1 1 . P e r tu r b a ti o n T h e o r y
1 1 . 1 T i m e - i n d e p e n d e n t p e r t u r b at i o n t h e o r y
1 1 . 1 . 1 N o n - d e g e n e r a t e c a s e
1 1 . 1 . 2 D e g e n e r a t e c a s e
1 1 . 1 . 3 T h e S t a r k e ff e c t
1 1 . 2 T i m e - d e p e n d e n t p e r t u r b at i o n t h e o r y
1 1 . 2 . 1 R e v i e w o f i n t e r a c t i o n p i c t u r e
1 1 . 2 . 2 D y s o n s e r i e s
1 1 . 2 . 3 F e r m i ’ s G o l d e n R u l e
1 1 . 1 Ti m e - i n d e p e n d e n t p e r t u r b a t i o n t h e o r y
B e c a u s e o f t h e c o m p le x it y o f m a n y p h y s ic a l p r o b le m s , v e r y f e w c a n b e s o lv e d e x a c t ly ( u n le s s t h e y in v o lv e o n ly s m a ll H ilb e r t s p a c e s ) . In p a r t ic u la r , t o a n a ly z e t h e in t e r a c t io n o f r a d ia t io n w it h m a t t e r w e w ill n e e d t o d e v e lo p a p p r o x im a t io n m e t h o d s 3 6 .
1 1 . 1 . 1 N o n - d e g e n e r a t e c a s e
W e h a v e a n H a m ilt o n ia n
H = H 0 + ǫ V
H
w h e r e w e k n o w t h e e ig e n v a lu e o f t h e u n p e r t u r b e d H a m ilt o n ia n 0 a n d w e w a n t t o s o lv e f o r t h e p e r t u r b e d c a s e
H H →
= 0 + ǫ V , in t e r m s o f a n e x p a n s io n in ǫ ( w it h ǫ v a r y in g b e t w e e n 0 a n d 1 ) . T h e s o lu t io n f o r ǫ 1 is t h e d e s ir e d s o lu t io n .
W e a s s u m e t h a t w e k n o w e x a c t ly t h e e n e r g y e ig e n k e t s a n d e ig e n v a lu e s o f H 0 :
( 0 )
L
H 0 | k ) = E k | k )
k
A s H 0 is h e r m it ia n , it s e ig e n k e t s f o r m a c o m p le t e b a s is k | k ) ( k | = 1 1 . W e a s s u m e a t fi r s t t h a t t h e e n e r g y s p e c t r u m is n o t d e g e n e r a t e ( t h a t is , a ll t h e E ( 0 ) a r e d iff e r e n t , in t h e n e x t s e c t io n w e w ill s t u d y t h e d e g e n e r a t e c a s e ) . T h e e ig e n s y s t e m f o r t h e t o t a l h a m ilt o n ia n is t h e n
( H 0 + ǫ V ) | ϕ k ) ǫ = E k ( ǫ ) | ϕ k ) ǫ
w h e r e ǫ = 1 is t h e c a s e w e a r e in t e r e s t e d in , b u t w e w ill s o lv e f o r a g e n e r a l ǫ a s a p e r t u r b a t io n in t h is p a r a m e t e r :
k
k
k
k
k
k
| ϕ k ) = ϕ ( 0 ) ) + ǫ ϕ ( 1 ) ) + ǫ 2 ϕ ( 2 ) ) + . . . , E k = E ( 0 ) + ǫ E ( 1 ) + ǫ 2 E ( 2 ) + . . .
3 6 A v e r y g o o d t r e a t me n t o f p e r t u r b a t i o n t h e o r y i s i n S a k u r a i ’ s b o o k – 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 ) , w h i c h w e f o l l o w h e r e .
k
k k
)
w h e r e o f c o u r s e ϕ ( 0 ) = | k ) . W h e n ǫ is s m a ll, w e c a n in f a c t a p p r o x im a t e t h e t o t a l e n e r g y E k b y E ( 0 ) . T h e e n e r g y s h if t d u e t o t h e p e r t u r b a t io n is t h e n o n ly ∆ k = E k − E ( 0 ) a n d w e c a n w r it e :
k ǫ k
( H 0 + ǫ V ) | ϕ k ) = ( E ( 0 ) + ∆ k ) | ϕ k ) → ( E ( 0 ) − H 0 ) | ϕ k ) = ( ǫ V − ∆ k ) | ϕ k )
T h e n , w e p r o j e c t o n t o ( k | :
( 0 )
( k | ( E k
— H 0 ) | ϕ k ) = ( k | ( ǫ V − ∆ k ) | ϕ k )
k
T h e L H S is z e r o s in c e ( k | H 0 | ϕ k ) = ( k | E ( 0 ) | ϕ k ) , a n d f r o m t h e R H S ( k | ( ǫ V − ∆ k ) | ϕ k ) = 0 w e o b t a in :
k
∆ = ǫ ( k | V | ϕ k )
( k | ϕ k )
→ ∆ k
= ǫ ( k | V | ϕ k )
k k
w h e r e w e s e t ( k | ϕ k ) = 1 ( a n o n - c a n o n ic a l n o r m a liz a t io n , a lt h o u g h , a s w e w ill s e e , it is a p p r o x im a t e ly v a lid ) . U s in g t h e e x p a n s io n a b o v e , w e c a n r e p la c e ∆ k b y ǫ E 1 + ǫ 2 E 2 + . . . a n d | ϕ k ) b y it s e x p a n s io n :
k
k
k k
ǫ E 1 + ǫ 2 E 2 + · · · = ǫ ( k | V ( | k ) + ǫ ϕ ( 1 ) ) + ǫ 2 ϕ ( 2 ) ) + . . . )
a n d e q u a t in g t e r m s o f t h e s a m e o r d e r in ǫ w e o b t a in :
E = (
k
n
k | V ϕ
( n
k
− 1 )
)
k
)
T h is is a r e c ip e t o fi n d t h e e n e r g y a t a ll o r d e r s b a s e d o n ly o n t h e k n o w le d g e o f t h e e ig e n s t a t e s o f lo w e r o r d e r s . H o w e v e r , t h e q u e s t io n s t ill r e m a in s : h o w d o w e fi n d ϕ ( n − 1 ) ?
W e c o u ld t h in k o f s o lv in g t h e e q u a t io n :
k
( E ( 0 ) − H 0 ) | ϕ k ) = ( ǫ V − ∆ k ) | ϕ k ) ( ∗ )
k
f o r | ϕ k ) , b y in v e r t in g t h e o p e r a t o r ( E ( 0 ) − H 0 ) a n d a g a in d o in g a n e x p a n s io n o f | ϕ k ) t o e q u a t e t e r m s o f t h e s a m e
o r d e r :
| k ) + ǫ ϕ ( 1 ) ) + · · · = ( E ( 0 ) − H 0 ) − 1 ( ǫ V − ∆ k ) ( | k ) + ǫ ϕ ( 1 ) ) + . . . )
k k k
k
k k
U n f o r t u n a t e ly t h is p r o m is in g a p p r o a c h is n o t c o r r e c t , s in c e t h e o p e r a t o r ( E ( 0 ) − H 0 ) − 1 is n o t a lw a y s w e ll d e fi n e d . S p e c ifi c a lly , t h e r e is a s in g u la r it y f o r ( E ( 0 ) − H 0 ) − 1 | k ) . W h a t w e n e e d is t o m a k e s u r e t h a t ( E ( 0 ) − H 0 ) − 1 is n e v e r
W e t h u s d e fi n e t h e p r o j e c t o r P k = 1 1 − | k ) ( k | =
k | h ) ( h | . T h e n w e c a n e n s u r e t h a t ∀ | ψ ) t h e p r o j e c t e d s t a t e
a p p lie d t o e ig e n s t a t e s o f t h e u n p e r t u r b e d H a m ilt o L n ia n , t h a t is , w e n e e d | ψ k ) = ( ǫ V − ∆ k ) | ϕ k ) / = / | k ) f o r a n y | ϕ k ) .
h / = /
| ψ ) ′ = P k | ψ ) is s u c h t h a t ( k | ψ ′ ) = 0 s in c e t h is is e q u a l t o
( k | P k | ψ ) = ( k | ψ ) − ( k | k ) ( k | ψ ) = 0
k
N o w , u s in g t h e p r o j e c t o r , ( E ( 0 ) − H 0 ) − 1 P k | ψ ) is w e ll d e fi n e d . W e t h e n t a k e t h e e q u a t io n ( ∗ ) a n d m u lt ip ly it b y P k
k
f r o m t h e le f t :
P k ( E ( 0 ) − H 0 ) | ϕ k ) = P k ( ǫ V − ∆ k ) | ϕ k ) .
