4. Energy L evel s
4 .1 .1 E n er g y in S qu a r e in fi n ite w ell (p a rt ic le in a b o x)
4 .1 .2 F in ite s qu a r e w ell
4.2 Q uan tum Me c hanic s in 3D: A ngular mome n tum
4 .2 .1 S c h r ¨ od in g er equ a tio n in s p h er ic a l c o o r d in a t es
4 .2 .2 A n g u la r mo men tu m o p er a t o r
4 .2 .3 S p in a n g u la r mo men tu m
4 .2 .4 A d d itio n o f a n g u la r mo men t u m
4.3 Solutions to the Sc hr o ¨ dinge r e quation in 3D
4 .3 .1 T h e H yd r o g en a to m
4 .3 .2 A to mic p er io d ic s tr u c tu r e
4 .3 .3 T h e H a r mo n ic Os c illa t o r P o ten tia l
4 .4 .1 B o s o n s , fer mio n s
4 .4 .2 E xc h a n g e o p er a to r
4 .4 .3 P a u li exc lu s io n p r in c ip le
4. 1 Bound p roblems
In th e p r ev iou s c h ap ter w e s tu d ied s tation ar y p r ob lems in wh ic h th e s y s tem is b es t d es cr ib ed as a (time-in d ep en d en t) w a v e, “s catter in g” an d “tu n n elin g” (th at is , s h o win g v ar iation on its in ten s it y ) b ecau s e of ob s tacles giv en b y c h an ges in th e p oten tial en er gy .
Alth ou gh th e p oten tial d eter min ed th e s p ace-d ep en d en t w a v efu n ction , th er e w as n o limitation imp os ed on th e p os s ib le w a v en u m b er s an d en er gies in v olv ed . An in fi n ite n u m b er of c ontinu ou s en er gies w er e p os s ib le s olu tion s to th e time- in d ep en d en t S c h r ¨ od in ger eq u ation .
In th is c h ap ter , w e w an t in s tead to d es cr ib e s y s tems wh ic h ar e b es t d es cr ib ed as p ar ticles con fi n ed in s id e a p oten tial. Th is t y p e of s y s tem w ell d es cr ib e atoms or n u clei wh os e con s titu en ts ar e b ou n d b y th eir m u tu al in ter action s . W e s h all s ee th at b ecau s e of th e p ar ticle con fi n emen t, th e s olu tion s to th e en er gy eigen v alu e eq u ation (i.e. th e time- in d ep en d en t S c h r ¨ od in ger eq u ation ) ar e n o w on ly a dis cr ete s et of p os s ib le v alu es (a d is cr ete s et os en er gy lev els ). Th e en er gy is th er efor e q u an ti zed . Cor r es p on d in gly , on ly a d is cr ete s et of eigen fu n ction s will b e s olu tion s , th u s th e s y s tem, if it’s in a s tation ar y s tate, can on ly b e fou n d in on e of th es e allo w ed eigen s tates .
W e will s tar t to d es cr ib e s imp le ex amp les . Ho w ev er , after lear n in g th e r elev an t con cep ts (an d math ematical tr ic k s ) w e will s ee h o w th es e s ame con cep ts ar e u s ed to p r ed ict an d d es cr ib e th e en er gy of atoms an d n u clei. Th is th eor y can p r ed ict for ex amp le th e d is cr ete emis s ion s p ectr u m of atoms an d th e n u clear b in d in g en er gy .
4.1.1 Energy i n Squa re i nfini te w el l ( pa rti c l e i n a b o x)
Th e s imp les t s y s tem to b e an aly zed is a p ar ticle in a b o x : clas s ically , in 3D , th e p ar ticle is s tu c k in s id e th e b o x an d can n ev er lea v e. An oth er clas s ical an alogy w ou ld b e a b all at th e b ottom of a w ell s o d eep th at n o matter h o w m u c h k in etic en er gy th e b all p os s es s , it will n ev er b e ab le to ex it th e w ell.
W e con s id er again a p ar ticle in a 1D s p ace. Ho w ev er n o w th e p ar ticle is n o lon ger fr ee to tr a v el b u t is con fi n ed to b e b et w een th e p os ition s 0 an d L . In or d er to con fi n e th e p ar ticle th er e m u s t b e an in fi n ite for ce at th es e b ou n d ar ies th at r ep els th e p ar ticle an d for ces it to s ta y on ly in th e allo w ed s p ace. Cor r es p on d in gly th er e m u s t b e an in fi n ite p oten tial in th e for b id d en r egion .
∞ ≤ ≤
Th u s th e p oten tial fu n ction is as d ep icted in F ig. 20 : V ( x ) = for x < 0 an d x > L ; an d V ( x ) = 0 for 0 x L . Th is las t con d ition mean s th at th e p ar ticle b eh a v es as a fr ee p ar ticle in s id e th e w ell (or b o x ) cr eated b y th e p oten tial.
V(x)
0 L x
F ig . 1 9 : P o ten tia l o f a n in fi n it e w ell
W e can th en wr ite th e en er gy eigen v alu e p r ob lem in s id e th e w ell:
k 2 ∂ 2 w n ( x )
H [ w n ] = − 2 m ∂ x 2 = E n w n ( x )
L ∀
O u ts id e th e w ell w e can n ot wr ite a p r op er eq u ation b ecau s e of th e in fi n ities . W e can s till s et th e v alu es of w n ( x ) at th e b ou n d ar ies 0 , L . Ph y s ically , w e ex p ect w n ( x ) = 0 in th e for b id d en r egion . In fact, w e k n o w th at ψ ( x ) = 0 in th e for b id d en r egion (s in ce th e p ar ticle h as zer o p r ob ab ilit y of b ein g th er e) 6 . Th en if w e wr ite an y ψ ( x ) in ter ms of th e en er gy eigen fu n ction s , ψ ( x ) = n c n w n ( x ) th is h as to b e zer o c n in th e for b id d en r egion , th u s th e w n h a v e to b e zer o.
A t th e b ou n d ar ies w e can th u s wr ite th e b ou n d ar y con d ition s 7 :
w n (0) = w n ( L ) = 0
W e can s olv e th e eigen v alu e p r ob lem in s id e th e w ell as d on e for th e fr ee p ar ticle, ob tain in g th e eigen fu n ction s
w n ′ ( x ) = A ′ e ik n x + B ′ e − ik n x ,
1 2 k 2
2 m
with eigen v alu es E n = n .
It is eas ier to s olv e th e b ou n d ar y con d ition s b y con s id er in g in s tead :
w n ( x ) = A s in ( k n x ) + B cos ( k n x ) .
W e h a v e:
w n (0) = A × 0 + B × 1 = B = 0
Th u s fr om w n (0) = 0 w e h a v e th at B = 0. Th e s econ d con d ition s tates th at
w n ( L ) = A s in ( k n L ) = 0
Th e s econ d con d ition th u s d o es n ot s et th e v alu e of A (th at can b e d on e b y th e n or malization con d ition ). In or d er to s atis fy th e con d ition , in s tead , w e h a v e to s et
nπ k n L = nπ → k n = L
for in teger n . Th is con d ition th en in tu r n s s ets th e allo w ed v alu es for th e en er gies :
n ≡ 1
k 2 k 2 k 2 π 2
E = n = n 2 E n 2
2 m 2 mL 2
wh er e w e s et E 1 =
1 2 π 2
2 m L 2
an d n is called a q u an tu m n u m b er (as s o ciated with th e en er gy eigen v alu e).
F r om th is , w e s ee th at on ly s ome v alu es of th e en er gies ar e allo w ed . Th er e ar e s till an in fi n ite n u m b er of en er gies ,
b u t n o w th ey ar e n ot a con tin u ou s s et. W e s a y th at th e en er gies ar e q u an ti zed . Th e q u an tization of en er gies (fi r s t
6 N o t e t h a t th is is tr u e b ec a u s e th e p o ten tia l is in fi n ite. T h e en er g y eig en v a lu e fu n c tio n (fo r th e H a milto n ia n o p er a to r ) is a lw a ys v a lid . T h e o n ly w a y fo r th e equ a tio n to b e v a lid o u ts id e th e w ell it is if w n ( x ) = 0
7 N o t e th a t in th is c a s e w e c a n n o t r equ ir e th a t th e fi r s t d er iv a tiv e b e c o n tin u o u s , s in c e th e p o ten tia l b ec o mes in fi n it y a t th e b o u n d a r y . I n th e c a s es w e exa min ed to d es c r ib e s c a tt er in g , th e p o ten tia l h a d o n ly d is c o n tin u it y o f th e fi r s t kin d .
V(x)
E n
0 L x
F ig . 2 0 : Qu a n tiz ed en er g y lev els ( E n fo r n = 0 − 4 ) in r ed . A ls o , in g r een t h e p o s itio n p r o b a b ilit y d is t rib u t io n | w n ( x ) | 2
th e p h oton en er gies in b lac k -b o d y r ad iation an d p h oto-electr ic eff ect, th en th e electr on en er gies in th e atom) is wh at ga v e qu antu m mec h an ics its n ame. Ho w ev er , as w e s a w fr om th e s catter in g p r ob lems in th e p r ev iou s c h ap ter , th e q u an tization of en er gies is n ot a gen er al p r op er t y of q u an tu m mec h an ical s y s tems . Alth ou gh th is is common (an d th e r u le an y time th at th e p ar ticle is b ou nd , or con fi n ed in a r egion b y a p oten tial) th e q u an tization is alw a y s a con s eq u en ce of a p ar ticu lar c h ar acter is tic of th e p oten tial. Th er e ex is t p oten tials (as for th e fr ee p ar ticle, or in gen er al for u n b ou n d p ar ticles ) wh er e th e en er gies ar e n ot q u an tized an d d o for m a con tin u u m (as in th e clas s ical cas e).
