig h t e x p e c t t h e n t h a t t h e o r d e r m a t t e r s le s s . W e t h u s c o n s id e r in fi n it e s im a l r o t a t io n s , w h e r e φ = ǫ → 0 :





1 0 0





ǫ 2

R ( ǫ ) ≈ 0 1 − − ǫ

2

x 2

0 ǫ 1 − ǫ

2



 1 − ǫ 2 0 ǫ 

R y ( ǫ ) = 

0 1 0

ǫ 2

− ǫ 0 1 − 2

2

 1 − ǫ 2 − ǫ 0 

2

R z ( ǫ ) = 

ǫ 1 − ǫ 2 0 

0 0 1

If w e t h e n c a lc u la t e f o r e x a m p le R x ( ǫ ) R y ( ǫ ) − R y ( ǫ ) R x ( ǫ ) w e o b t a in :

 1 − ǫ 2 0 ǫ 

2

x y

2

R ( ǫ ) R ( ǫ ) = 

2

ǫ 2 1 − ǫ

− ǫ ( 1 − ǫ

2 ) 

2 2

 − ǫ ( 1 − ǫ 2 ) ǫ ( 1 − ǫ 2 ) 2 

2

ǫ 2

2

 1 − ǫ 2 ǫ 2 ǫ − ǫ 3 

R y ( ǫ ) R x ( ǫ ) = 

0 1 − 2

− ǫ 

 − ǫ

ǫ − ǫ 3

( 1 − ǫ 2 ) 2 

a n d

2 2

2

 0 − ǫ 2 ǫ 3 

x

y

x

ǫ 3

2

0

R ( ǫ ) R ( ǫ ) − R ( ǫ ) R ( ǫ ) =   ǫ 2 0

ǫ 3  

2 2

T h u s w e s e e t h a t

∝

1 . If w e ig n o r e d t e r m s ǫ 2 a n d h ig h e r , t h e r o t a t io n s d o c o m m u t e .

− −

2 . A t t h e s e c o n d o r d e r in ǫ w e c a n w r it e t h e r e s u lt a s R x ( ǫ ) R y ( ǫ ) R y ( ǫ ) R x ( ǫ ) = R z ( ǫ 2 ) 1 1 . T h is r e s u lt s t a n d s a ls o f o r c y c lic p e r m u t a t io n s o f t h e s u b s c r ip t s . T h e s e c o m m u t a t io n r e la t io n s h ip s a r e a g u id e in fi n d in g c o m m u t a t io n r e la t io n s h ip s t h a t t h e e q u iv a le n t Q M r o t a t io n o p e r a t o r s s h o u ld o b e y .

4 . 2 . 2 Q M a n g u l a r m o m e n t u m a s g e n e r a t o r o f r o t a t i o n s

D

In Q M w e c a n a s w e ll d e fi n e r o t a t io n s , a s w e a lr e a d y d id f o r c la s s ic a l m e c h a n ic s . A lt h o u g h w e w ill fi r s t s t u d y e x a m p le s f o r T L S , r o t a t io n s c a n b e d e fi n e d f o r a n y s y s t e m ( e v e n h ig h e r d im e n s io n a l s y s t e m s ) . G e n e r a lly , a r o t a t io n w ill b e r e p r e s e n t e d b y a n o p e r a t o r ( R α ( φ ) ) a s s o c ia t e d t o a c la s s ic a l r o t a t io n R α ( φ ) . W e fi r s t d e fi n e t h e a c t io n o f a n in fi n it e s im a l r o t a t io n . T o d o s o w e d e fi n e t h e a n g u la r m o m e n t u m o p e r a t o r J in t e r m s o f t h e in fi n it e s im a l r o t a t io n :

D ( R n ( δ φ ) ) = 1 1 − i δ φ J J · J n

w h e r e J n is a u n it v e c t o r . A fi n it e r o t a t io n c a n b e f o u n d b y r e p e a t in g m a n y in fi n it e s im a l r o t a t io n s . F o r e x a m p le , f o r a r o t a t io n a b o u t z :

z

1

z

1

z

D ( R ( ϕ ) ) = lim � 1 − i J ϕ � N = 1 − i J ϕ − 1 J 2 ϕ 2 · · · = e x p ( − i J ϕ )

N → ∞ N 2

( N o t e t h a t h e r e a g a in I t o o k I = 1 .) T h e a n g u la r m o m e n t u m c a n t h u s b e c o n s id e r e d a s t h e g e n e r a t o r o f r o t a t io n s .