k k
S in c e P k c o m m u t e s w it h H 0 ( a s | k ) is a n e ig e n s t a t e o f H 0 ) w e h a v e P k ( E ( 0 ) − H 0 ) | ϕ k ) = ( E ( 0 ) − H 0 ) P k | ϕ k ) a n d w e
c a n r e w r it e t h e e q u a t io n a s
k
P k | ϕ k ) = ( E ( 0 ) − H 0 ) − 1 P k ( ǫ V − ∆ k ) | ϕ k )
W e c a n f u r t h e r s im p lif y t h is e x p r e s s io n , n o t in g t h a t P k | ϕ k ) = | ϕ k ) − | k ) ( k | ϕ k ) = | ϕ k ) − | k ) ( s in c e w e a d o p t e d t h e n o r m a liz a t io n ( k | ϕ k ) = 1 ) . F in a lly w e o b t a in :
k
| ϕ k ) = | k ) + ( E ( 0 ) − H 0 ) − 1 P k ( ǫ V − ∆ k ) | ϕ k ) ( ∗ ∗ )
k
0
E 0
|
h / = k |
E k − |
h |
|
N o w u s in g t h e e x p a n s io n |
||
|
( 1 ) 1 2 |
( 1 ) |
T h is e q u a t io n is n o w r e a d y t o b e s o lv e d b y u s in g t h e p e r t u r b a t io n e x p a n s io n . T o s im p lif y t h e e x p r e s s io n , w e d e fi n e t h e o p e r a t o r R k
k
k
0
R = ( E ( 0 ) − H ) − 1 P
= L | h ) ( h |
| k ) + ǫ | ϕ k
) + · · · = | k ) + R k ǫ ( V − E k − ǫ E k − . . . ) ( | k ) + ǫ | ϕ k
) + . . . )
w e c a n s o lv e t e r m b y t e r m t o o b t a in :
k
k
1 s t o r d e r : | ϕ ( 1 ) ) = R k ( V − E 1 ) | k ) = R k ( V − ( k | V | k ) ) | k ) = R k V | k )
| )
k h
( w h e r e w e u s e d t h e e x p r e s s io n f o r t h e fi r s t o r d e r e n e r g y a n d t h e f a c t t h a t R k k = 0 b y d e fi n it io n ) . W e c a n n o w c a lc u la t e t h e s e c o n d o r d e r e n e r g y , s in c e w e k n o w t h e fi r s t o r d e r e ig e n s t a t e :
E 2 = ( k | V | ϕ ( 1 ) ) = ( k | V R V | k ) = ( k | V L
| h ) ( h |
V | k )
o r e x p lic it ly
k k k
h / = k
E 0 − E 0
E k =
2
L
| V |
k h
2
0 0
h / = k
E − E
k h
T h e n t h e s e c o n d o r d e r e ig e n s t a t e is
k
2 n d o r d e r : ϕ 2 = R k V R k V | k )
A . F o r m a l S o l u ti o n
W e c a n a ls o fi n d a m o r e f o r m a l e x p r e s s io n t h a t c a n y ie ld t h e s o lu t io n t o a ll o r d e r s . W e r e w r it e E q . ( * * ) u s in g R k
a n d o b t a in
| ϕ k ) = | k ) + R k ( ǫ V − ∆ k ) | ϕ k ) = R k H 1 | ϕ k )
w h e r e w e d e fi n e d H 1 = ( ǫ V − ∆ k ) . T h e n b y it e r a t io n w e c a n w r it e :
| ϕ k ) = | k ) + R k H 1 ( | k ) + R k H 1 | ϕ k ) ) = | k ) + R k H 1 | k ) + R k H 1 R k H 1 | ϕ k )
a n d in g e n e r a l:
n
| ϕ k ) = | k ) + R k H 1 | k ) + R k H 1 R k H 1 | k ) + · · · + ( R k H 1 )
| k ) + . . .
T h is is j u s t a g e o m e t r ic s e r ie s , w it h f o r m a l s o lu t io n :
| ϕ k ) = ( 1 1 − R k H 1 ) − 1 | k )
B . N o r m a l i za ti o n
In d e r iv in g t h e T IP T w e in t r o d u c e d a n o n - c a n o n ic a l n o r m a liz a t io n ( k | ϕ k ) = 1 , w h ic h im p lie s t h a t t h e p e r t u r b e d s t a t e | ϕ k ) is n o t n o r m a liz e d . W e c a n t h e n d e fi n e a p r o p e r ly n o r m a liz e d s t a t e a s
| ϕ k )
k
k
| ψ k ) = v ( ϕ | ϕ )
s o t h a t ( k | ψ k ) = 1 / v ( ϕ k | ϕ k ) . W e c a n c a lc u la t e p e r t u r b a t iv e ly t h e n o r m a liz a t io n f a c t o r ( ϕ k | ϕ k ) :
( ϕ k | ϕ k ) = ( k + ǫ ϕ k + . . . | k + ǫ ϕ
+ . . . ) = 1 + ǫ ✟ ( k | ϕ k ) + .. + ǫ ( ϕ k | ϕ k ) + .. = 1 + ǫ
1 1 ✟ 1 ✟
h k
2 1 1
2 L | V k h | 2
h = /
k
( E 0 − E 0 ) 2
N o t ic e t h a t t h e s t a t e is c o r r e c t ly n o r m a liz e d u p t o t h e s e c o n d o r d e r in ǫ .
C . A n ti - c r o s s i n g
k h
C o n s id e r t w o le v e ls , h a n d k w it h e n e r g ie s E 0 a n d E 0 a n d a s s u m e t h a t w e a p p ly a p e r t u r b a t io n V w h ic h c o n n e c t s o n ly t h e s e t w o s t a t e s ( t h a t is , V is s u c h t h a t ( l | V | j ) = 0 a n d it is d iff e r e n t t h a n z e r o o n ly f o r t h e t r a n s it io n f r o m h
t o k : ( h | V | k ) = / 0 .)
If t h e p e r t u r b a t io n is s m a ll, w e c a n a s k w h a t a r e t h e p e r t u r b e d s t a t e e n e r g ie s .
( 2 ) L | V k j | 2 | V k h | 2
T h e fi r s t o r d e r is z e r o b y t h e c h o ic e o f V , t h e n w e c a n c a lc u la t e t h e s e c o n d o r d e r :
a n d s im ila r ly
E k =
E 0 E 0 = E 0 E 0
− −
j / = k k j k h
E h
=
E 0 E 0 = E 0 E 0 = − E k
j / = h h j h k
.
( 2 ) L | V h j | 2 | V k h | 2 ( 2 )
− −
k
h
k h
T h is o p p o s it e e n e r g y s h if t w ill b e m o r e im p o r t a n t ( m o r e n o t ic e a b le ) w h e n t h e e n e r g ie s o f t h e t w o le v e ls E 0 a n d E 0 a r e c lo s e t o e a c h o t h e r . In d e e d , in t h e a b s e n c e o f t h e p e r t u r b a t io n , t h e t w o e n e r g y le v e ls w o u ld “ c r o s s ” w h e n E 0 = E 0 . If w e a d d t h e p e r t u r b a t io n , h o w e v e r , t h e t w o le v e ls a r e r e p e lle d w it h o p p o s it e e n e r g y s h if t s . W e d e s c r ib e w h a t is h a p p e n in g a s a n “ a n t i- c r o s s in g ” o f t h e le v e ls : e v e n a s t h e le v e ls b e c o m e c o n n e c t e d b y a n in t e r a c t io n , t h e le v e ls n e v e r m e e t ( n e v e r h a v e t h e s a m e e n e r g y ) s in c e e a c h le v e l g e t s s h if t e d b y t h e s a m e a m o u n t in o p p o s it e d ir e c t io n s .
D . E x a m p l e : T L S e n e r g y s p l i tti n g f r o m p e r tu r b a ti o n
k
C o n s id e r t h e H a m ilt o n ia n H = ω σ z + ǫ Ω σ x . F o r ǫ = 0 t h e e ig e n s t a t e s a r e | k ) = { | 0 ) , | 1 ) } a n d e ig e n v a lu e s E 0 = ± ω .
W e a ls o k n o w h o w t o s o lv e e x a c t ly t h is s im p le p r o b le m b y d ia g o n a liz in g t h e e n t ir e m a t r ix :
E 1 , 2 = ± v ω 2 + ǫ 2 Ω 2 ,
| ϕ 1 ) = c o s ( ϑ / 2 ) | 0 ) + s in ( ϑ / 2 ) | 1 ) , | ϕ 2 ) = c o s ( ϑ / 2 ) | 1 ) − s in ( ϑ / 2 ) | 0 ) w it h ϑ = a r c t a n ( ǫ Ω / ω ) F o r ǫ ≪ 1 w e c a n e x p a n d in s e r ie s t h e s e r e s u lt s t o fi n d :
E 1 , 2 ≈ ± ( ω +
ǫ 2 Ω 2
2 ω
+ . . . )
ϑ ǫ Ω ϑ ǫ Ω
| ϕ 1 ) ≈ | 0 ) + 2 | 1 ) = | 0 ) + 2 ω | 1 ) | ϕ 2 ) ≈ | 1 ) − 2 | 0 ) = | 1 ) − 2 ω | 0 )
A s a n e x e r c is e , w e c a n fi n d a s w e ll t h e r e s u lt s o f T IP T . F ir s t w e fi n d t h a t t h e fi r s t o r d e r e n e r g y s h if t is z e r o , s in c e
k
E 1 = ( k | V | k ) = ( 0 | ( Ω σ x ) | 0 ) = 0 ( a n d s a m e f o r ( 1 | ( Ω σ x ) | 1 ) ) . T h e n w e c a n c a lc u la t e t h e fi r s t o r d e r e ig e n s t a t e :
1
1
0
1
z
x
2 ω
x
2 ω
ϕ 1 = | 0 ) + ( E 0 − H ) − 1 P V | 0 ) = | 0 ) + [ ω ( 1 1 − σ ) ] − 1 | 1 ) ( 1 | ǫ Ω σ | 0 ) = | 0 ) + 1 ǫ Ω | 1 ) ( 1 | σ | 0 ) = | 0 ) + ǫ Ω | 1 )
2 2
s im ila r ly , w e fi n d ϕ 1 = | 1 ) − ǫ Ω | 0 ) . F in a lly , t h e s e c o n d o r d e r e n e r g y s h if t is E 2 = | V 1 2 | = ( ǫ Ω )
in a g r e e m e n t
1 2
2 2 ω
w it h t h e r e s u lt f r o m t h e s e r ie s e x p a n s io n .