F in ally w e calcu late th e n or malization of th e en er gy eigen fu n ction s :
� � L
∞ L
dx | w n | 2 = 1 → A 2 s in ( k n x ) 2 dx =
A 2 = 1 → A =
2
− ∞ 0 2 L
J p ( x ) dx = 1 or J | ψ ( x ) | dx = 1.
Notice th at b ecau s e th e s y s tem is b ou n d in s id e a w ell d efi n ed r egion of s p ace, th e n or malization con d ition h as n o w a v er y clear p h y s ical mean in g (an d th u s w e m u s t alw a y s ap p ly it): if th e s y s tem is r ep r es en ted b y on e of th e eigen fu n ction s (an d it is th u s s tation ar y ) w e k n o w th at it m u s t b e fou n d s omewh er e b et w een 0 an d L . Th u s th e p r ob ab ilit y of fi n d in g th e s y s tem s omewh er e in th at r egion m u s t b e on e. Th is cor r es p on d s to th e con d ition
L L 2
0 0
F in ally , w e h a v e
w n ( x ) = L s in k n x, k n = L , E n = 2 mL 2 n
2
nπ
k 2 π 2
2
L L
No w as s u me th at a p ar ticle is in an en er gy eigen s tate, th at is ψ ( x ) = w n ( x ) for s ome n : ψ ( x ) = I 2 s in ( nπ x ) . W e
p lot in F ig. 21 s ome p os s ib le w a v efu n ction s .
0
5
0
L
.5
.0
1.
0.
-0
-1
F ig . 2 1 : E n er g y eig en fu n c tio n s . B lu e: n =1 , M a u v e n =2 , B r o w n n =1 0 , Gr een n =1 0 0
Con s id er for ex amp le n = 1.
? Qu estion : W h at d o es an en er gy meas u r emen t y ield ? W h at is th e p r ob ab ilit y of th is meas u r emen t?
2 m
( E = 1 2 π 2
with p r ob ab ilit y 1)
≤ ≤
? Q u estion : wh at d o es a p os tion meas u r emen t y ield ? W h at is th e p r ob ab ilit y of fi n d in g th e p ar ticle at 0 x L ? an d at x = 0 , L ?
| |
→ ∞
? Qu estion : W h at is th e d iff er en ce in en er gy b et w een n an d n + 1 wh en n ? An d wh at ab ou t th e p os ition p r ob ab ilit y w n 2 at lar ge n ? W h at d o es th at s a y ab ou t a p os s ib le clas s ical limit?
∝
In th e limit of lar ge q u an tu m n u m b er s or s mall d eBr oglie w a v elen gth λ 1 /k on a v er age th e q u an tu m mec h an ical d es cr ip tion r eco v er s th e clas s ical on e ( B ohr c or r es p ondenc e pr incipl e ).
4.1.2 Fi ni te s qua re w el l
W e n o w con s id er a p oten tial wh ic h is v er y s imilar to th e on e s tu d ied for s catter in g (comp ar e F ig. 15 to F ig. 22 ), bu t th at r ep r es en ts a comp letely d iff er en t s itu ation . Th e p h y s ical p ictu r e mo d eled b y th is p oten tial is th at of a b ou n d p ar ticle. S p ecifi cally if w e con s id er th e cas e wh er e th e total en er gy of th e p ar ticle E 2 < 0 is n egativ e, th en clas s ically w e w ou ld ex p ect th e p ar ticle to b e tr ap p ed in s id e th e p oten tial w ell. Th is is s imilar to wh at w e alr ead y s a w wh en s tu d y in g th e in fi n ite w ell. Her e h o w ev er th e h eigh t o f th e w ell is fi n ite, s o th at w e will s ee th at th e q u an tu m mec h an ical s olu tion allo ws for a fi n ite p en etr ation of th e w a v efu n ction in th e clas s ically for b id d en r egion .
? Q u estion : W h at is th e ex p ect b eh a v ior of a clas s ical p ar ticle? (con s id e r for ex amp le a s n o wb oar d er in a h alf-p ip e. If s h e d o es n ot h a v e en ou gh s p eed s h e’s n ot goin g to b e ab le to ju mp o v er th e s lop e, an d will b e con fi n ed in s id e).
V(x)
E 1 =+E
- a
a
x
R eg ion I
R eg ion II
R eg ion III
- V H
E 2 =-E
F ig . 2 2 : P o t en tia l o f a fi n it e w ell. T h e p o ten t ia l is n o n -z er o a n d equ a l t o − V H in th e r eg io n − a ≤ x ≤ a .
F or a q u an tu m mec h an ical p ar ticle w e w an t in s tead to s olv e th e S c h r ¨ od in ger eq u ation . W e con s id er t w o cas es . In th e fi r s t cas e, th e k in etic en er gy is alw a y s p os itiv e:
2 m d x 2
2 2
− 2 m d x 2 = ( E + V
) ψ ( x ) in Region I I
− 1 2 d 2 ψ ( x ) = E ψ ( x ) in Region I
1 d ψ ( x )
2 m d x 2
2 d 2
H ψ ( x ) = − 2 ψ ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) →
H
2 mdx
− 1 2 d 2 ψ ( x ) = E ψ ( x ) in Region I I I
—
| | ≤
s o w e ex p ect to fi n d a s olu tion in ter ms of tr a v elin g w a v es . Th is is n ot s o in ter es tin g, w e on ly n ote th at th is d es cr ib es th e cas e of an u n b ou n d p ar ticle. Th e s olu tion s will b e s imilar to s catter in g s olu tion s (s ee math ematica d emon s tr ation ). In th e s econ d cas e, th e k in etic en er gy is gr eater th an zer o for x a an d n egativ e oth er wis e (s in ce th e total en er gy is n egativ e). Notice th at I s et E to b e a p os itiv e q u an tit y , an d th e s y s tem’s en er gy is E . W e als o as s u me th at E < V H . Th e eq u ation s ar e th u s r ewr itten as :
− = ( V
2 m d x 2
1 2 d 2 ψ ( x )
— E ) ψ ( x ) in Region I I
− 1 2 d 2 ψ ( x ) = − E ψ ( x ) in Region I
2 m d x 2
2 d 2
H ψ ( x ) = − ψ ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) →
2
2 m
d x 2
H
2 mdx
− 1 2 d 2 ψ ( x ) = − E ψ ( x ) in Region I I I
Th en w e ex p ect w a v es in s id e th e w ell an d an imagin ar y momen tu m (y ield in g ex p on en tially d eca y in g p r ob ab ilit y of fi n d in g th e p ar ticle) in th e ou ts id e r egion s . M or e p r ecis ely , in th e 3 r egion s w e fi n d :
Region I Region I I Region I I I
1 2
k ′ = iκ, k = I 2 m ( V H + E 2 )
k ′ = iκ,
1 2 1 2
1 2
1 2
κ = I − 2 m E 2 = I 2 m E = I 2 m ( V H − E )
κ = I 2 m E
2
8
4
6
An d th e w a v efu n ction is
(Notice th at in th e fi r s t r egi
F ig . 2 3 : c o t z (R ed ) a n d z c o t z (B la c k)
|
Region I |
Region I I |
Region I I I |
|
|
C ′ e − κ | x | |
A ′ e ik x |
+ B ′ e − ik x |
D ′ e − κx |
|
on I can wr ite ei |
th er C ′ e |
− κ | x | or C ′ e κx . Th |
e fi r s t n otation |
mak es it clear th at w e h a v e
an ex p on en tial d eca y ). W e n o w w an t to matc h th e b ou n d ar y con d ition s in or d er to fi n d th e co efficien ts . Als o, w e r emem b er fr om th e in fi n ite w ell th at th e b ou n d ar y con d ition s ga v e u s n ot th e co efficien t A , B b u t a con d ition on th e allo w ed v alu es of th e en er gy . W e ex p ect s ometh in g s imilar h er e, s in ce th e in fi n ite cas e is ju s t a limit of th e p r es en t cas e.
—
− − −
F ir s t w e n ote th at th e p oten tial is an ev en fu n ction of x . Th e d iff er en tial op er ator is als o an ev en fu n ction of x . Th en th e s olu tion h as to eith er b e o d d or ev en for th e eq u ation to h old . Th is mean s th at A an d B h a v e to b e c h os en s o th at ψ ( x ) = A ′ e ik x + B ′ e − ik x is eith er ev en or o d d . Th is is ar r an ged b y s ettin g ψ ( x ) = A cos ( k x ) [ev en s olu tion ] or ψ ( x ) = A s in ( k x ) [o d d s olu tion ]. Her e I c h o os e th e o d d s olu tion , ψ ( x ) = ψ ( x ). Th at als o s ets C ′ = D ′ an d w e r ewr ite th is con s tan t as C ′ = D ′ = C .