A . R o ta ti o n s p r o p e r ti e s

– Id e n t it y : ∃ 1 1 : 1 1 D = D 1 1 = D

– C lo s u r e : D 1 D 2 is a ls o a r o t a t io n D 3 .

– In v e r s e : ∃ a n d in v e r s e s u c h t h a t D D − 1 = 1 1

– A s s o c ia t iv it y ( D 1 D 2 ) D 3 = D 1 ( D 2 D 3 )

B . C o m m u ta ti o n

In a n a lo g y w it h t h e c la s s ic a l c a s e , w e c a n w r it e t h e c o m m u t a t io n f o r t h e in fi n it e s im a l r o t a t io n s :

D x ( ǫ ) D y ( ǫ ) − D y ( ǫ ) D x ( ǫ )

= ( 1 1 − i J

ǫ − 1 J 2 ǫ 2 ) ( 1 1 − i J ǫ − 1 J 2 ǫ 2 ) − ( 1 1 − i J ǫ − 1 J 2 ǫ 2 ) ( 1 1 − i J ǫ − 1 J 2 ǫ 2 ) = − ( J J

— J J

) ǫ 2 + O ( ǫ 3 )

D − −

x

2

x

y

2 y

y

2

y

x

2

x

y

x

a n d e q u a t e t h is t o z ( ǫ 2 ) 1 1 = i J z ǫ 2 . W it h t h is a n a lo g y w e j u s t if y t h e d e fi n it io n o f a n g u la r m o m e n t u m o p e r a t o r s a s o p e r a t o r s t h a t g e n e r a t e t h e r o t a t io n s a n d o b e y t h e c o m m u t a t io n r e la t io n s h ip s :

[ J i , J j ] = i I ǫ i j k J k

2

C . S p i n - 1

A lt h o u g h a n g u la r m o m e n t u m o p e r a t o r s h a v e s o m e c la s s ic a l a n a lo g y , t h e y a r e m o r e g e n e r a l, a s t h e y d e s c r ib e f o r e x a m p le p h y s ic a l p r o p e r t ie s t h a t h a v e n o c la s s ic a l c o u n t e r p a r t s , s u c h a s t h e s p in . In p a r t ic u la r , t h e lo w e s t d im e n s io n in w h ic h t h e c o m m u t a t io n r e la t io n s h ip s a b o v e h o ld is 2 . T h e a n g u la r m o m e n t u m S f o r a T L S is r e p r e s e n t e d b y t h e o p e r a t o r s :

2 2

2

S x = 1 σ x = 1 ( | 0 ) � 1 | + | 1 ) � 0 | ) S y = 1 σ y = i ( | 0 ) � 1 | − | 1 ) � 0 | ) S z = 1 σ z = 1 ( | 0 ) � 0 | − | 1 ) � 1 | )

w h e r e { σ x , σ y , σ z } a r e c a lle d P a u li o p e r a t o r s o r P a u li m a t r ic e s . T h e P a u li m a t r ic e s h a v e t h e f o llo w in g p r o p e r t ie s :

α

1 . σ 2 = 1 1

2 . σ i σ j + σ j σ i = 0 , t h a t is , t h e y a n t ic o m m u t e .

3 . σ i σ j = − σ j σ i = i σ k ( f r o m t h e p r e v io u s p r o p e r t y )

4 . H e r m it ic it y : σ i † = σ i

5 . Z e r o t r a c e : T r { σ i } = 0

6 . D e t e r m in a n t d e t ( σ i ) = − 1 .

? Q u e s t i o n : S h o w t h a t S s a t i s fi e s t h e c o mm u t a t i o n r e l a t i o n s h i p .