1 E 0 − E 0 2 ω
W e c a n a ls o lo o k a t t h e le v e l a n t i- c r o s s in g : If w e v a r y t h e e n e r g y ω a r o u n d z e r o , t h e t w o e n e r g y le v e ls c r o s s e a c h o t h e r .
Eigenvalues
Ω
F i g . 1 8 : Le v e l a n t i c r o s s i n g : E i g e n v a l u e s o f t h e H a mi l t o n i a n H = ω σ z + ǫ Ω σ x a s a f u n c t i o n o f ω . D a s h e d l i n e s : Ω = 0 . R e d l i n e s : Ω = / 0 s h o w i n g t h e a n t i c r o s s i n g .
1 1 . 1 . 2 D e g e n e r a t e c a s e
H
If t h e r e a r e d e g e n e r a t e ( o r q u a s i- d e g e n e r a t e ) e ig e n v a lu e s o f t h e u n p e r t u r b e d H a m ilt o n ia n 0 , t h e e x p a n s io n u s e d a b o v e is n o lo n g e r v a lid . T h e r e a r e t w o p r o b le m s :
| ) →
| ) | )
1 . If k ′ , k ′ ′ , ... h a v e t h e s a m e e ig e n v a lu e , w e c a n c h o o s e a n y c o m b in a t io n o f t h e m a s t h e u n p e r t u r b e d e ig e n k e t . B u t t h e n , if w e w e r e t o fi n d t h e p e r t u r b e d e ig e n k e t ψ k , t o w h ic h s t a t e w o u ld t h is g o t o w h e n ǫ 0 ?
2 . T h e t e r m R k = P k c a n b e s in g u la r f o r t h e d e g e n e r a t e e ig e n v a lu e s .
k
E ( 0 ) − H 0
H d . W e c a n t h e n d e fi n e t h e p r o j e c t o r s Q d =
i
d
k ∈ H
| k i ) ( k i | a n d P d = 1 1 − Q d . T h e s e p r o j e c t o r s a ls o d e fi n e s u b s p a c e s
A s s u m e t h e r e is a d - f o ld d e g e n e r a c y o f t h e e i L g e n v a lu e E d , w it h t h e u n p e r t u r b e d e ig e n k e t s { | k i ) } f o r m in g a s u b s p a c e
o f t h e t o t a l H ilb e r t s p a c e H t h a t w e w ill c a ll H d ( s p a n n e d b y Q d ) a n d H d ¯ ( s p a n n e d b y P d ) . N o t ic e t h a t b e c a u s e o f t h e ir n a t u r e o f p r o j e c t o r s , w e h a v e t h e f o llo w in g id e n t it ie s :
d d
P 2 = P d , Q 2 = Q d , P d Q d = Q d P d = 0 a n d P d + Q d = 1 1 .
W e t h e n r e w r it e t h e e ig e n v a lu e e q u a t io n a s :
( H 0 + ǫ V ) | ϕ k ) = E k | ϕ k ) → H 0 ( Q d + P d ) | ϕ k ) + ǫ V ( Q d + P d ) | ϕ k ) = E k ( Q d + P d ) | ϕ k )
→ ( Q d + P d ) H 0 | ϕ k ) + ǫ V ( Q d + P d ) | ϕ k ) = E k ( Q d + P d ) | ϕ k )
H H
w h e r e w e u s e d t h e f a c t t h a t [ 0 , Q d ] = [ 0 , P d ] = 0 s in c e t h e p r o j e c t o r s a r e d ia g o n a l in t h e H a m ilt o n ia n b a s is . W e t h e n m u lt ip ly f r o m t h e le f t b y Q d a n d P d , o b t a in in g 2 e q u a t io n s :
1 . P d × [( Q d + P d ) H 0 | ϕ k ) + ǫ V ( Q d + P d ) | ϕ k ) ] = P d × ( E k ( Q d + P d ) | ϕ k ) )
→ H 0 P d | ϕ k ) + ǫ P d V ( Q d + P d ) | ϕ k ) = E k P d | ϕ k )
2 . Q d × [( Q d + P d ) H 0 | ϕ k ) + ǫ V ( Q d + P d ) | ϕ k ) ] = Q d × ( E k ( Q d + P d ) | ϕ k ) )
→ H 0 Q d | ϕ k ) + ǫ Q d V ( Q d + P d ) | ϕ k ) = E k Q d | ϕ k )
a n d w e s im p lif y t h e n o t a t io n b y s e t t in g | ψ k ) = P d | ϕ k ) a n d | χ k ) = Q d | ϕ k )
H 0 | ψ k ) + ǫ P d V ( | χ k ) + | ψ k ) ) = E k | ψ k ) H 0 | χ k ) + ǫ Q d V ( | χ k ) + | ψ k ) ) = E k | χ k )
w h ic h g iv e s a s e t o f c o u p le d e q u a t io n s in | ψ k ) a n d | χ k ) :
1 . ǫ P d V | χ k ) = ( E k − H 0 − ǫ P d V P d ) | ψ k )
2 . ǫ Q d V | ψ k ) = ( E k − H 0 − ǫ Q d V Q d ) | χ k )
N o w ( E k − H 0 − ǫ P d V P d ) − 1 is fi n a lly w e ll d e fi n e d in t h e P d s u b s p a c e , s o t h a t w e c a n s o lv e f o r | ψ k ) f r o m ( 1 . ) :
| ψ k ) = ǫ P d ( E k − H 0 − ǫ P d V P d ) − 1 P d V | χ k )
a n d b y in s e r t in g t h is in ( 2 . ) w e fi n d
( E k − H 0 − ǫ Q d V Q d ) | χ k ) = ǫ 2 Q d V P d ( E k − H 0 − ǫ P d V P d ) − 1 P d V | χ k ) .
If w e k e e p o n ly t h e fi r s t o r d e r in ǫ in t h is e q u a t io n w e h a v e :
[( E k − E d ) − ǫ Q d V Q d ] | χ k ) = 0 w h ic h is a n e q u a t io n d e fi n e d o n t h e s u b s p a c e H d o n ly .
W e n o w c a ll U d = Q d V Q d t h e p e r t u r b a t io n H a m ilt o n ia n in t h e H d s p a c e a n d ∆ k = ( E k − E d ) 1 1 d , t o g e t :
( ∆ k − ǫ U d ) | χ k ) = 0
) ) )
O f t e n it is p o s s ib le t o j u s t d ia g o n a liz e U d ( if t h e d e g e n e r a t e s u b s p a c e is s m a ll e n o u g h , f o r e x a m p le f o r a s im p le d o u b le d e g e n e r a c y ) a n d n o t ic e t h a t o f c o u r s e ∆ k is a lr e a d y d ia g o n a l. O t h e r w is e o n e c a n a p p ly p e r t u r b a t io n t h e o r y t o t h is
i i i
) )
s u b s p a c e . T h e n w e w ill h a v e f o u n d s o m e ( e x a c t o r a p p r o x im a t e ) e ig e n s t a t e s o f U d , k ( 0 ) , s .t . U d k ( 0 ) = u i k ( 0 )
i i
a n d H 0 k ( 0 ) = E d k ( 0 ) , ∀ i . T h u s , t h is s t e p s e t s w h a t u n p e r t u r b e d e ig e n s t a t e s w e s h o u ld c h o o s e in t h e d e g e n e r a t e s u b s p a c e , h e n c e s o lv in g t h e fi r s t is s u e o f d e g e n e r a t e p e r t u r b a t io n t h e o r y .
W e n o w w a n t t o lo o k a t t e r m s ∝ ǫ 2 in
2
k
0
d
k
d
d
k
0
✘ ǫ P d V P d )
d
k
( E − H − ǫ U ) | χ ) = ǫ Q V P ( E − H − ✘ ✘ ✘ − 1 P V | χ )
w h e r e w e n e g le c t e d t e r m s h ig h e r t h a n s e c o n d o r d e r . R e a r r a n g in g t h e t e r m s , w e h a v e :
E k | χ k ) = [ H 0 + ǫ U d + ǫ 2 Q d V P d ( E k − H 0 ) − 1 P d V ] | χ k ) → ( H ˜ 0 + V ˜ ) | χ k ) = E k | χ k )
w it h
H ˜ 0 = H 0 + ǫ U d V ˜ = ǫ Q d V P d ( E k − H 0 ) − 1 P d V Q d
k
If t h e r e a r e n o d e g e n e r a c ie s le f t in H ˜ 0 , w e c a n s o lv e t h is p r o b le m b y T IP T a n d fi n d χ ( n ) ) .