W e th en h a v e:
Region I Region I I Region I I I
ψ ( x ) = − C e κ x ψ ( x ) = A s in ( k x ) ψ ( x ) = C e − κ x ψ ′ ( x ) = − κC e κ x ψ ′ ( x ) = k A cos ( k x ) ψ ′ ( x ) = − κC e − κ x
− −
{
S in ce w e k n o w th at ψ ( x ) = ψ ( x ) (o d d s olu tion ) w e can con s id er th e b ou n d ar y matc h in g con d ition on ly at x = a . Th e t w o eq u ation s ar e:
A s in ( k a ) = C e − κ a
Ak cos ( k a ) = − κC e − κ a
—
S u b s titu tin g th e fi r s t eq u ation in to th e s econ d w e fi n d : Ak cos ( k a ) = κA s in ( k a ). Th en w e ob tain an eq u ation n ot for th e co efficien t A (as it w as th e cas e for th e in fi n ite w ell) b u t a con s tr ain t on th e eigen v alu es k an d κ :
κ = − k cot ( k a )
Th is is a con d ition on th e eigen v alu es th at allo ws on ly a s u b s e t of s olu tion s . Th is eq u ation can n ot b e s olv ed an aly t ically , w e th u s s ear c h for a s olu tion gr ap h ically (it cou ld b e d on e of cou r s e n u mer ically !).
/
1
1 2 1 2
0
1 2
1
0
0
T o d o s o, w e fi r s t mak e a c h an ge of v ar iab le, m u ltip ly in g b oth s id es b y a an d s ettin g k a = z , κa = z 1 . Notice th at
z 2 = 2 m E a 2 an d z 2 = 2 m ( V H − E ) a 2 . S ettin g z 2 = 2 m V H a 2 , w e h a v e z 2 = z 2 − z 2 or κa = z 2 − z 2 . Th en w e can
π
k a c ot(k a)
2π 3π
k a
√z 0 -( k a)
2 2
k a tan(k a)
√ z 2 -( k a) 2
0
π/2
3 π/2
5π/ 2
k a
y y
0
0
F ig . 2 4 : Gr a p h ic s o lu t io n o f th e e ig en v a lu e eq u a tio n . Left: o dd s o lu tio n s ; R ig h t: e v e n s o lu tio n s . T h e r ed c u r v es o f d iffer en t to n e a r e t h e fu n c tio n − z 2 − z 2 (left ) o r z 2 − z 2 (r ig h t ) fo r d iffer en t (in c r ea s in g ) v a lu es o f z 0 . C r o s s in g s (s o lu t io n s ) a r e
ma r k ed b y a b la c k d o t .
R eg ion I
R eg i o n II
R eg ion III
F ig . 2 5 : Left: Od d s o lu tio n fo r th e fi n it e b a r r ier p o ten tia l, fo r t w o p o t en t ia l d ep th . Gr o u n d s ta t e o f th e w a v efu n c t io n . T h e w a v efu n c tio n is a s in u s o id a l in R eg io n I I (B la c k) a n d a n exp o n en tia l d ec a y in r eg io n s I a n d I I I (B lu e). N o tic e th a t fo r th e s h a llo w er p o ten tia l (d a s h ed lin es ) th e w a v efu n c tio n ju s t b a r ely “ fi t” in s id e th e w ell. R ig h t: Od d s o lu tio n , fo r la r g er k v ec to r (h ig h er qu a n tu m n u m b er ), a llo w in g t w o o s c illa tio n s .
0
r ewr ite th e eq u ation κa = − k a cot ( k a ) → z 1 = − z cot ( z ) as / z 2 − z 2 = − z cot ( z ), or :
0
I z 2 − z 2 = − z cot ( z )
/
I
2 m V H a 2
1 2
0
Th is is a tr an s cen d en tal eq u ation for z (an d h en ce E ) as a fu n ction of z 0 , wh ic h giv es th e d ep th of th e w ell (v ia V H ). T o fi n d s olu tion s w e p lot b oth s id es of th e eq u ation an d lo ok for cr os s in gs . Th at is , w e p lot y 1 ( z ) = − z 2 − z 2 ,
wh ic h r ep r es en t a q u ar ter cir cle (as z is p os itiv e) of r ad iu s z
0
=
an d y 2 ( z ) = z cot ( z ).
O b s . 1 Th e co efficien t A ( an d th u s C an d D ) can b e fou n d (on ce th e eigen fu n ction s h a v e b een fou n d n u mer ically or gr ap h ically ) b y imp os in g th at th e eigen fu n ction is n or malized .
O b s . 2 Notice th at th e fi r s t r ed cu r v e n ev er cr os s es th e b lu e cu r v es . T h at mean s th at th er e ar e n o s olu tion s . If
z 0 < π / 2 th er e ar e n o s olu tion s (Th at is , if th e w ell is to o s h allo w th er e ar e n o b ou n d s olu tion s , th e p ar ticle can
es cap e). O n ly if V H >
1 2 π 2
≤ ≤
m a 2 8
th er e’s a b ou n d s olu tion .
O b s . 3 Th er e’s a fi n ite n u m b er of s olu tion s , giv en a v alu e of z 0 > π / 2. F or ex amp le, for π / 2 z 0 3 π / 2 th er e’s on ly
≤ ≤
on e s olu tion , 2 for 3 π / 2 z 0 5 π / 2, etc.
Remem b er h o w ev er th at w e on ly con s id er ed th e o dd s olu tion s . A b ou n d s olu tion is alw a y s p os s ib le if w e con s id er th e ev en s olu tion s ., s in ce th e eq u ation to b e s olv ed is
0
κa = k a tan ( k a ) = I z 2 − z 2 .
=
Imp or tan tly , w e fou n d th at for th e o d d s olu tion th er e is a min im u m s ize of th e p oten tial w ell (wid th an d d ep th ) th at s u p p or ts b ou n d s tates . Ho w can w e es timate th is s ize? A b ou n d s tate r eq u ir es a n egativ e total en er gy , or a k in etic
en er gy s maller th an th e p oten tial: E
k
w a v elen gth , λ = 2 π :
k in
1 2 k 2
2 m
< V H
. Th is p os es a con s tr ain t on th e w a v en u m b er k an d th u s th e
2 π k
H
λ ≥ √ 2 mV
2
Ho w ev er , in or d er to s atis fy th e b ou n d ar y con d ition s (th at con n ect th e os cillatin g w a v efu n ction to th e ex p on en tially d eca y on e) w e n eed to fit at leas t h alf of a w a v elen gth in s id e th e 2 a wid th of th e p oten tial., 1 λ ≤ 2 a . Th en w e ob tain
F ig . 2 6 : E v en s o lu tio n fo r th e fi n ite b a r r ier p o ten t ia l. T h e w a v efu n c t io n is ∝ c o s ( k x ) in R eg io n I I (B la c k) a n d a n exp o n en tia l d ec a y in r eg io n s I a n d I I I (B lu e). Left: a n y w a v efu n c tio n c a n “ fi t” in t h e w ell a n d s a tis fy th e b o u n d a r y c o n d itio n (th er e’s n o min im u m w ell d ep th a n d w id th ). R ig h t , w a v efu n c tio n w ith a h ig h er qu a n t u m n u m b er , s h o w in g t w o o s c illa tio n s
a r elation s h ip b et w een th e min im u m p oten tial d ep th an d wid th
2 π k k 2 π 2
√ 2 mV H
≤ λ ≤ 4 a → V H ≥ ma 2 8
{ }
Alth ou gh w e s olv ed a 1D p r ob lem, th e s q u ar e w ell r ep r es en ts a 3D p r ob lem as w ell. Con s id er for ex amp le a s p h er ical w ell in 3D : Th e p oten tial is zer o in s id e a r egion of r ad iu s a an d is V H for r > a . Th en w e can r ewr ite th e time- in d ep en d en t S c h r ¨ od in ger eq u ation in 3D for th is p oten tial in s p h er ical co or d in ates an d u s e s ep ar ation of v ar iab les ( r , ϑ , ϕ ). Becau s e of s y mmetr y , th e w a v efu n ction is a con s tan t in ϑ an d ϕ , th u s w e will h a v e to s olv e ju s t a s in gle d iff er en tial eq u ation for th e r ad ial v ar iab le, v er y s imilar to wh at fou n d h er e. W e m u s t th en c h o os e th e o d d -p ar it y s olu tion in or d er to ob tain a fi n ite w a v efu n ction at r = 0. Th u s in 3D , on ly th e o d d s olu tion s ar e p os s ib le an d w e n eed a min im u m p oten tial w ell d ep th in or d er to fi n d a b ou n d s tate.