1 . S h o w i t b y m u l t i p l y i n g t h e o p e r a t o r s .

2 . W r i t e d o w n t h e ma t r i x f o r m a n d p e r f o r m ma t r i x m u l t i p l i c a t i o n s .

W e c a n n o w c h e c k w h a t is t h e a c t io n o f t h e s p in o p e r a t o r s o n t h e T L S s t a t e v e c t o r | ψ ) = α | 0 ) + β | 1 ) :

σ x | ψ ) = α | 1 ) + β | 0 ) σ y | ψ ) = i α | 1 ) − i β | 0 ) σ z | ψ ) = α | 0 ) − β | 1 )

| )

in p a r t ic u la r , σ x s w a p t h e t w o c o m p o n e n t s ( s p in fl ip ) a n d σ z in v e r t t h e s ig n o f t h e 1 c o m p o n e n t ( p h a s e s h if t ) , w h ile

σ y d o e s b o t h .

2

D . S p i n - 1 r o ta ti o n s

2

D

α

W e c a n n o w lo o k a t r o t a t io n s o f s p in - 1 . In p a r t ic u la r w e w a n t t o c a lc u la t e α ( ϕ ) = e − i S α ϕ . F o r t h is w e r e m e m b e r t h e p r o p e r t y : σ 2 = 1 1 . W it h t h is , a n d u s in g a T a y lo r e x p a n s io n it is e a s y t o s h o w t h a t w e h a v e

e − i S α ϕ = c o s ( ϕ / 2 ) 1 1 − i s in ( ϕ / 2 ) S α

? Q u e s t i o n : C a l c u l a t e t h e e x p o n e n t i a l .

✭ ✭ ✭ ✭

L L

x

y

z

F r o m ( σ · n ) 2 = ( σ x n x + σ y n y + σ z n z ) 2 = σ 2 n 2 + n x n y ✭ ( σ x ✭ σ y + σ y σ x ) + · · · + n 2 1 1 + n 2 1 1 = 1 1 a n d t h e T a y l o r e x p a n s i o n w e

o b t a i n e − i ϕ σ · n = 1 1 ( − i ϕ ) n / n ! + σ · n ( − i ϕ ) n / n ! = 1 1 c o s ϕ + σ · n s i n ϕ .

n e v e n n o d d

4 . 2 . 3 E x a m p l e o f T w o - L e v e l S y s t e m : N e u t r o n I n t e r f e r o m e t r y

N o w w e c a n r e v is it t h e T L S e x a m p le s w e h a v e s e e n e a r lie r . In p a r t ic u la r w e n o t ic e t h a t t h e p o la r iz a t io n r o t a t o r is r e p r e s e n t e d b y r o t a t io n o p e r a t o r s , in p a r t ic u la r r o t a t io n s a r o u n d t h e x - a x is e − i θ S x .

C o n s id e r a n o t h e r v e r y s im p le s y s t e m , a n e u t r o n in t e r f e r o m e t e r , s u c h a s t h e M a c h - Z e h n d e r in t e r f e r o m e t e r .

|U 

BS

|L 

1

BS

2

F i g . 3 : N e u t r o n I n t e r f e r o me t e r

W e s e n d in a b e a m o f n e u t r o n s . T h e fi r s t b e a m s p lit t e r d iv id e s t h e n e u t r o n fl u x in t o t w o p a r t s , t h a t w ill g o in t o t h e u p p e r a r m o r t h e lo w e r a r m . T h u s t h e s t a t e o f t h e s y s t e m is a t t h is p o in t in t im e