F o r e x a m p le , t o fi r s t o r d e r , w e h a v e
j i k ( 0 )
k , i
χ ( 1 ) )
L k ( 0 ) | V ˜ | k ( 0 ) ) )
=
j / / = i
ǫ ( u i − u j )
j
a n d u s in g t h e e x p lic it f o r m o f t h e m a t r ix e le m e n t V ˜ i j = ( k ( 0 ) | V ˜ | k ( 0 ) ) ,
j i
) L k ( 0 ) V | h ) ( h | V k ( 0 ) )
V ˜ = k ( 0 ) ǫ 2 V P ( E 0 − H ) − 1 P V k ( 0 )
—
= ǫ 2
j i
w e o b t a in :
i j j
d d 0 d i
E ( 0 ) E ( 0 )
h ∈ / H d d h
χ ( 1 ) )
= ǫ L ( k j | V | h ) ( h | V | k i )
k ( 0 ) )
k , i
( 0 ) ( 0 )
i d h
j = / /
( u i − u j ) E ( 0 ) − E ( 0 ) j
F in a lly , w e n e e d t o a d d | χ ) a n d | ψ ) t o fi n d t h e t o t a l v e c t o r :
E ( 0 ) − E ( 0 )
) L L D k ( 0 ) V | h )
) ( h | V k ( 0 ) )
k
E 0 − E ( 0 ) ( u i − u j ) j
d h j / = i d
ϕ ( 1 ) =
( h | V | k i ) | h ) + ǫ
h ∈ / H d
j k ( 0 ) i
h
( 0 )
k
j
ϕ ( 1 )
=
i
E 0 − E ( 0 )
| h ) + ǫ
j
( u i − u j )
k ( 0 )
) L ( h | V | k )
h ∈ / H d
d h
L ( k | V | h ) )
j = /
i
E x a m p l e : D e g e n e r a te T L S
C o n s id e r t h e H a m ilt o n ia n H = ω σ z + ǫ Ω σ x . W e a lr e a d y s o lv e d t h is H a m ilt o n ia n , b o t h d ir e c t ly a n d w it h T IP T . N o w c o n s id e r t h e c a s e ω ≈ 0 a n d a s lig h t ly m o d ifi e d H a m ilt o n ia n :
H = ( ω 0 + ω ) | 0 ) ( 0 | + ( ω 0 − ω ) | 1 ) ( 1 | + ǫ Ω σ x = ω 0 1 1 + ω σ z + ǫ Ω σ x .
2
W e c o u ld s o lv e e x a c t ly t h e s y s t e m f o r ω = 0 , s im p ly fi n d in g E 0 , 1 = ω 0 ± ǫ Ω a n d | ϕ ) 0 , 1 = |± ) = √ 1 ( | 0 ) ± | 1 ) ) . W e
c a n a ls o a p p ly T IP T .
| ) | )
H o w e v e r t h e t w o e ig e n s t a t e s 0 , 1 a r e ( q u a s i- ) d e g e n e r a t e t h u s w e n e e d t o a p p ly d e g e n e r a t e p e r t u r b a t io n t h e o r y . In p a r t ic u la r , a n y b a s is a r is in g f r o m a r o t a t io n o f t h e s e t w o b a s is s t a t e s c o u ld b e a p r io r i a g o o d b a s is , s o w e n e e d fi r s t t o o b t a in t h e c o r r e c t z e r o t h o r d e r e ig e n v e c t o r s . In t h is v e r y s im p le c a s e w e h a v e H d = H ( t h e t o t a l H ilb e r t s p a c e ) a n d H d ¯ = 0 , o r in o t h e r w o r d s , Q d = 1 1 , P d = 0 . W e fi r s t n e e d t o d e fi n e a n e q u a t io n in t h e d e g e n e r a t e s u b s p a c e o n ly :
( ∆ k − ǫ U d ) | χ k ) = 0
w h e r e U d = Q d V Q d . H e r e w e h a v e : U d = V = Ω σ x . T h u s w e o b t a in t h e c o r r e c t z e r o t h o r d e r e ig e n v e c t o r s f r o m d ia g o n a liz in g t h is H a m ilt o n ia n . N o t s u r p r is in g ly , t h e y a r e :
ϕ
( 0 ) ) 1
0 , 1
= |± ) = √ 2 ( | 0 ) ± | 1 ) ) .
w it h e ig e n v a lu e s : E 0 , 1 = ω 0 ± ǫ Ω . W e c a n n o w c o n s id e r h ig h e r o r d e r s , f r o m t h e e q u a t io n :
( H ˜ 0 + V ˜ ) | χ k ) = E k | χ k )
w it h H ˜ 0 = ω 0 1 1 + ǫ Ω σ x a n d V ˜ = 0 . T h u s in t h is c a s e , t h e r e a r e n o h ig h e r o r d e r s a n d w e s o lv e d t h e p r o b le m .
E x a m p l e : S p i n - 1 s y s te m
W e c o n s id e r a s p in - 1 s y s t e m ( t h a t is , a s p in s y s t e m w it h S = 1 d e fi n e d in a 3 - d im e n s io n a l H ilb e r t s p a c e ) . T h e m a t r ix r e p r e s e n t a t io n f o r 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 s S x a n d S z in t h is H ilb e r t s p a c e a r e :
1
S x = √ 2
0 1 0
, S z =
1 0 0
0 0 0
1 0 1
0 1 0 0 0 − 1
2
T h e H a m ilt o n ia n o f t h e s y s t e m is H = H 0 + ǫ V w it h
G iv e n t h a t
H 0 = ∆ S z ; V = S x + S z
S z = 0 0 0
0 0 1
2 1 0 0
T h e m a t r ix r e p r e s e n t a t io n o f t h e t o t a l H a m ilt o n ia n is :
√
∆ + ǫ ǫ 0
ǫ
2
2 2
0 ǫ
2
H = √
0 √ ǫ
√
∆ − ǫ
P o s s ib le e ig e n s t a t e s o f t h e u n p e r t u r b e d H a m ilt o n ia n a r e | + 1 ) , | 0 ) , | − 1 ) :
1 0 0
| + 1 ) = 0 , | 0 ) = 1 , |− 1 ) = 0 ,
0 0 1
| ) | − )
w it h e n e r g ie s + ∆ , 0 , + ∆ r e s p e c t iv e ly . H o w e v e r , a n y c o m b in a t io n o f + 1 a n d 1 is a v a lid e ig e n s t a t e , f o r e x a m p le w e c o u ld h a v e c h o s e n :
1 1
1 1
1
2
| + 1 ) = √ 2 0 ,
|− 1 ) = √ 0
− 1
T h is is t h e c a s e b e c a u s e t h e t w o e ig e n s t a t e s a r e d e g e n e r a t e . S o h o w d o w e c h o o s e w h ic h a r e t h e c o r r e c t e ig e n s t a t e s t o z e r o t h o r d e r 3 7 ? W e n e e d t o fi r s t c o n s id e r t h e t o t a l H a m ilt o n ia n in t h e d e g e n e r a t e s u b s p a c e .
z
T h e d e g e n e r a t e s u b s p a c e is t h e s u b s p a c e o f t h e t o t a l H ilb e r t s p a c e H s p a n n e d b y t h e b a s is | + 1 ) , |− 1 ) ; w e c a n c a ll t h is s u b s p a c e H Q . W e c a n o b t a in t h e H a m ilt o n ia n in t h is s u b s p a c e b y u s in g t h e p r o j e c t o r o p e r a t o r Q : H Q = Q H Q , w it h Q = | + 1 ) ( + 1 | + |− 1 ) ( − 1 | = S 2 . T h e n :
2 2
H Q = Q ( ∆ S z + ǫ ( S z + S x ) ) Q = ∆ S z + ǫ S z
( N o t ic e t h is c a n b e o b t a in e d b y d ir e c t m a t r ix m u lt ip lic a t io n o r m u lt ip ly in g t h e o p e r a t o r s ) . In m a t r ix f o r m :
H Q =
∆ + ǫ 0 0
0 0 0
0 0 ∆ − ǫ
→ H Q =
∆ + ǫ
(
0
0
)
∆ − ǫ
| ) |− )
w h e r e in t h e la s t lin e I r e p r e s e n t e d t h e m a t r ix in t h e 2 - d im e n s io n a l s u b p s a c e H Q . W e c a n n o w e a s ily s e e t h a t t h e c o r r e c t e ig e n v e c t o r s f o r t h e u n p e r t u r b e d H a m ilt o n ia n w e r e t h e o r ig in a l + 1 a n d 1 a f t e r a ll. F r o m t h e H a m ilt o n ia n in t h e H Q s u b s p a c e w e c a n a ls o c a lc u la t e t h e fi r s t o r d e r c o r r e c t io n t o t h e e n e r g y f o r t h e s t a t e s in t h e d e g e n e r a t e
s u b s p a c e . T h e s e a r e j u s t E ( 1 ) − E ( 0 ) = + ǫ a n d E ( 1 ) − E ( 0 ) = − ǫ .