4. 2 Quantum M ec hanic s in 3D: Angula r momentum
4.2.1 Sc hr o ¨ di nger equati on i n s pheri c al c o o rdi nates
W e n o w go b ac k to th e time-in d ep en d en t S c h r ¨ od in ger eq u ation
− 2 m ∇
+ V ( x , y , z ) ψ ( x ) = E ψ ( x )
( k 2 2 )
W e h a v e alr ead y s tu d ied s ome s olu tion s to th is eq u ation s – fo r s p ecifi c p oten tials in one dim ens ion . No w w e w an t to s olv e Q M p r ob lems in 3D . S p ecifi cally , w e lo ok at 3D p r ob lems wh er e th e p oten tial V ( x x ) is is otr op ic, th at is , it on ly d ep en d s on th e d is tan ce fr om th e or igin . Th en , in s tead of u s in g car tes ian co or d in ates x x = { x , y , z } , it is con v en ien t
to u s e s p h er ical co or d in ates x x = { r , ϑ , ϕ } :
↔
x = r s in ϑ cos ϕ
r = / x 2 + ( y 2 + z 2 )
y = r s in ϑ s in ϕ
ϑ = ar ctan z / / x 2 + y 2
x = r cos ϑ
ϕ = ar ctan ( y /x )
z
θ
r
x
φ
y
F ig . 2 7 : S p h er ic a l C o o r d in a tes
F ir s t, w e ex p r es s th e Lap lacian ∇ 2 in s p h er ical co or d in ates :
∇
r
2 1 ∂
( 2 ∂ )
1 ∂ (
∂ )
1 ∂ 2
= r 2 ∂ r
∂ r + r 2 s in ϑ ∂ ϑ
s in ϑ
∂ ϑ
+ r 2 s in 2 ϑ ∂ ϕ 2
T o lo ok for s olu tion s , w e u s e again th e s ep ar ation of v ar iab le meth o d s , wr itin g ψ ( x x ) = ψ ( r , ϑ , ϕ ) = R ( r ) Y ( ϑ , ϕ ):
− 2 m r 2 d r
r
d r + r 2 s in ϑ ∂ ϑ
s in ϑ
∂ ϑ
+ r 2 s in 2 ϑ ∂ ϕ 2
+ V ( r ) R Y = E R Y
k 2 [ Y d ( 2 d R ) R ∂ ( ∂ Y ) R ∂ 2 Y ]
W e th en d iv id e b y R Y /r 2 an d r ear r an ge th e ter ms as
− 2 m R d r
r
k 2 [ 1 d
( 2 d R )] 2
d r
k 2 [
1 ∂ (
∂ Y )
1 ∂ 2 Y ]
2 m
+ r ( V − E ) = 2 mY s in ϑ ∂ ϑ
s in ϑ ∂ ϑ + s in 2 ϑ ∂ ϕ 2
Eac h s id e is a fu n ction of r on ly an d ϑ , ϕ , s o th ey m u s t b e in d ep en d en tly eq u al to a con s tan t C th at w e s et (for r eas on s to b e s een later ) eq u al to C = − 1 2 l ( l + 1). W e ob tain t w o eq u ation s :
1 d d R
R d r d r
(
r
2
)
—
2 m r 2
2
( V − E ) = l ( l + 1)
1 ∂
s in ϑ ∂ ϑ
(
s in ϑ +
∂ Y
∂ ϑ
)
1 ∂ 2 Y
s in 2 ϑ ∂ ϕ 2
= − l ( l + 1) Y
an d
Th is las t eq u ation is th e an gu lar eq u ation . Notice th at it can b e con s ider ed an eigen v alu e eq u ation for an op er ator
s i n ϑ ∂ ϑ ∂ ϑ s i n 2 ϑ ∂ ϕ 2
1 ∂ ( s in ϑ ∂ ) +
1 ∂ 2 . W h at is th e mean in g of th is op er ator ?
4.2.2 A ngul a r momentum op erato r
×
W e tak e on e s tep b ac k an d lo ok at th e an gu lar momen tu m op er ator . F r om its clas s ical for m L x = x r p x w e can d efi n e th e Q M op er ator :
x ˆ ˆ ˆ ˆ x ˆ
In car tes ian co or d in ates th is r ead s
L = x r × p x = − i x r × ∇
x
L ˆ = y ˆ p ˆ
— p ˆ z ˆ = − i k ( y ∂
— ∂ z )
z
y
∂ z ∂ y
y
L ˆ = z ˆ p ˆ
— p ˆ x ˆ = − i k ( z ∂
— ∂ x )
z
∂ x ∂ z
z
L ˆ = x ˆ p ˆ
— p ˆ y ˆ = − i k ( z ∂
— ∂ y )
x
y
x
∂ y ∂ x
S ome v er y imp or tan t p r op er ties of th is v ector op er ator r egar d its comm u tator . Con s id er for ex amp le [ L ˆ x , L ˆ y ]: [ L ˆ x , L ˆ y ] = [ y ˆ p ˆ z − p ˆ y z ˆ , z ˆ p ˆ x − p ˆ z x ˆ ] = [ y ˆ p ˆ z , z ˆ p ˆ x ] − [ p ˆ y z ˆ , z ˆ p ˆ x ] − [ y ˆ p ˆ z , p ˆ z x ˆ ] + [ p ˆ y z ˆ , p ˆ z x ˆ ]
No w r emem b er th at [ x i , x j ] = [ p i , p j ] = 0 an d [ x i , p j ] = i δ ij . Als o [ AB , C ] = A [ B , C ] + [ A, C ] B . Th is s imp lifi es
matter s a lot
ˆ ˆ
c c c
c c c ˆ
[ L x , L y ] = y ˆ [ p ˆ z , z ˆ ] p ˆ x − c [ p ˆ y c z ˆ , z ˆ p ˆ x ] − c [ y ˆ p ˆ c z , p ˆ z x ˆ ] + p ˆ y [ z ˆ , p ˆ z ] x ˆ = i k ( x ˆ p ˆ y − y ˆ p ˆ x ) = i k L z
By p er for min g a cy clic p er m u tation of th e in d ex es , w e can s h o w th at th is h old s in gen er al:
[ L ˆ a , L ˆ b ] = i k L ˆ c
O b s . S in ce th e d iff er en t comp on en ts of th e an gu lar momen tu m d o n ot c omm u te, th ey d o n ot p os s es s common eigen v alu es an d th er e is an u n cer tain t y r elation for th em. If for ex amp le I k n o w with ab s olu te p r ecis ion th e an gu lar momen tu m alon g th e z d ir ection , I can n ot h a v e an y k n o wled ge of th e comp on en ts alon g x an d y .
? Qu estion : : wh at is th e u n cer tain t y r elation for th e x an d y comp on en ts ?
ΔL x ΔL y ≥ 2 | ( L z ) |
1
1
? Qu estion : As s u me w e k n o w with cer tain t y th e an gu lar momen tu m alon g th e z d ir ection . W h at is th e u n cer tain t y in th e an gu lar momen tu m in th e x an d y d ir ection s ?
x
z
2
y
y
z
2
x
z
F r om th e u n cer tain t y r elation s , ΔL ΔL ≥ | ( L ) | an d ΔL ΔL ≥ | ( L ) | , w e h a v e th at if ΔL = 0 (p er fect
k n o wled ge) th en w e h a v e a comp lete u n cer tain t y in L x an d L y .
{ }
x y z
O b s . Con s id er th e s q u ar ed len gth of th e an gu lar momen tu m v ector L ˆ 2 = L ˆ 2 + L ˆ 2 + L ˆ 2 . W e can sh o w th at [ L ˆ a , L ˆ 2 ] = 0 (for a = x , y , z ). Th u s w e can alw a y s k n o w th e len gth of th e an gu lar momen tu m p lu s on e of its comp on en ts .
( )
( )
F or ex amp le, c h o os in g th e z-comp on en t, w e can r ep r es en t th e an gu lar momen tu m as a con e, of len gth L , p r o jection on th e z-ax is L z an d with comp lete u n cer tain t y of its p r o jection alon g x an d y .
W e n o w ex p r es s th e an gu lar momen tu m u s in g s p h er ical co or d in ates . Th is s imp lifi es p ar ticu lar ly h o w th e azim u th al an gu lar momen tu m L ˆ z is ex p r es s ed :
x
∂ ϑ ∂ ϕ
L ˆ = i k ( s in ϕ ∂
∂
)
+ cot ϑ cos ϕ ,
L ˆ = − i k ( cos ϕ ∂
∂
)
— cot ϑ s in ϕ ,
y
Th e for m of L ˆ 2 s h ou ld b e familiar :
L ˆ z
∂ ϑ ∂ ϕ
∂
= − i k ∂ ϕ
L
= − k
s in ϑ ∂ ϑ + s in 2 ϑ ∂ ϕ 2
ˆ 2 2 [ 1 ∂ (
s in ϑ ∂ ϑ
∂ )
1 ∂ 2 ]
3 k
2 k k
- k
- 2 k
- 3 k
F ig . 2 8 : Gr a p h ic a l r ep r es en ta tio n o f th e a n g u la r mo men tu m, w ith fi xed L z a n d L 2 , b u t c o mp lete u n c er ta in t y in L x a n d L y .
as y ou s h ou ld r ecogn ize th e an gu lar p ar t of th e 3D S c h r ¨ od in ger eq u ation . W e can th en wr ite th e eigen v alu e eq u ation s for th es e t w o op er ator s :
an d
L ˆ 2 Φ ( ϑ , ϕ ) = k 2 l ( l + 1) Φ ( ϑ , ϕ )
L ˆ z Φ ( ϑ , ϕ ) = k m z Φ ( ϑ , ϕ )
wh er e w e alr ead y u s ed th e fact th at th ey s h ar e common eigen fu n ction s (th en , w e can lab el th es e eigen fu n ction s b y l
an d m z : Φ l ,m z ( ϑ , ϕ ).
− −
Th e allo w ed v alu es for l an d m z ar e in teger s s u c h th at l = 0 , 1 , 2 , . . . an d m z = l , . . . , l 1 , l . Th is r es u lt can b e
in fer r ed fr om th e comm u tation r elation s h ip . F or in ter es ted s tu d en ts , th e d er iv ation is b elo w.