2 2

| ψ ) 1 = α | U ) + β | L ) , α + β = 1

√

W e a s s u m e t h a t t h e fl u x o f n e u t r o n s is s o lo w ( n e u t r o n s c a n b e v e r y s lo w ) s o t h a t o n ly o n e n e u t r o n is p r e s e n t a t a n y t im e in s id e t h e in t e r f e r o m e t e r . T h e lo w e r a n d u p p e r b e a m s a r e t h e n r e fl e c t e d a t t h e m ir r o r s a n d r e c o m b in e d a t t h e s e c o n d b e a m s p lit t e r , a f t e r w h ic h t h e n e u t r o n fl u x is m e a s u r e d a t o n e a r m . If w e a s s u m e t h a t b o t h b e a m s p lit t e r w o r k s in t h e s a m e w a y , d e liv e r in g a n e q u a l fl u x t o e a c h a r m ( t h a t is , t h e t r a n s m is s io n a n d r e fl e c t io n a r e t h e s a m e ) , t h e n w e h a v e | ψ ) 1 = ( | U ) + | L ) ) / 2 a n d | ψ ) 2 = | U ) .

√ √

? Q u e s t i o n : W h a t i s t h e p r o p a g a t o r d e s c r i b i n g t h e a c t i o n o f t h e B e a m s p l i t t e r ?

,

U B S | U ) = ( | U ) + | L ) ) / 2 a n d w e a l s o k n o w t h a t U B S ( | U ) + | L ) ) / 2 = | U ) . W e c a n v e r i f y t h a t

U B S

1 1 1

= √ 2 1 − 1

p e r f o r ms a s w e w a n t . I n p a r t i c u l a r , n o t i c e t h a t U B S U B S = 1 1 .

T h u s , i f o u r o b s e r v a b l e i s t h e n u m b e r o f n e u t r o n i n t h e u p p e r a r m , t h e me a s u r e m e n t a l w a y s r e t u r n s 1 w i t h c e r t a i n t y ( p r o b a b i l i t y

= 1 ) .

Le t ’ s n o w c o n s i d e r t h e c a s e i n w h i c h w e w a n t t o me a s u r e a t p o i n t 1 h o w ma n y n e u t r o n s a r e i n t h e u p p e r a r m. T h e o b s e r v a b l e i s j u s t t h e p r o j e c t o r o n t o t h e u p p e r a r m O b = P U = | U ) ( U | a n d w e w i l l d e t e c t o n e n e u t r o n ( o r z e r o n e u t r o n s ) w i t h p r o b a b i l i t y

2 2

1 . I n f a c t p ( U ) = ( ψ | U ) 2 = 1 . A l s o , t h e a v e r a g e v a l u e o f t h e n u m b e r o f n e u t r o n i n t h e u p p e r a r m i s 1 / 2 a s w e l l , s i n c e

1

( ) | ( | | ) |

O b = ψ P U ψ = 2 A f t e r t h e me a s u r e m e n t , t h e s t a t e i s p r o j e c t e d o n t o t h e u p p e r a r m , i f w e d i d d e t e c t a p h o t o n , o r t h e l o w e r a r m , o t h e r w i s e . W e a s s u m e t h a t t h e n e u t r o n i s f r e e t o c o n t i n u e o n i t s p a t h a f t e r t h e me a s u r e me n t a n d w e p e r f o r m a s e c o n d me a s u r e m e n t a f t e r t h e s e c o n d b e a ms p l i t t e r .

√

2

? Q u e s t i o n : W h a t i s t h e p r o b a b i l i t y o f m e a s u r i n g 1 n e u t r o n i n t h e u p p e r p a r t i n t h i s c a s e ? N o w t h e s t a t e a t 2 i s | ψ ) 2 = ( | U ) ± | L ) ) / 2 , h e n c e p ( U ) = ( ψ | U ) = 1 .

4 . 2 . 4 S p i n o r b e h a v i o r

B y c a lc u la t in g e − i S α ϕ S β e i S α ϕ w e s e e t h a t t h e r o t a t io n s o f t h e o p e r a t o r g iv e t h e f o llo w in g r e s u lt :

x x y

S → S z S c o s ( ϕ ) − S s in ( ϕ ) ,

z z

y y x

S → S z S c o s ( ϕ ) + S s in ( ϕ ) , S → S z S

T h e s e a r e t h e s a m e r o t a t io n r u le s w e w o u ld h a v e e x p e c t e d c la s s ic a lly . In p a r t ic u la r , t a k in g t h e e x p e c t a t io n v a lu e s , w e