+ 1 + 1
− 1 − 1
( 1 )
N o w w e w a n t t o c a lc u la t e t h e fi r s t o r d e r c o r r e c t io n t o t h e e ig e n s t a t e s |± 1 ) . T h is w ill h a v e t w o c o n t r ib u t io n s : | ψ ) ± 1 =
± 1
± 1
Q | ψ ) ( 1 ) + P | ψ ) ( 1 ) w h e r e P = 1 1 − Q = | 0 ) ( 0 | is t h e c o m p le m e n t a r y p r o j e c t o r t o Q . W e fi r s t c a lc u la t e t h e fi r s t t e r m
in t h e f o llo w in g w a y . W e r e d e fi n e a n u n p e r t u r b e d H a m ilt o n ia n in t h e s u b s p a c e H Q :
z
H ˜ 0 = H Q = Q H Q = ∆ S 2 + ǫ S z
a n d t h e p e r t u r b a t io n in t h e s a m e s u b s p a c e is ( f o llo w in g S a k u r a i) :
V ˜ = V
= ǫ Q ( V P ( ∆ − H ) − 1 P V ) Q = ǫ Q � ( S
+ S ) | 0 ) ( 0 | ( ∆ | 0 ) ( 0 | ) − 1 | 0 ) ( 0 | ( S
ǫ
�
+ S ) Q = Q S P S Q
Q
In m a t r ix f o r m :
0
V Q = 2 ∆
ǫ
z x
0 0 0
1 0 1
→
V Q = 2 ∆
1 0 1
z x
= 2 ∆ ( 1 1 + σ x )
ǫ ( 1 1 ) ǫ
1 1
∆ x x
N o w t h e p e r t u r b e d e ig e n s t a t e s c a n b e c a lc u la t e d a s :
k
Q | ψ ) ( 1 ) = | k ) + ǫ
h V Q k
L | )
E − E
( | | ) h
( 1 ) ( 1 )
h ∈ H Q = k k h
3 7 H e r e b y c o r r e c t e i g e n s t a t e s I m e a n s t h e e i g e n s t a t e s t o w h i c h t h e e i g e n s t a t e s o f t h e t o t a l H a mi l t o n i a n w i l l t e n d t o w h e n
ǫ → 0
In o u r c a s e :
Q | ψ ) ( 1 ) = | + 1 ) + ǫ ( − 1 | V Q | + 1 ) |− 1 ) = | + 1 ) + ǫ ǫ ( − 1 | ( 1 1 + σ x ) | + 1 ) |− 1 ) = | + 1 ) + ǫ |− 1 ) ,
E − E
+ 1 ( 1 ) ( 1 ) 2 ∆ 2 ǫ 4 ∆
+ 1 − 1
( 1 ) ǫ
− 1
4 ∆
Q ψ = |− 1 ) − | 1 )
In o r d e r t o c a lc u la t e P ψ ( 1 ) w e c a n j u s t u s e t h e u s u a l f o r m u la f o r n o n - d e g e n e r a t e p e r t u r b a t io n t h e o r y , b u t s u m m in g
± 1
o n ly o v e r t h e s t a t e s o u t s id e H Q . H e r e t h e r e ’s o n ly o n e o f t h e m | 0 ) , s o :
± 1 E ± 1 − E 0 2 ∆
P | ψ ) ( 1 ) = ǫ ( 0 | V |± 1 ) | 0 ) = √ ǫ
| 0 )
F in a lly , t h e e ig e n s t a t e s t o fi r s t o r d e r a r e :
( 1 ) ǫ ǫ
| ψ + 1 ) = | + 1 ) + 4 ∆ |− 1 ) + √ 2 ∆ | 0 )
a n d
( 1 ) ǫ ǫ
| ψ − 1 ) = |− 1 ) − 4 ∆ | 1 ) + √ 2 ∆ | 0 )
±
T h e e n e r g y s h if t t o s e c o n d o r d e r is c a lc u la t e d f r o m ∆ ( 2 ) = L
| ( h | V |± 1 ) | 2
:
∆ − E ( 0 )
h ∈ / H Q h
( 2 ) |( 0 | V | + 1 ) | 2 ǫ 2
a n d
∆ + 1 = ∆ = 2 ∆
∆ = =
( 2 ) |( 0 | V |− 1 ) | 2 ǫ 2
− 1 ∆ 2 ∆
T o c a lc u la t e t h e p e r t u r b a t io n e x p a n s io n f o r | 0 ) a n d it s e n e r g y , w e u s e n o n - d e g e n e r a t e p e r t u r b a t io n t h e o r y , t o fi n d :
+ 1
( 1 ) ( ( + 1 | V | 0 ) ( − 1 | V | 0 ) ) ǫ | + 1 ) + |− 1 )
∆ ( 1 ) = ( 0 | V | 0 ) = 0
0 ∆
a n d ∆ ( 2 ) = − ǫ 2 .
| ψ + 1 ) = | 0 ) + ǫ
− ∆ | + 1 ) +
− ∆ |− 1 ) = − ∆ √ 2
1 1 . 1 . 3 T h e S t a r k e ff e c t
W e a n a ly z e t h e in t e r a c t io n o f a h y d r o g e n a t o m w it h a ( c la s s ic a l) e le c t r ic fi e ld , t r e a t e d a s a p e r t u r b a t io n 3 8 . D e p e n d in g o n t h e h y d r o g e n ’s s t a t e , w e w ill n e e d t o u s e T IP T o r d e g e n e r a t e T IP T , t o fi n d e it h e r a q u a d r a t ic o r lin e a r ( in t h e fi e ld ) s h if t o f t h e e n e r g y . T h e s h if t in e n e r g y is u s u a lly c a lle d S t a r k s h if t o r S t a r k e ff e c t a n d it is t h e e le c t r ic a n a lo g u e o f t h e Z e e m a n e ff e c t , w h e r e t h e e n e r g y le v e l is s p lit in t o s e v e r a l c o m p o n e n t s d u e t o t h e p r e s e n c e o f a m a g n e t ic fi e ld . M e a s u r e m e n t s o f t h e S t a r k e ff e c t u n d e r h ig h fi e ld s t r e n g t h s c o n fi r m e d t h e c o r r e c t n e s s o f t h e q u a n t u m t h e o r y o v e r t h e B o h r m o d e l.
| |
S u p p o s e t h a t a h y d r o g e n a t o m is s u b j e c t t o a u n if o r m e x t e r n a l e le c t r ic fi e ld , o f m a g n it u d e E , d ir e c t e d a lo n g t h e
z - a x is . T h e H a m ilt o n ia n o f t h e s y s t e m c a n b e s p lit in t o t w o p a r t s . N a m e ly , t h e u n p e r t u r b e d H a m ilt o n ia n ,
e
p 2 H 0 = 2 m
3 8 T h i s s e c t i o n f o l l o w s P r o f . F i t z p a t r i c k o n l i n e l e c t u r e s
e 2
—
,
4 π ǫ 0 r
a n d t h e p e r t u r b in g H a m ilt o n ia n
H 1 = e | E | z .
| )
N o t e t h a t t h e e le c t r o n s p in is ir r e le v a n t t o t h is p r o b le m ( s in c e t h e s p in o p e r a t o r s a ll c o m m u t e w it h H 1 ) , s o w e c a n ig n o r e t h e s p in d e g r e e s o f f r e e d o m o f t h e s y s t e m . H e n c e , t h e e n e r g y e ig e n s t a t e s o f t h e u n p e r t u r b e d H a m ilt o n ia n a r e c h a r a c t e r iz e d b y t h r e e q u a n t u m n u m b e r s – t h e r a d ia l q u a n t u m n u m b e r n , a n d t h e t w o a n g u la r q u a n t u m n u m b e r s l a n d m . L e t u s d e n o t e t h e s e s t a t e s a s t h e n l m , a n d le t t h e ir c o r r e s p o n d in g e n e r g y e ig e n v a lu e s b e t h e E n lm . W e u s e T IP T t o c a lc u la t e t h e e n e r g y s h if t t o fi r s t a n d s e c o n d o r d e r .
A . T h e q u a d r a ti c S ta r k e ff e c t
.
W e fi r s t w a n t t o s t u d y t h e p r o b le m u s in g n o n - d e g e n e r a t e p e r t u r b a t io n t h e o r y , t h u s a s s u m in g t h a t t h e u n p e r t u r b e d s t a t e s a r e n o n - d e g e n e r a t e . A c c o r d in g t o T IP T , t h e c h a n g e in e n e r g y o f t h e e ig e n s t a t e c h a r a c t e r iz e d b y t h e q u a n t u m n u m b e r s n , l , m in t h e p r e s e n c e o f a s m a ll e le c t r ic fi e ld is g iv e n b y
2
∆ E n lm = e | E | ( n , l , m | z | n , l , m ) + e
| E | 2
L
n ′ , l ′ , m ′ = n , l, m
|( n , l , m | z | n ′ , l ′ , m ′ ) | 2 E n lm − E n ′ l ′ m ′
( | | )
T h is e n e r g y - s h if t is k n o w n a s t h e S t a r k e ff e c t . T h e s u m o n t h e r ig h t - h a n d s id e o f t h e a b o v e e q u a t io n s e e m s v e r y c o m p lic a t e d . H o w e v e r , it t u r n s o u t t h a t m o s t o f t h e t e r m s in t h is s u m a r e z e r o . T h is f o llo w s b e c a u s e t h e m a t r ix e le m e n t s n , l , m z n ′ , l ′ , m ′ a r e z e r o f o r v ir t u a lly a ll c h o ic e s o f t h e t w o s e t s o f q u a n t u m n u m b e r n , l , m a n d n ′ , l ′ , m ′ . L e t u s t r y t o fi n d a s e t o f r u le s w h ic h d e t e r m in e w h e n t h e s e m a t r ix e le m e n t s a r e n o n - z e r o . T h e s e r u le s a r e u s u a lly r e f e r r e d t o a s t h e s e le c t io n r u le s f o r t h e p r o b le m in h a n d .
N o w , s in c e [ L z , z ] = 0 , it f o llo w s t h a t
( n , l , m | [ L z , z ] | n ′ , l ′ , m ′ ) = ( n , l , m | L z z − z L z | n ′ , l ′ , m ′ ) = l ( m − m ′ ) ( n , l , m | z | n ′ , l ′ , m ′ ) = 0 .
H e n c e , o n e o f t h e s e le c t io n r u le s is t h a t t h e m a t r ix e le m e n t ( n , l , m | z | n ′ , l ′ , m ′ ) is z e r o u n le s s
m ′ = m .