De riv ation of the e ige n v alue s . A s s u me t h a t th e eig en v a lu es o f L 2 a n d L z a re u n kn o w n , a n d c a ll th em λ a n d µ . W e in tr o d u c e t w o n ew o p er a to r s , th e r a is in g a n d lo w er in g o p er a to r s L + = L x + iL y a n d L − = L x − iL y . T h e c o mm u ta t o r w ith L z is [ L z , L ± ] = ± k L ± (w h ile th ey o f c o u r s e c o mm u t e w ith L 2 ). N o w c o n s id er t h e fu n c tio n f ± = L ± f , w h er e f is a n eig en fu n c tio n o f L 2 a n d L z :
a n d
L 2 f ± = L ± L 2 f = L ± λf = λf ±
L z f ± = [ L z , L ± ] f + L ± L z f = ± k L ± f + L ± µf = ( µ ± k ) f ±
T h en f ± = L ± f is a ls o a n eig en fu n c tio n o f L 2 a n d L z . F u r th er mo r e, w e c a n k eep fi n d in g eig en fu n c tio n s o f L z w ith h ig h er a n d h ig h er eig en v a lu es µ ′ = µ + k + k + . . . , b y a p p lyin g t h e L + o p er a to r (o r lo w er a n d lo w er w ith L − ), w h ile th e L 2 eig en v a lu e is fi xed . Of c o u r s e th er e is a limit, s in c e w e w a n t µ ′ ≤ λ . T h en t h er e is a ma xim u m eig en fu n c tio n s u c h th a t L + f M = 0 a n d w e s et th e c o r r es p o n d in g eig en v a lu e to k l M . N o w n o tic e th a t w e c a n w r ite L 2 in s tea d o f b y u s in g L x ,y b y u s in g L ± :
z
L 2 = L − L + + L 2 + k L z
U s in g th is r ela tio n s h ip o n f M w e fi n d :
L 2 f m = λf m → ( L − L + + L 2 + k L z ) f M = [0 + k 2 l 2
+ k ( k l M )] f M → λ = k 2 l M ( l M + 1 )
z M
I n th e s a me w a y , th er e is a ls o a min im u m eig en v a lu e l m a n d eig en fu n c tio n s .t. L − f m = 0 a n d w e c a n fi n d λ = k 2 l m ( l m − 1 ). S in c e λ is a lw a ys th e s a me, w e a ls o h a v e l m ( l m − 1 ) = l M ( l M + 1 ), w ith s o lu tio n l m = − l M (th e o th er s o lu tio n w o u ld h a v e l m > l M ). F in a lly w e h a v e fo u n d th a t th e eig en v a lu es o f L z a re b et w een + k l a n d − k l w ith in teg er in c rea s es , s o th a t l = − l + N g ivin g l = N / 2 : th a t is , l is eit h er a n in teg er o r a n h a lf-in teg er . W e t h u s s et λ = k 2 l ( l + 1 ) a n d µ = k m , m = − l , − l + 1 , . . . , l .
�
W e can gath er s ome in tu ition ab ou t th e eigen v alu es if w e s olv e fi r s t th e s econ d eq u ation , fi n d in g
∂ Φ l ,m im z ϕ
− i k ∂ ϕ = k m z Φ ( ϑ , ϕ ) , Φ l ,m ( ϑ , ϕ ) = Θ l ( ϑ ) e
wh er e, b ecau s e of th e p er io d icit y in ϕ , m z can on ly tak e on in teger v alu es (p os itiv e an d n egativ e) s o th at Φ l m ( ϑ , ϕ + 2 π ) = Φ l m ( ϑ , ϕ ).
—
If w e s olv e th e fi r s t eq u ation , w e w ou ld fi n d for eac h eigen v alu e l th er e ar e man y eigen fu n ction s . W h at is th e d egen er acy of th e eigen v alu e l ? W e k n o w th at giv en l , m z can tak e man y v alu es (b et w een l an d l ), in p ar ticu lar 2 l + 1 v alu es . Th is is th e d egen er acy of l .
? Qu estion : W h at ar e th e p os s ib le v alu es of L ˆ x if l = 7 an d m z = 5?
− −
W e k n o w th at w e can d efi n e q u an tu m n u m b er s m x ( y ) s u c h th at th ey tak e in teger n u m b er s m x ( y ) = l , . . . , l 1 , l . Als o, w e h a v e th e r elation amon g th e ex p ectation v alu es :
x y z z x y
\ L ˆ 2 ) = \ L ˆ 2 + L ˆ 2 + L ˆ 2 ) → l ( l + 1) = m 2 + \ L ˆ 2 + L ˆ 2 ) / k 2
s o in gen er al
Th en h er e w e h a v e
\ L ˆ 2 ) ≤ k 2 [ l ( l + 1) − m 2 ]
x
x z
\ L ˆ 2 ) ≤ k 2 (56 − 25) = 31 k 2
x x ,i x ,m ax
L
h a v e L x ,m ax ≤ 5 k (with 5 th e clos es t in teger to √ 31). O ften , b ecau s e of s y mmetr y , w e h a v e \ L ˆ 2 ) = \ L ˆ 2 ) an d,
If L ˆ x cou ld on ly tak e its max im u m v alu e (with p r ob ab ilit y on e) w e w ou ld h a v e \ L ˆ 2 ) = P i L 2 = L 2 th u s w e
x y
x z
\ L ˆ 2 ) = k 2 [ l ( l + 1) − m 2 ] / 2
th u s r es tr ictin g ev en fu r th er th e max im u m v alu e of L x .
4.2.3 Spi n angul a r momentum
Th e q u an tization of an gu lar momen tu m ga v e th e r es u lt th at th e an gu lar momen tu m q u an tu m n u m b er w as d efi n ed b y in teger v alu es . Th er e is an oth er q u an tu m op er ator th at h as th e s ame comm u tation r elation s h ip as th e an gu lar momen tu m b u t h as n o clas s ical cou n ter p ar t an d can as s u me h alf-in teger v alu es . It is called th e in tr in s ic s p in an gu lar
x
S
ˆ momen tu m (or for s h or t, s p in ). Becau s e it is n ot a clas s ical p r op er ties , w e can n ot wr ite s p in in ter ms of p os ition
an d momen tu m op er ator . Th e s p in is d efi n ed in an ab s tr act s p in s p ace (n ot th e u s u al p h as e s p ace).
Ev er y elemen tar y p ar ticle h as a s p ecifi c an d imm u tab le v alu e of th e in tr in s ic s p in q u an tu m n u m b er s (with s d eter min in g th e eigen v alu es of S ˆ 2 , k 2 s ( s + 1)), wh ic h w e call th e s p in of th at p ar ticu lar s p ecies : p i mes on s h a v e s p in 0; electr on s h a v e s p in 1 / 2; p h oton s h a v e s p in 1; gr a v iton s h a v e s p in 2; an d s o on . By con tr as t, th e or b ital an gu lar momen tu m q u an tu m n u m b er l of a p ar ticle can a p r ior i tak e on an y (in teger ) v alu e, an d l will c h an ge wh en th e s y s tem is p er tu r b ed .
2
Th e eigen v ector s of th e s p in op er ator s ar e n ot s p h er ical h ar mon ics . Actu ally , s in ce th e s p in is n ot d efi n ed in ter ms of p os ition an d momen tu m, th ey ar e n ot a fu n ction of p os ition an d ar e n ot d efi n ed on th e u s u al p h as e s p ace. Th e eigen s tates ar e in s tead d es cr ib ed b y lin ear v ector s , for ex amp le, t w o-d imen s ion al v ector s for th e s p in - 1 . Th u s th e
op er ator s will b e as w ell r ep r es en ted b y matr ices .
2
W e alr ead y s a w th e op er ator s d es cr ib in g th e s p in - 1 op er ators an d w e ev en calcu lated th eir eigen v alu es an d eigen v ec
tor s (s ee s ection 2.2 )
W e can th en als o d efi n e th e total an gu lar momen tu m, wh ic h is th e s u m of th e u s u al an gu lar momen tu m (called th e
or bital an gu lar momen tu m) an d th e s p in :
x ˆ x ˆ x ˆ
J = L + S
W h at is th e mean in g of th e s u m of t w o an gu lar momen tu m op er ator s an d wh at ar e th e eigen v alu es an d eigen fu n ction s of th e r es u ltin g op er ator s ?
4.2.4 A ddi ti on of angul a r momentum
W e h a v e s een ab o v e th at an y elemen tar y p ar ticle p os s es an in tr in s ic s p in . Th en , w e can alw a y s d efi n e th e total an gu lar momen tu m as th e s u m of th e or b ital an gu lar momen tu m an d th e in tr in s ic s p in . Th is is an ex amp le of ad d ition of an gu lar momen tu m. Th en of cou r s e w e cou ld als o con s id er t w o d is tin ct p ar ticles an d as k wh at is th e total or b ital an gu lar momen tu m of th e t w o p ar ticles (or of mor e p ar ticles ). Th er e ar e th u s man y cas es of ad d ition of an gu lar momen tu m, for ex amp le:
x ˆ x ˆ x ˆ
1. J = L + S
x ˆ x ˆ x ˆ
2. L = L 1 + L 2
x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ x ˆ
3. J = J 1 + J 2 = L 1 + S 1 + L 2 + S 2
x ˆ x ˆ x ˆ x ˆ
4. S = S 1 + S 2 + S 3
5. . . .
z
z z
| )
Con s id er for ex amp le th e s econ d cas e. A p os s ib le s tate of th e t w o p ar ticles can b e d es cr ib ed b y th e eigen v al u es /eigen fu n ction s of eac h p ar ticle an gu lar momen tu m. F or ex amp le w e cou ld s p ecify l 1 an d m 1 as w ell as l 2 an d m 2 (I will fr om n o w on ju s t wr ite m 1 for m 1 etc.). Th en a s tate cou ld b e for ex amp le wr itten in D ir ac’s n otation as l 1 , m 1 , l 2 , m 2 . Th is h o w ev er d o es n ot tell u s an y th in g ab ou t th e total s y s tem an d its an gu lar momen tu m. S ometime th is q u an tit y is mor e in ter es tin g (for ex amp le if th e t w o p ar ticles ar e in ter actin g, th eir total an gu lar momen tu m is b ou n d to d eter min e th eir en er gy , an d n ot th e s tate of eac h p ar ticle alon e).