| ) | ) | )

s e e t h a t t h e y c o r r e s p o n d e x a c t ly t o t h e r o t a t io n s in 3 D o f a v e c t o r , w it h a p e r io d ic it y o f 2 π . T h in g s a r e a b it d iff e r e n t ( a n d m o r e s u r p r is in g ) if w e c o n s id e r in s t e a d t h e s t a t e r o t a t io n . C o n s id e r t h e r o t a t io n o f t h e s t a t e ψ = α 0 + β 1 w it h r e s p e c t t o S z :

e − i J z ϕ | ψ ) = e − i ϕ / 2 α | 0 ) + e i ϕ / 2 β | 1 ) ,

| ) − | )

n o w t h e a n g le o f r o t a t io n s e e m s t o b e ϕ / 2 . T h is h a s a n in t e r e s t in g c o n s e q u e n c e : if w e r o t a t e b y ϕ = 2 π in s t e a d o f r e t u r n in g t o t h e in it ia l s t a t e , a s w e w o u ld h a v e e x p e c t e d , w e o b t a in e − i J z 2 π ψ = ψ . T h is is t h e s o - c a lle d s pi no r b e h a v io r . N o t ic e t h a t f r o m a s im p le m e a s u r e m e n t t h is m in u s s ig n ( w h ic h is e q u iv a le n t t o a g l o b a l p h a s e ) is ir r e le v a n t , h e n c e w e o b t a in t h e s a m e e x p e c t a t io n v a lu e s f o r t h e a n g u la r m o m e n t a a s b e f o r e . W e w ill s e e in P - S e t o n t h a t e x p e r im e n t s c a n b e d e v is e d t o s h o w t h e s p in o r b e h a v io r ( b u t t h e y n e e d t o u s e m o r e t h a n o n e s p in ) .

4 . 2 . 5 T h e S U ( 2 ) a n d S O ( 3 ) g r o u p s

A g r o u p G is a fi n it e o r in fi n it e s e t o f e le m e n t s t o g e t h e r w it h a b in a r y o p e r a t io n ( c a lle d t h e g r o u p o p e r a t io n ) t h a t t o g e t h e r s a t is f y t h e f o u r f u n d a m e n t a l p r o p e r t ie s o f c lo s u r e , a s s o c ia t iv it y , t h e id e n t it y p r o p e r t y , a n d t h e in v e r s e p r o p e r t y . A r o t a t io n g r o u p is a g r o u p in w h ic h t h e e le m e n t s a r e o r t h o g o n a l m a t r ic e s w it h d e t e r m in a n t 1 . In t h e c a s e o f t h r e e - d im e n s io n a l s p a c e , t h e r o t a t io n g r o u p is k n o w n a s t h e s p e c ia l o r t h o g o n a l g r o u p o r S O ( 3 ) 9 . T h e s p e c ia l u n it a r y g r o u p S U ( 2 ) is t h e s e t o f 2 b y 2 u n it a r y m a t r ic e s w it h d e t e r m in a n t + 1 [it is a s u b g r o u p o f t h e u n it a r y g r o u p U ( 2 ) ]. T h e t w o g r o u p s S O ( 3 ) a n d S U ( 2 ) b o t h r e p r e s e n t r o t a t io n s , h o w e v e r t h e r e is a o n e - t o - t w o c o r r e s p o n d e n c e f o r a g iv e n R ∈ S O ( 3 ) t h e r e a r e 2 U ∈ S U ( 2 ) . T h is is b e c a u s e a 2 π a n d 4 π r o t a t io n s a r e t h e s a m e in S O ( 3 ) b u t t h e y a r e

1 1 a n d − 1 1 in S U ( 2 ) .

9 F o r a m o r e r i g o r o u s a n d e x t e n s i v e e x p l a n a t i o n s e e J . J . S a k u r a i “ M o d e r n Q u a n t u m M e c h a n i c s ” , A d d i s o n - W e s l e y ( 1 9 9 4 ) , p a g e 1 6 8

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22.51 Quantum Theory of Radiation Interactions

Fall 201 2

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