T h e s e le c t io n r u le f o r l c a n b e s im ila r ly c a lc u la t e d f r o m p r o p e r t ie s o f t h e t o t a l a n g u la r m o m e n t u m L 2 a n d it s c o m m u t a t o r w it h z . W e o b t a in t h a t t h e m a t r ix e le m e n t is z e r o u n le s s
l ′ = l ± 1 .
2
.
A p p lic a t io n o f t h e s e s e le c t io n r u le s t o t h e p e r t u r b a t io n e q u a t io n s h o w s t h a t t h e lin e a r ( fi r s t o r d e r ) t e r m is z e r o , w h ile t h e s e c o n d o r d e r t e r m y ie ld s
∆ E n lm = e 2
| E |
L |( n , l , m | z | n ′ , l ′ , m ) | 2
n ′ , l ′ = l ± 1
E n lm − E n ′ l ′ m
O n ly t h o s e t e r m s w h ic h v a r y q u a d r a t ic a lly w it h t h e fi e ld - s t r e n g t h h a v e s u r v iv e d . H e n c e , t h is t y p e o f e n e r g y - s h if t o f a n a t o m ic s t a t e in t h e p r e s e n c e o f a s m a ll e le c t r ic fi e ld is k n o w n a s t h e q u a d r a t ic S t a r k e ff e c t .
N o w , t h e e le c t r ic p o la r iz a b ilit y o f a n a t o m is d e fi n e d in t e r m s o f t h e e n e r g y - s h if t o f t h e a t o m ic s t a t e a s f o llo w s :
2
∆ E = − 1 α | E | 2 .
H e n c e , w e c a n w r it e
α n lm = 2 e 2
L |( n , l , m | z | n ′ , l ′ , m ) | 2
.
n ′ , l ′ = l ± 1
E n ′ l ′ m − E n lm
A lt h o u g h w r it t e n f o r a g e n e r a l s t a t e , t h e e q u a t io n s a b o v e a s s u m e t h e r e is n o d e g e n e r a c y o f t h e u n p e r t u r b e d e ig e n v a lu e s . H o w e v e r , t h e u n p e r t u r b e d e ig e n s t a t e s o f a h y d r o g e n a t o m h a v e e n e r g ie s w h ic h o n ly d e p e n d o n t h e r a d ia l q u a n t u m n u m b e r n , t h u s t h e y h a v e h ig h ( a n d in c r e a s in g w it h n ) o r d e r o f d e g e n e r a c y . W e c a n t h e n o n ly a p p ly t h e a b o v e r e s u lt s t o t h e n = 1 e ig e n s t a t e ( s in c e f o r n ≥ 1 t h e r e w ill b e c o u p lin g t o d e g e n e r a t e e ig e n s t a t e s w it h t h e s a m e
v a lu e o f n b u t d iff e r e n t v a lu e s o f l ) . T h u s , a c c o r d in g t o n o n - d e g e n e r a t e p e r t u r b a t io n t h e o r y , t h e p o la r iz a b ilit y o f t h e g r o u n d - s t a t e ( i.e ., n = 1 ) o f a h y d r o g e n a t o m is g iv e n b y
α = 2 e .
2 L |( 1 , 0 , 0 | z | n , 1 , 0 ) | 2
n > 1
E n − E 1
H e r e , w e h a v e m a d e u s e o f t h e f a c t t h a t E n 1 0 = E n 0 0 = E n .
T h e s u m in t h e a b o v e e x p r e s s io n c a n b e e v a lu a t e d a p p r o x im a t e ly b y n o t in g t h a t
0
e 2
w h e r e a 0
E n = − 8 π ǫ
m e 2
= 4 π ǫ 0 k 2 is t h e B o h r r a d iu s . H e n c e , w e c a n w r it e
e
a n 2 ,
0
0
0
3 e 2
w h ic h im p lie s t h a t t h e p o la r iz a b ilit y is
E n − E 1 ≥ E 2 − E 1 = 4 8 π ǫ a ,
1 6
α < 4 π ǫ 0 a 0
3
|( 1 , 0 , 0 | z | n , 1 , 0 ) | 2 .
L
n > 1
3
H o w e v e r , t h a n k s t o t h e s e le c t io n r u le s w e h a v e , L n > 1 |( 1 , 0 , 0 | z | n , 1 , 0 ) | 2 = ( 1 , 0 , 0 | z 2 | 1 , 0 , 0 ) = 1 ( 1 , 0 , 0 | r 2 | 1 , 0 , 0 ) ,
w h e r e w e h a v e m a d e u s e o f t h e f a c t t h e t h e g r o u n d - s t a t e o f h y d r o g e n is s p h e r ic a lly s y m m e t r ic . F in a lly , f r o m
0
( 1 , 0 , 0 | r 2 | 1 , 0 , 0 ) = 3 a 2 w e c o n c lu d e t h a t
1 6
0
α < 4 π ǫ 0
3
a 3 ≃ 5 . 3 4 π ǫ 0
0
a 3 .
T h e e x a c t r e s u lt ( w h ic h c a n b e o b t a in e d b y s o lv in g S c h r d in g e r ’s e q u a t io n in p a r a b o lic c o o r d in a t e s ) is
9
0
α = 4 π ǫ 0
2
a 3 = 4 . 5 4 π ǫ 0
0
a 3 .
B . T h e l i n e a r S ta r k e ff e c t
≥
− −
W e n o w e x a m in e t h e e ff e c t o f a n e le c t r ic fi e ld o n t h e e x c it e d e n e r g y le v e ls n 1 o f a h y d r o g e n a t o m . F o r in s t a n c e , c o n s id e r t h e n = 2 s t a t e s . T h e r e is a s in g le l = 0 s t a t e , u s u a lly r e f e r r e d t o a s 2 s , a n d t h r e e l = 1 s t a t e s ( w it h m = 1 , 0 , 1 ) , u s u a lly r e f e r r e d t o a s 2 p . A ll o f t h e s e s t a t e s p o s s e s s t h e s a m e e n e r g y , E 2 = e 2 / ( 3 2 π ǫ 0 a 0 ) . B e c a u s e o f t h e d e g e n e r a c y , t h e t r e a t m e n t a b o v e is n o lo n g e r v a lid a n d in o r d e r t o a p p ly p e r t u r b a t io n t h e o r y , w e h a v e t o r e c u r t o d e g e n e r a t e p e r t u r b a t io n t h e o r y .
W e fi r s t n e e d t o U d = Q d V Q d , w h e r e Q d is t h e p r o j e c t o r o b t a in e d f r o m t h e d e g e n e r a t e 2 s a n d 2 p s t a t e s ( t h a t is , t h e o p e r a t o r t h a t p r o j e c t in t o t h e d e g e n e r a t e s u b s p a c e ) . T h is o p e r a t o r is ,
0 ( 2 , 0 , 0 | z | 2 , 1 , 0 ) 0 0
d | | ( | | )
U = e E 2 , 1 , 0 z 2 , 0 , 0 0 0 0
0 0 0 0
0 0 0 0
0 ( 2 , 0 , 0 | z | 2 , 1 , 0 ) ,
→ ( 2 , 1 , 0 z 2 , 0 , 0 0
)
( | | )
| ) | ) | ) | − )
w h e r e t h e r o w s a n d c o lu m n s c o r r e s p o n d t o t h e 2 , 0 , 0 , 2 , 1 , 0 , 2 , 1 , 1 a n d 2 , 1 , 1 s t a t e s , r e s p e c t iv e ly a n d in t h e s e c o n d s t e p w e r e d u c e t h e o p e r a t o r t o t h e d e g e n e r a t e s u b s p a c e o n ly . T o s im p lif y t h e m a t r ix w e u s e d t h e s e le c t io n r u le s , w h ic h t e ll u s t h a t t h e m a t r ix e le m e n t o f b e t w e e n t w o h y d r o g e n a t o m s t a t e s is z e r o u n le s s t h e s t a t e s p o s s e s s t h e s a m e n q u a n t u m n u m b e r , a n d l q u a n t u m n u m b e r s w h ic h d iff e r b y u n it y . It is e a s ily d e m o n s t r a t e d , f r o m t h e e x a c t f o r m s o f t h e 2 s a n d 2 p w a v e - f u n c t io n s , t h a t
( 2 , 0 , 0 | z | 2 , 1 , 0 ) = ( 2 , 1 , 0 | z | 2 , 0 , 0 ) = 3 a 0 .
| | − | |
( 0 ) ) | 2 , 0 , 0 ) + | 2 , 1 , 0 ) 1 ( 1 )
It c a n b e s e e n , b y in s p e c t io n , t h a t t h e e ig e n v a lu e s o f U d a r e u 1 = 3 e a 0 E , u 2 = 3 e a 0 E , w it h c o r r e s p o n d in g e ig e n v e c t o r s
k 1 = √ 2 = √ 2 1 ,
2
2
√ 2
k ( 0 ) ) = | 2 , 0 , 0 ) √ − | 2 , 1 , 0 ) =
1 ( 1 )
− 1
In t h e a b s e n c e o f a n e le c t r ic fi e ld , a ll o f t h e s e s t a t e s p o s s e s s t h e s a m e e n e r g y , E 2 . T h e fi r s t - o r d e r e n e r g y s h if t s in d u c e d
b y a n e le c t r ic fi e ld a r e g iv e n b y
∆ E 1 = + 3 e a 0 | E | ,
∆ E 2 = − 3 e a 0 | E | ,
| |
T h u s , t h e e n e r g ie s o f s t a t e s 1 a n d 2 a r e s h if t e d u p w a r d s a n d d o w n w a r d s , r e s p e c t iv e ly , b y a n a m o u n t 3 e a 0 E in t h e p r e s e n c e o f a n e le c t r ic fi e ld . S t a t e s 1 a n d 2 a r e o r t h o g o n a l lin e a r c o m b in a t io n s o f t h e o r ig in a l 2 s a n d 2 p ( m = 0 ) s t a t e s . N o t e t h a t t h e e n e r g y s h if t s a r e lin e a r in t h e e le c t r ic fi e ld - s t r e n g t h , s o t h is is a m u c h la r g e r e ff e c t t h a t t h e q u a d r a t ic e ff e c t d e s c r ib e d in t h e p r e v io u s s e c t io n .