A . Cou p led a n d u n cou p led rep resen ta tion s
Th e s u m of an gu lar momen tu m s atis fy th e gen er al comm u tation r u les , [ L 2 , L z ] = 0, [ L x , L y ] = i L z etc. W e can th en
x ˆ 2
als o d efi n e th e eigen v alu es (an d eigen fu n ction s ) of th e total an gu lar momen tu m L , for ex amp le l (for L ) an d m (for
L z ). Ho w ev er , s in ce w e on ly h a v e 2 q u an tu m n u m b er s , w e ex p ect th e eigen fu n ction s to b e d egen er ate an d w e s till n eed to fi n d t w o mor e q u an tu m n u m b er s . Eq u iv alen tly , wh at w e n eed to d o is to fi n d a comp lete s et of comm u tin g ob s er v ab les , s u c h th at an eigen fu n ction (common to all th es e ob s er v ab les ) is w ell d efi n ed –n o am b igu it y in it– b y th e s et of eigen v alu es (or q u an tu m n u m b er s ) of th e ob s er v ab les .
Th e fi r s t q u es tion w e can as k is : ar e th es e eigen fu n ction s goin g to b e in common with th e s in gle p ar ticle op er ator s ? T o d eter min e th is , w e n eed to lo ok at th e comm u tation of th e op er ator s .
1
No w w e k n o w th at [ L 2 , L z , 1 ] = 0, b u t wh at ab ou t [ L 2 , L z , 1 ]?
L 1 + L 2
= L 1 + L 2 + 2 L 1
L 2 . Th en th e comm u tator is :
W e fi r s t ex p r es s L 2 ex p licitly : L 2 = | x ˆ
x ˆ | 2
2 2 x ˆ
· x ˆ
[ L 2 , L z , 1 ] = [ L 2 + L 2 + 2( L x , 1 L x , 2 + L y , 1 L y , 2 + L 1 L 2 ) , L z , 1 ]
1 2 z z
= [2( L x , 1 L x , 2 + L y , 1 L y , 2 ) , L z , 1 ] = 2 i (( L y , 1 L x , 2 − L x , 1 L y , 2 ) / = 0
1
Th u s th e t w o op er ator s d o n ot comm u te an d d o n ot s h ar e common eigen fu n ction s . W h at ab ou t L 2 ?
[ L 2 , L 2 ] = [ L 2 + L 2 + 2( L x , 1 L x , 2 + L y , 1 L y , 2 + L 1 L 2 ) , L 2 ] = 0
1 1 2 z z 1
s in ce [ L 2 , L a, 1 ] = 0. Th is mean s th at th er e ar e common eigen fu n ction s of L 2 , L 2 , L 2 an d L z . Th es e op er ator s ar e a
1 1 2
comp lete s et of comm u tin g ob s er v ab les . An eigen fu n ction is th u s w ell d efi n ed b y th e s et of eigen v alu es l , m, l 1 an d l 2
| )
an d w e can wr ite th e eigen s tates as ψ l ,m ,l 1 ,l 2 or l , m, l 1 , l 2 .
Th er e ar e th en t w o p os s ib le r ep r es en tation s of th e com b in ed s y s tem (t w o p os s ib le b as is to r ep r es en t a gen er al s tate):
R ep resen tati on Ei gen states Comp l ete set of comm u ti n g ob ser v ab l es
1
2
Un cou p led | l 1 , m 1 , l 2 , m 2 ) , L 2 , L 2 , L 1 ,z an d L 2 ,z
1
2
Cou p led | l , m, l 1 , l 2 ) , L 2 , L 2 , L 2 an d L z .
Ho w d o w e go fr om on e b as is to th e oth er ? As u s u al th is is d on e b y e x p r es s in g eac h v ector in on e b as is as a lin ear com b in ation of v ector s in th e oth er b as is :
| l , m, l 1 , l 2 ) =
m L 1 ,m 2
c
l m 1 ,m 2
| l 1 , m 1 , l 2 , m 2 )
Notice th at s in ce th e total an gu lar momen tu m in th e z d ir ection m u s t b e m , w e limit th e s u m to ter ms s .t. m 1 + m 2 =
m .
m 1 ,m 2
W h at ar e th e co efficien ts c l ?
S in ce th e t w o r ep r es en tation s ar e t w o or th ogon al b as is , w e h a v e th at ( l 1 ′ , m ′ 1 , l 2 ′ , m 2 ′ | l 1 , m 1 , l 2 , m 2 ) = 0 u n les s all th e in d ex es ar e eq u al. Th en th e co efficien t can b e calcu lated (as u s u al!) fr om th e in n er p r o d u ct of | l 1 , m 1 , l 2 , m 2 ) an d
| l , m, l 1 , l 2 ) :
c
l m 1 ,m 2
= ( l 1 , m 1 , l 2 , m 2 | l , m, l 1 , l 2 )
Th es e co efficien ts ar e called th e Cleb s c h -G or d on co efficien ts .
B . A d d ition ru les: Tw o p a rticles
In d es cr ib in g th e en er gy lev els of atoms an d n u clei it is v er y con v en ien t to b e ab le to fi n d th e allo w ed v alu es of ( l , m ) giv en th e v alu es of ( l 1 , l 2 ) for t w o giv en p ar ticles (e.g. electr on s , p r oton s or n eu tr on s ). In d eed , w e s a w th at th e op er ator L ˆ 2 ap p ear s in th e Hamilton ian of th e s y s tem. Th u s its eigen v alu e k 2 l ( l + 1) will b e imp or tan t in d eter min in g th e s y s tem en er gy .
Ev en if w e can n ot fi x th e v alu e of l if w e on ly k n o w l 1 , m 1 , l 2 an d m 2 w e can at leas t r es tr ict th e p os s ib le v alu es of l . In or d er to d o s o, on e h as to an aly ze th e p os s ib le max im u m len gth of th e total an gu lar momen tu m an d th e d egen er acy of th e eigen v alu es .
− −
− −
1) Max i m u m l : F or t w o p ar ticles with q u an tu m n u m b er s l 1 an d l 2 w e k n o w th at in th e cou p led r ep r es en tation w e can n ot fi x th e v alu es of m 1 an d m 2 . Ho w ev er , w e k n o w th at giv en l 1 an d l 2 on ly s ome v alu es of m 1 an d m 2 ar e allo w ed (e.g. m 1 = l 1 , l 1 + 1 , . . . , l 1 ). Th en th e max im u m v alu es of m 1 an d m 2 ar e m 1 = l 1 an d m 2 = l 2 . Th is als o d eter min es th e max im u m v alu e of m : m m ax = l 1 + l 2 . Bu t m its elf can on ly tak e v alu es m = l , . . . , l 1 , l . Th en th e max im u m v alu e of l is m m ax = l m ax .
Th u s , wh at w e ju s t p r o v ed is th at l ≤ l 1 + l 2 .
s tate cou ld b e als o wr itten (in th e u n cou p led r ep r es en tation ) as c l
| l 1 , m 1 , l 2 , m 2 ) , th e d egen er acy of
2) Mi n i m u m l : T o fi n d th e min im u m l v alu e w e n eed to lo ok at th e L d egen er acy of th e s tate | l , m, l 1 , l 2 ) . S in ce th is
m 1 ,m 2
th e s tate m u s t b e th e s ame. W h at ar e th e t w o d egen er acies ?
m 1 + m 2 = m
W e k n o w th at for a giv en an gu lar momen tu m op er ator L ˆ with total an gu lar momen tu m q u an tu m n u m b er l , th er e ar e 2 l + 1 s tates with th e s ame an gu lar momen tu m k 2 l ( l + 1).
Th en , con s id er in g th e u n cou p led r ep r es en tation w e h a v e D = (2 l 1 + 1)(2 l 2 + 1) p os s ib le s tates with l 1 an d l 2 .
l = l m in
In th e cou p led r ep r es en tation in s tead w e h a v e to con s id er all th e s tates th at h a v e an allo w ed l : D = L l 1 + l 2 (2 l + 1).
W e w an t th es e t w o q u an tities to b e eq u al. No w r emem b er th at L K k = K ( K + 1) . Th en L l 1 + l 2 (2 l + 1) = (1 + l 1 +
k = 1 2 l = l m in
m in m in
l 2 ) 2 − l 2 , s o th at l 2 = (1 + l 1 + l 2 ) 2 − (2 l 1 + 1)(2 l 2 + 1) = ( l 1 − l 2 ) 2 .
Us in g th e d egen er acy con d ition w e th u s p r o v ed th at l ≥ | l 1 − l 2 | .
Th e ad d ition r u le s tates th u s th at
Th e total an gu lar momen tu m q u an tu m n u m b er is b ou n d ed b y | l 1 − l 2 | ≤ l ≤ l 1 + l 2
Ex amp le: Con s id er t w o s p in s -1/2 p ar ticles (for ex amp le t w o electr on s with zer o or bital an gu lar momen tu m). S in ce w e
)
)
)
c h o os e s p in - 1 w e h a v e on ly 1 p os s ib le v alu e s = 1 an d t w o v alu es for m z : m z = ± 1 . W e can omit wr itin g ex p licitly
th e s q u an tu m n u m b er (s in ce it’s alw a y s 2 , an d w e wr ite u n cou p led r ep r es en tation is th en giv en b y :
+ 2 , + 2
=
+ 2
an d
+ 2 , − 2
=
− 2
. A b as is for th e
+ 1 , + 1 , + 1 , + 1 ) = + 1 , + 1 )
2 2 2 2 2 2
)
)
+ 1 , + 1 , + 1 , − 1 = + 1 , − 1
)
)
| s 1 , m 1 , s 2 , m 2 ) =
2 2 2 2 2 2
+ 1 , − 1 , + 1 , + 1 = − 1 , + 1
2
2
2
2 2 2 ) )
2
2
2
2
2
2
+ 1 , − 1 , + 1 , − 1 = − 1 , − 1
—
Con s id er n o w th e cou p led r ep r es en tation . Th e p os s ib le v alu es for s ar e 1 or 0. In th e fi r s t cas e, w e h a v e 3 p os s ib le v alu es for m = 1 , 0 , 1. W h ile th e s econ d on ly h as m = 0. Again , s in ce th e v alu es of s 1 an d s 2 ar e fi x ed w e d o n ot wr ite th em:
2
2
0 , 0 , 1 , 1 ) = ) | 0 , 0 )
1 , − 1 , 1 , 1 = | 1 , − 1 )
2
2
1 , 0 , 1 , 1 ) = | 1 , 0 )
| s , m, s 1 , s 2 ) =
2 ) 2
2
2
1 , 1 , 1 , 1 = | 1 , 1 )
In th is p ar ticu lar ex amp le it is eas y to calcu late th e Cleb s c h -G or d on co efficien ts an d w e fi n d th e r elation s b et w een th e t w o r ep r es en tation s :
| − ) − | − )
+ 1 , 1 1 , + 1
2 2 2 2
| 0 , 0 ) = √ 2
2
2
| 1 , − 1 ) = − 1 , − 1 )
√ 2
2 2 2 2
| 1 , 0 ) =
| 1 , 1 ) =
2
2
+ 1 , + 1 )
C. A d d ition ru les: ma n y p a rticles
Th e ad d ition r u les can b e gen er alized to man y p ar ticles , b y ju s t r ep etitiv ely ap p ly in g th e t w o-p ar ticle r u les . W e th en fi n d for N p ar ticles :
k = 1
- l m ax = L N l k
- l m in = max { 0 , 2 l N − l m ax }
wh er e l N is th e lar ges t of th e { l k } .
4. 3 Solutions to the Sc hr o ¨ dinger equation in 3D
W e n o w go b ac k to th e S c h r ¨ od in ger eq u ation in s p h er ical co or d in ates an d w e con s id er th e an gu lar an d r ad ial eq u ation s ep ar ately to fi n d th e en er gy eigen v alu es an d eigen fu n ction s .
D. A n gu la r Eq u a tion
Th e an gu lar eq u ation w as fou n d to b e:
l
1 ∂
( s in ϑ
∂ Y m ( ϑ , ϕ )
l
l
)
+
1 ∂ 2 Y m ( ϑ , ϕ )
= − l ( l + 1) Y m
( ϑ , ϕ )
s in ϑ ∂ ϑ ∂ ϑ s in 2 ϑ ∂ ϕ 2
Notice th at th is eq u ation d o es n ot d ep en d at all on th e p oten tial, th u s it will b e common to all p r ob lems with an is otr op ic p oten tial.
)]
W e can s olv e th e eq u ation b y u s in g again s ep ar ation of v ar iab les : Y ( ϑ , ϕ ) = Θ ( ϑ ) Φ ( ϕ ). By m u ltip ly in g b oth s id es of th e eq u ation b y s in 2 ( ϑ ) / Y ( ϑ , ϕ ) w e ob tain :
1
Θ ( ϑ )
s in ϑ d
[
d ϑ
s in ϑ d Θ
(
d ϑ
+ l ( l + 1) s in 2
1 d 2 Φ
ϑ = − Φ ( ϕ ) d ϕ 2
As u s u al w e s ep ar ate th e t w o eq u ation s in th e d iff er en t v ar iab les an d in tr o d u ce a con s tan t C = m 2 :
d 2 Φ 2
d ϕ 2 = − m Φ ( ϕ )
d ( d Θ ) [ ]
s in ϑ s in ϑ = m 2 − l ( l + 1) s in 2 ϑ Θ ( ϑ )
d ϑ d ϑ
± ±
Th e fi r s t eq u ation is eas ily s olv ed to giv e Φ ( ϕ ) = e im ϕ with m = 0 , 1 , 2 , . . . s in ce w e n eed to imp os e th e p er io d icit y of Φ , s u c h th at Φ ( ϕ + 2 π ) = Φ ( ϕ ).
l
Th e s olu tion s to th e s econ d eq u ation s ar e as s o ciated Legen d r e P oly n omials : Θ ( ϑ ) = AP m (cos ϑ ), th e fi r s t few of wh ic h ar e in tab le 1 . Notice th at, as p r ev iou s ly fou n d wh en s olv in g for th e eigen v alu es of the an gu lar momen tu m, w e h a v e th at m = − l , − l + 1 , . . . , l , with l = 0 , 1 , . . . .
|
l \ m 0 |
0 |
1 |
2 |
3 |
|
P 0 = 1 0 P 0 = cos ϑ 1 P 0 = 1 (3 cos 2 ϑ − 1) 2 2 P 0 = 1 (5 cos 3 ϑ − 3 cos ϑ ) 3 2 |
||||
|
1 2 3 |
P 1 = s in ϑ 1 P 1 = 3 cos ϑ s in ϑ 2 P 1 = 3 (5 cos 2 ϑ − 1) s in ϑ 3 2 |
P 2 = 3 s in 2 ϑ 2 P 2 = 15 cos ϑ s in 2 ϑ 3 |
P 3 = 15 s in 3 ϑ 3 |
T a b le 1 : Leg en d r e P o lyn o mia ls
Th e n or malized an gu lar eigen fu n ction s ar e th en S p h er ical Har mon ic fu n ction s , giv en b y th e n or malized Legen d r e p oly n omial times th e s olu tion to th e eq u ation in ϕ , (s ee als o T ab le 2 )
l
4 π ( l + m )!
l
Y m ( ϑ , ϕ ) = (2 l + 1 ) ( l − m )! P m (cos ϑ ) e im ϕ
As w e ex p ect fr om eigen fu n ction s , th e S p h er ical Har mon ics ar e or th ogon al:
∫ 4 π
l
dΩ Y m
( ϑ , ϕ ) Y m ′
l ′
( ϑ , ϕ ) = δ l ,l ′ δ m ,m ′
( V − E ) = l ( l + 1) R ( r )
E. Th e ra d ia l eq u a tion
W e n o w tu r n to th e r ad ial eq u ation :
r
—
d ( 2 d R ( r ) )
d r
d r
2 m r 2
k 2
0 I 1
Y 2 − 2 ( ϑ , φ ) = 1 I 15 s in 2 ϑ e − 2 iφ
4 2 π
I
Y 0 ( ϑ , φ ) = 4 π
Y − 1 ( ϑ , φ ) = 1 I 15 s in ϑ cos ϑ e − iφ
I
1 8 π
2
Y − 1 ( ϑ , φ ) = 3 s in ϑ e − iφ 2
2 2 π
0 1 5
Y 0 ( ϑ , φ ) = I 3
cos ϑ
Y 2 ( ϑ , φ ) = 4
π (3 cos
I
ϑ − 1)
I
1 4 π
3
2
2 2 π
2
4 2 π
Y 1 ( ϑ , φ ) = − 1 15 s in ϑ cos ϑ e iφ
1
iφ
Y 1 ( ϑ , φ ) = − 8 π s in ϑ e
Y 2 ( ϑ , φ ) = 1 I 15 s in 2 ϑ e 2 iφ
T a b le 2 : S p h er ic a l H a r mo n ic s
T o s imp lify th e solu tion, w e in tr o d u ce a d iff er en t fu n ction u ( r ) = r R ( r ). Th en th e eq u ation r ed u ces to:
− 2 m d r 2 + V + 2 m r 2
u ( r ) = E u ( r )
k 2 d 2 u [ k 2 l ( l + 1) ]
Th is eq u ation is v er y s imilar to th e S c h r ¨ od in ger eq u ation in 1D if w e d efi n e an effe ctive p otential V ′ ( r ) = V ( r ) +
1 2 l ( l + 1) . Th e s econ d ter m in th is eff ectiv e p oten tial is called th e cen tri fu gal ter m.
2 m r 2
S olu tion s can b e fou n d for s ome for ms of th e p oten tial V ( r ), b y fi r s t calcu latin g th e eq u ation s olu tion s u n,l ( r ), th en
fi n d in g R n,l ( r ) = u n,l ( r ) /r an d fi n ally th e w a v efu n ction
l
Ψ n,l ,m ( r , ϑ , ϕ ) = R n,l ( r ) Y m ( ϑ , ϕ ) .
Notice th at w e n eed 3 q u an tu m n u m b er s ( n, l , m ) to d efi n e th e eigen fu n ction s of th e Hamilton ian in 3D .