T h e e n e r g ie s o f s t a t e s 2 p ( m = 1 ) a n d 2 p ( m = - 1 ) ( w h ic h a r e o u t s id e t h e d e g e n e r a t e s u b s p a c e ) a r e n o t a ff e c t e d t o fi r s t o r d e r ( a s w e a lr e a d y s a w a b o v e f o r t h e n o n - d e g e n e r a t e c a s e ) . O f c o u r s e , t o s e c o n d - o r d e r t h e e n e r g ie s o f t h e s e s t a t e s a r e s h if t e d b y a n a m o u n t w h ic h d e p e n d s o n t h e s q u a r e o f t h e e le c t r ic fi e ld - s t r e n g t h , t h e q u a d r a t ic s h if t f o u n d p r e v io u s ly . N o t e t h a t t h e lin e a r S t a r k e ff e c t d e p e n d s c r u c ia lly o n t h e d e g e n e r a c y o f t h e 2 s a n d 2 p s t a t e s . T h is d e g e n e r a c y is a s p e c ia l p r o p e r t y o f a p u r e C o u lo m b p o t e n t ia l, a n d , t h e r e f o r e , o n ly a p p lie s t o a h y d r o g e n a t o m . T h u s , a lk a li m e t a l a t o m s d o n o t e x h ib it t h e lin e a r S t a r k e ff e c t .
1 1 . 2 Ti m e - d e p e n d e n t p e r t u r b a t i o n t h e o r y
1 1 . 2 . 1 R e v i e w o f i n t e r a c t i o n p i c t u r e
W h e n fi r s t s t u d y in g t h e t im e e v o lu t io n o f Q M s y s t e m s , o n e a p p r o a c h w a s t o s e p a r a t e t h e H a m ilt o n ia n m u c h in t h e s a m e w a y w e d id a b o v e f o r T IP T . W e w r o t e ( s e e S e c t io n 5 .2 ) :
H = H 0 + V ( t )
w h e r e H 0 is a ” s o lv a b le ” H a m ilt o n ia n o f w h ic h w e a lr e a d y k n o w t h e e ig e n - d e c o m p o s it io n ,
0
H 0 | k ) = E k | k ) ,
L
( s o t h a t it is e a s y t o c a lc u la t e e .g . U 0 = e − i H 0 t ) a n d V ( t ) is a p e r t u r b a t io n t h a t d r iv e s a n in t e r e s t in g ( a lt h o u g h
0
u n k n o w n ) d y n a m ic s . H e r e w e e v e n a llo w f o r t h e p o s s ib ilit y t h a t V is t im e - d e p e n d e n t . F o r a n y s t a t e | ψ ) = k c k ( 0 ) | k )
k
k
k
t h e e v o lu t io n c a n b e w r it t e n a s | ψ ) = L c ( t ) e − i E t | k ) . T h is c o r r e s p o n d t o e x p lic it ly w r it in g d o w n t h e e v o lu t io n d u e
( w h ile E 0 d o n o t p la y a r o le ) .
H H
t o t h e k n o w n H a m ilt o n ia n ( if = 0 t h e n w e w o u ld h a v e c k ( t ) = c k ( 0 ) a n d t h e e v o lu t io n w o u ld b e g iv e n b y o n ly t h e p h a s e f a c t o r s ) . In o t h e r w o r d s , if w e w a n t t o c o m p a r e t h e s t a t e e v o lu t io n w it h t h e in it ia l e ig e n s t a t e s , b y c a lc u la t in g t h e o v e r la p |( k | ψ ( t ) ) | 2 , w e w o u ld b e r e a lly in t e r e s t e d o n ly in t h e d y n a m ic s d r iv e n b y V s in c e |( k | ψ ( t ) ) | 2 = | c k ( t ) | 2
k
W e d e fi n e s t a t e s in t h e in t e r a c t io n p ic t u r e b y
| ψ ) I = U 0 ( t ) † | ψ ) = e i H 0 t | ψ )
S im ila r ly w e d e fi n e t h e c o r r e s p o n d in g in t e r a c t io n p ic t u r e o p e r a t o r s a s :
A I ( t ) = U 0 † A U 0 → V I ( t ) = U 0 † V U 0
W e c a n n o w d e r iv e t h e d iff e r e n t ia l e q u a t io n g o v e r n in g t h e e v o lu t io n o f t h e s t a t e in t h e in t e r a c t io n p ic t u r e , s t a r t in g f r o m S c h r ¨ o d in g e r e q u a t io n .
∂ t
∂ t
∂ t
0
∂ t
i ∂ | ψ ) I = i ∂ ( U 0 † | ψ ) ) = i ( ∂ U 0 † | ψ ) + U † ∂ | ψ ) )
In s e r t in g ∂ t U 0 = i H 0 U 0 a n d i ∂ t | ψ ) = H 0 | ψ ) , w e o b t a in
∂ t
0
0
0
0
i ∂ | ψ ) = U † H | ψ ) − U † ( H
0
+ V ) | ψ ) = U † V | ψ ) .
In s e r t in g t h e id e n t it y 1 1 = U 0 U 0 † , w e o b t a in = U 0 † V U 0 U 0 † | ψ ) = V I | ψ ) I :
i
∂ | ψ ) I
∂ t
= V I ( t ) | ψ ) I
T h is is a S c h r ¨ o d in g e r - lik e e q u a t io n f o r t h e v e c t o r in t h e in t e r a c t io n p ic t u r e , e v o lv in g u n d e r t h e a c t io n o f t h e o p e r a t o r
V I ( t ) o n ly .
1 1 . 2 . 2 D y so n s e r i e s
B e s id e s e x p r e s s in g t h e S c h r ¨ o d in g e r e q u a t io n in t h e in t e r a c t io n p ic t u r e , w e c a n a ls o w r it e t h e e q u a t io n f o r t h e p r o p a g a t o r t h a t d e s c r ib e s t h e e v o lu t io n o f t h e s t a t e :
d t
I
I
I
I
I
d U I = − i V U , | ψ ( t ) ) = U ( t ) | ψ ( 0 ) )
S in c e V I ( t ) is t im e - d e p e n d e n t , w e c a n o n ly w r it e f o r m a l s o lu t io n s f o r U I . O n e e x p r e s s io n is g iv e n b y t h e D y s o n s e r ie s .
T h e d iff e r e n t ia l e q u a t io n is e q u iv a le n t t o t h e in t e g r a l e q u a t io n
1
t
—
U I ( t ) = 1 1 i V I ( t ′ ) U I ( t ′ ) d t ′
0
B y it e r a t in g , w e c a n fi n d a f o r m a l s o lu t io n t o t h is e q u a t io n :
− −
1 t 1 t 1 t ′
U I ( t ) = 1 1 i d t ′ V I ( t ′ ) + ( i ) 2 d t ′ d t ′ V I ( t ′ ) V I ( t ′ ′ ) + . . .
0 0 0
1 t 1 t ( n − 1 )
—
+ ( i ) n d t ′ . . . d t ( n ) V I ( t ′ ) . . . V I ( t ( n ) ) + . . .
0 0
T h is is t h e D y s o n s e r ie s .
1 1 . 2 . 3 F e r m i ’ s G o l d e n R u l e
| ) H H | ) | )
L
T h e p r o b le m t h a t w e t r y t o s o lv e v ia T D P T is t o c a lc u la t e t h e t r a n s it io n p r o b a b ilit y f r o m a n in it ia l s t a t e t o a fi n a l s t a t e . C o n s id e r a n in it ia l s t a t e i w h ic h is a n e ig e n s t a t e o f 0 ( 0 i = E i i ) . T h e n in t h e in t e r a c t io n p ic t u r e w e h a v e t h e e v o lu t io n
| i ( t ) ) I = U I ( t ) | i ) = c k ( t ) | k ) , w it h c k ( t ) = ( k | U I ( t ) | i )
k
W e c a n in s e r t t h e p e r t u r b a t io n e x p a n s io n f o r U I ( t ) t o o b t a in a n e x p a n s io n f o r c k ( t ) :
c k ( t ) = ( k | 1 1 − i
t t
1 ′ ′ ′
1
V I ( t ) U I ( t ) d t | i ) = ( k | 1 1 − i
t
1
d t ′ V I ( t ′ ) + ( − i ) 2
d t ′
1 t ′
d t ′ V I ( t ′ ) V I ( t ′ ′ ) + . . . | i )
0 0 0 0
In t h e e x p a n s io n w e w ill o b t a in t e r m s s u c h a s ( k | V I ( t ) | i ) t h a t w e c a n s im p lif y s in c e :
( k | V I ( t ) | i ) = ( k | ( U 0 † V ( t ) U 0 ) | i ) = ( U 0 k | V ( t ) | U 0 i ) = ( k | e i ω k t V ( t ) e − i ω i t | i ) = ( k | V | i ) e i ω k i t = V k i ( t ) e i ω k i t w h e r e w e d e fi n e d ω j = E j / l a n d ω k i = ω k − ω i . U s in g t h e s e r e la t io n s h ip s a n d t h e s e r ie s e x p a n s io n w e o b t a in :
k
c ( 0 ) ( t ) = ( k | 1 1 | i ) = δ k i
′
k 0
I
0
k i
k h
c ( 1 ) ( t ) = − i J t ( k | V ( t ′ ) | i ) d t ′ = − i J t V ( t ′ ) e i ω k i t d t ′
k 0 0
h i
c ( 2 ) ( t ) = − J t d t ′ J t ′ d t ′ ′ V
( t ′ ) V
( t ′ ′ ) e i ω
t ′ e i ω
t ′ ′
k h
h i
F r o m t h is e x p a n s io n w e c a n c a lc u la t e t h e t r a n s it io n p r o b a b ilit y a s P ( i → k ) = | c k ( t ) | 2 .
s in
W e fi r s t c o n s id e r t h e c a s e w h e r e t h e p e r t u r b a t io n V is t im e - in d e p e n d e n t a n d it is t u r n e d o n a t t h e t im e t = 0 . T h e n w e h a v e
c k ( t ) = − i V k i e
k i
d t
= ( 1 − e
k i ) = − 2 i e
k i
( 1 ) 1 t i ω t ′ ′
V k i
i ω t
V k i i ω
t / 2
( ω k i t )
0 ω k i ω k i 2
T h e n t o fi r s t o r d e r p e r t u r b a t io n , t h e t r a n s it io n p r o b a b ilit y is
P ( i → k ) =
4 | V k i | 2
ω
2
k i
s in
2 ( ω k i t )
2
W e c a n p lo t t h is t r a n s it io n p r o b a b ilit y a s a f u n c t io n o f t h e e n e r g y s e p a r a t io n ω k i b e t w e e n t h e t w o s t a t e s . W e w o u ld e x p e c t t h a t if t h e s e p a r a t io n in e n e r g y is s m a lle r , it w ill b e e a s ie r t o m a k e t h e t r a n s it io n . T h is is in d e e d t h e c a s e , s in c e P h a s t h e s h a p e o f a s in c f u n c t io n s q u a r e .
N o t ic e t h a t t h e p e a k h e ig h t is p r o p o r t io n a l t o t 2 , w h ile t h e z e r o s a p p e a r a t 2 k π / t , t h a t is , t h e p e a k w id t h is p r o p o r t io n a l t o 1 / t ( t h e o t h e r p e a k s a r e q u it e s m a ll) . T h is m e a n s t h a t t h e p r o b a b ilit y is s ig n ifi c a n t ly d iff e r e n t t h a n
0.25
0.20
0.15
0.10
0.05
– 6
– 4
– 2
2
4
6
F i g . 1 9 : T r a n s i t i o n p r o b a b i l i t y
≤ ∼
z e r o o n ly f o r ω k i t 2 π . In t e r m s o f e n e r g y , w e h a v e t h a t ∆ t∆ E l ( w h e r e w e d e fi n e d ∆ t a s t h e d u r a t io n o f t h e in t e r a c t io n ) , o r in o t h e r w o r d s , w e c a n h a v e a c h a n g e o f e n e r g y in t h e s y s t e m o n ly a t s h o r t t im e s , w h ile a t lo n g t im e s w e r e q u ir e q u a s i- c o n s e r v a t io n o f e n e r g y . C o n s id e r t h e lim it o f t h e s in c f u n c t io n :
s in ( ω t/ 2 )
lim = π δ ( ω )
t → ∞ ω
T h e n , f r o m f ( x ) δ ( x ) = f ( 0 ) a n d s in c ( 0 ) = 1 , w e o b t a in
( ) ( )
s in ( ω t/ 2 ) 2 s in ( ω t/ 2 ) s in ( ω t/ 2 ) s in ( ω t/ 2 ) s in ( ω t/ 2 ) t π t
lim = lim = π δ ( ω ) = π δ ( ω ) = δ ( ω )
t → ∞ ω ω t → ∞ ω ω ω t/ 2 2 2
W e h a v e t h e n f o u n d t h e t r a n s it io n p r o b a b ilit y a t lo n g t im e :
2
2
k i
P ( i → k ) t → → ∞ π t δ ( ω ) 4 | V | ,
w h ic h c o n fi r m s t h e f a c t t h a t in t h e lo n g - t im e lim it w e n e e d t o e n f o r c e e n e r g y c o n s e r v a t io n . A b e t t e r d e fi n e d q u a n t it y is t h e r a t e o f t r a n s it io n :
W ( i → k ) = 2 π | V k i | 2 δ ( ω ) .
k i
k
k i 0 k
k i
N o t ic e t h a t f o r ω = 0 , f r o m c ( 1 ) ( t ) = − i V J t e i ω k i t ′ d t ′ w e o b t a in c ( 1 ) ( t ) = − i V
t a n d t h u s t h e p r o b a b ilit y
N o w w e c o n s id e r a c o n t in u u m o f fi n a l s t a t e s , a ll w it h e n e r g y E k f ≈ E i . T h e n t h e p r o b a b ilit y J o f a t r a n s it io n t o t h is
| c k ( t ) | 2 = | V k i | 2 t 2 . T h e r e is a q u a d r a t ic d e p e n d e n c e o n t im e f o r a s in g le fi n a l s t a t e .
c o n t in u u m is g iv e n b y t h e s u m o f t h e p r o b a b ilit y f o r e a c h in d iv id u a l s t a t e : P f = L k | c k | 2 → d E k ρ ( E k ) | c k | 2 , w h e r e
w e d e fi n e d t h e d e n s it y o f s t a t e s ρ ( E k ) , s u c h t h a t ρ ( E k ) d E k is t h e n u m b e r o f s t a t e s w it h e n e r g y b e t w e e n E k a n d
E k + d E k . W e c a n t h e n r e w r it e t h e p r o b a b ilit y a s
P i → f = 4 1 d E ρ ( E ) s in U s in g t h e lim it o f t h e s in c f u n c t io n , w e fi n d
2 ( E − E i ) t | V k i | 2
( )
2 ( E − E i ) 2
1 | k i |
π t V 2
i
P i → f = 4 d E ρ ( E ) δ ( E − E i ) 2 ( E − E ) 2
| | ≈ | |
S in c e a ll t h e s t a t e s a r e in a n e ig h b o r h o o d o f t h e e n e r g y , w e e x p e c t V k i 2 V ¯ k i 2 o v e r t h e r a n g e o f e n e r g y o f in t e r e s t . T h u s b y e v a lu a t in g t h e in t e g r a l ( w it h t h e d e lt a f u n c t io n ) w e o b t a in t h e t r a n s it io n p r o b a b ilit y :
P i → f = 2 | V k i | 2 π tρ ( E k ) | E k ≈ E i
k
∫
S im ila r ly , w e c a n c a lc u la t e t h e t r a n s it io n r a t e t o a c o n t in u u m o f s t a t e s . F r o m t h e e x p r e s s io n f o r a s in g le s t a t e , W i → k = 2 π | V k i | 2 δ ( E k − E i ) , w e in t e g r a t e o v e r a ll fi n a l e n e r g ie s , W i → f = W i → k ρ ( E k ) d E k , w h e r e f is t h e c o n t in u u m o f s t a t e s k s u c h t h a t E k ≈ E i . T h e n w e o b t a in t h e t r a n s it io n r a t e :
W = | V
2 π
l
k i k E ≈ E
| ρ ( E )
2
|
k i
T h is is F e r m i’s G o ld e n R u le .
V i r tu a l T r a n s i ti o n s
If t h e m a t r ix e le m e n t o f t h e in t e r a c t io n c o n n e c t in g t w o g iv e n s t a t e is z e r o , w e h a v e s e e n f r o m t h e e x p r e s s io n a b o v e t h a t n o t r a n s it io n is p o s s ib le , t o fi r s t o r d e r .
k
Σ V V ∫
H o w e v e r , c o n s id e r c ( 2 ) ( t ) . T h is is g iv e n b y
c ( 2 ) ( t ) = −
Σ V k h
V h i
t
∫
d t ′
∫ t ′
′ ′ ′
d t ′ ′ e i ω k h t e i ω h i t = i
t k h h i
′ ′
d t ′ ( e i ω k i t − e i ω k h t )
k
h 0 0
h ω k i 0
s a m ` e a ˛ s ¸ b x e f o r e ` ≈ ˛ ¸ 0 x
If E h = /
E k , E i , t h e s e c o n d t e r m o s c illa t e s r a p id ly a n d g o e s t o z e r o . F in a lly w e h a v e :
2
2 π Σ V V
V +
l ω k i
W i → k =
k i
k h h i
δ ( E k − E i )
o r f o r a c o n t in u u m
V +
W i → f =
l
k i
2 π
h
2
k h h i
ρ ( E k ) | E k ≈ E
Σ V V
i
ω k i
h
N o t ic e t h a t e v e n if V i k = 0 , w e c a n s t ill h a v e a t r a n s it io n t o k , v ia v i r t u a l t r a n s it io n s t o in t e r m e d ia t e s t a t e s , w h ic h a r e c o n n e c t e d t o t h e t w o r e le v a n t le v e ls .
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22.51 Quantum Theory of Radiation Interactions
Fall 201 2
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