F or ex amp le w e can h a v e a s imp le s p h er ical w ell: V ( r ) = 0 for r < r 0 an d V ( r ) = V 0 oth er wis e. In th e cas e of l = 0, th is is th e s ame eq u ation as for th e s q u ar e w ell in 1D . Notice h o w ev er th at s in ce th e b ou n d ar y con d ition s n eed to b e s u c h th at R ( r ) is fi n ite for all r , w e n eed to imp os e th at u ( r = 0) = 0, h en ce on ly th e o d d s olu tion s ar e accep tab le (as w e h ad an ticip ated ). F or l > 0 w e can fi n d s olu tion s in ter ms of Bes s el fu n ction s
r − V
2
Tw o oth er imp or tan t ex amp les of p oten tial ar e th e h ar mon ic os cillator p oten tial V ( r ) = V 0 r 2
0
0 (wh ic h is an
ap p r o x imation for an y p oten tial clos e to its min im u m) an d th e Cou lom b p oten tial V ( r ) = − e 2
1 , wh ic h d es cr ib es
th e atomic p oten tial an d in p ar ticu lar th e Hy d r ogen atom.
4.3.1 T he Hydrogen atom
4 π ǫ 0 r
κ = I − , an d th e qu an tities :
W e w an t to s olv e th e r ad ial eq u ation for th e Cou lom b p oten tial, or at leas t fi n d th e eigen v alu es of th e eq u ation . Notice w e ar e lo ok in g for b ou n d s tates , th u s th e total en er gy is n egativ e E < 0. Th en w e d efi n e th e r eal q u an tit y
2 m E 8
1 2
Boh r r ad iu s : a 0 =
4 π ǫ 0 k 2
m e e 2
k 2
0
, Ry d b er g con s tan t: R = 2 ma 2
an d λ 2 = R . Th e v alu es of th e t w o con s tan ts ar e a
| E | 0
= 5 . 29 × 10 − 11 m an d R = 13 . 6 eV (th u s λ is a d imen s ion les s
p ar ameter ). Th e Boh r r ad iu s giv es th e d is tan ce at wh ic h th e k in etic en er gy of an electr on (clas s ically ) or b itin g
2
ar ou n d th e n u cleu s eq u als th e Cou lom b in ter action : 1 m v 2 = 1 e . In th e s emi-clas s ical Boh r mo d el, th e an gu lar
2 e 4 π ǫ 0 r
momen tu m L = m e v r is q u an tized , with lo w es t v alu e L = k , th en b y in s er tin g in th e eq u ation ab o v e, w e fi n d r = a 0 . W e will s ee th at th e Ry d b er g en er gy giv es in s tead th e min im u m en er gy for th e h y d r ogen .
W e fu r th er ap p ly a c h an ge of v ar iab le ρ = 2 κr , an d w e r ewr ite th e r ad ial eq u ation as :
d ρ 2 =
4 − ρ +
u ( ρ )
d 2 u [ 1 λ l ( l + 1) ]
ρ 2
Th er e ar e t w o limitin g cas es :
d ρ 2 4
d ρ 2 ρ 2
F or ρ → 0, th e eq u ation r ed u ces to d 2 u = l ( l + 1) u , with s olu tion u ( ρ ) ∼ ρ l + 1 . F or ρ → ∞ w e h a v e d 2 u = u ( ρ ) , giv in g u ( ρ ) ∼ e − ρ/ 2 .
8 N o t e th a t t h e d efi n itio n o f th e B o h r r a d iu s is s lig h tly d iffer en t if th e C o u lo m b p o ten tia l is n o t exp r es s ed in S I u n its b u t in c g s u n its
L L
A gen er al s olu tion can th en b e wr itten as u ( ρ ) = e − ρ/ 2 ρ l + 1 S ( ρ ) (with S to b e d eter min ed ). W e th en ex p an d S ( ρ ) in s er ies as S ( ρ ) = j ∞ = 0 s j ρ j an d tr y to fi n d th e co efficien ts s j . By in s er tin g u ( ρ ) = e − ρ/ 2 ρ l + 1 j ∞ = 0 s j ρ j in th e
eq u ation w e h a v e:
[ d 2 d ]
F r om wh ic h w e ob tain :
ρ d ρ 2 + (2 l + 2 − ρ ) d ρ − ( l + 1 − λ ) S ( ρ ) = 0
L [ ρ { j ( j + 1) s j + 1 ρ j − 1 } + (2 l + 2 − ρ ) { ( j + 1) s j + 1 ρ j } − ( l + 1 − λ ) { s j ρ j } ] = 0
j
j
(wh er e th e ter ms in b r ac k ets cor r es p on d to th e d er iv ativ es of S ( ρ )). Th is eq u ation d efi n es a r ecu r s iv e eq u ation for th e co efficien ts s j :
s j + 1
= j + l + 1 − λ s j ( j + 1) + (2 l + 2)( j + 1)
→ → ∞
If w e w an t th e fu n ction u ( ρ ) to b e w ell d efi n ed , w e m u s t imp os e th at u ( ρ ) 0 for ρ . Th is imp os es a max im u m v alu e for j , j m ax , s u c h th at all th e h igh er co efficien ts s j > j m a x ar e zer o.
W e th u s imp os e th at s j m a x + 1 = 0, s ettin g th e n u mer ator of th e for m u la ab o v e to zer o.
Th en w e h a v e th at j m ax + l + 1 − λ = 0. Bu t th is is an eq u ation for λ , wh ic h in tu r n s d eter min es th e en er gy eigen v alu e:
λ = j m ax + l + 1 .
W e th en r en ame th e p ar ameter λ th e p r in cip al q u an tu m n u m b er n , s in ce it is an in teger (as j an d l ar e in teger s ).
n 2
Th en th e en er gy is giv en b y E = − R an d th e allo w ed ener gies ar e giv en b y th e famou s Boh r for m u la :
E n = − n 2 2 k 2
1 m
e
(
e
2
4 π ǫ
0
)
2
—
O b s . : Note th at th e en er gy is on ly d eter min ed b y th e p r in cip al q u an tu m n u m b er . W h at is th e d egen er acy of th e n q u an tu m n u m b er ? W e k n o w th at th e fu ll eigen fu n ction is s p ecifi ed b y k n o win g th e an gu lar momen tu m L 2 an d on e of its comp on en ts (e.g. L z ). F r om th e eq u ation ab o v e, n = j m ax + l + 1, w e s ee th at for eac h n , l can v ar y fr om l = 0 to l = n 1. Th en w e als o h a v e 2 l + 1 m v alu es for eac h l (an d 2 s p in s tates for eac h m ). F in ally , th e d egen er acy is th en giv en b y
L 2
n − 1
2(2 l + 1) = 2 n
l = 0
4.3.2 A tomi c p eri o di c s truc ture
W e calcu lated th e en er gy lev els for th e Hy d r ogen atom. Th is will giv e u s s p ectr os cop y in for mation ab ou t th e ex cited s tates th at w e can ex cite u s in g, for ex amp le, las er ligh t. Ho w can w e u s e th is in for mation to in fer th e s tr u ctu r e of th e atoms ?
A n eu tr al atom, of atomic n u m b er Z, con s is ts of a h ea v y n u cleu s , with electr ic c h ar ge Ze, s u r r ou n d ed b y Z electr on s (mas s m an d c h ar ge -e). Th e Hamilton ian for th is s y s tem is
2
L Z [ k 2
1 Z e 2 ]
1 1 L e 2
0
j
H = − 2 m ∇ j − 4 π ǫ r
j = 1
+
Z
2 4 π ǫ
0 j / = k
| x r j
— x r k |
Th e fi r s t ter m is s imp ly th e k in etic en er gy of eac h electr on in th e atom. Th e s econ d ter m is th e p oten tial en er gy of th e jth electr on in th e electr ic fi eld cr eated b y th e n u cleu s . F in ally th e las t s u m (wh ic h r u n s o v er all v alu es of j an d k ex cep t j = k ) is th e p oten tial en er gy as s o ciated with th e m u tu al r ep u ls ion of th e electr on s (th e factor of 1/2 in fr on t cor r ects for th e fact th at th e s u mmation cou n ts eac h p air t wice).
G iv en th is Hamilton ian , w e w an t to fi n d th e en er gy lev els (an d in p ar ticu lar th e gr ou n d s tate, wh ic h will giv e u s th e s tab le atomic con fi gu r ation ). W e th en n eed to s olv e S c h r ¨ od in ger ’s eq u ation . Bu t wh at w ou ld an eigen s tate of th is eq u ation n o w b e?
Con s id er for ex amp le Heliu m, an atom with on ly t w o electr on s . Neglectin g for th e momen t s p in , w e can wr ite th e w a v efu n ction as Ψ ( x r 1 , x r 2 , t ) (an d s tation ar y w a v efu n ction s , ψ ( x r 1 , x r 2 )), th at is , w e h a v e a fu n ction of th e s p atial co or d in ates of b oth electr on s . Th e p h y s ical in ter p r etation of th e w a v efu n ction is a s imp le ex ten s ion of th e on e-p ar ticle w a v efu n ction : | ψ ( x r 1 , x r 2 ) | 2 d 3 x r 1 d 3 x r 2 is th e p r ob ab ilit y of fi n d in g con temp or an eou s ly th e t w o electr on s at th e p os ition s
J | |
x r 1 a n d x r 2 , r es p ectiv ely . Th e w a v efu n ction m u s t th en b e n or malized as ψ ( x r 1 , x r 2 ) 2 d 3 x r 1 d 3 x r 2 = 1. Th e gen er alization to man y electr on s (or mor e gen er ally to man y p ar ticles ) is th en ev id en t.
T o d eter min e th e gr ou n d s tate of an atom w e will th en h a v e to s olv e th e S c h r ¨ od in ger eq u ation
H ψ ( x r 1 , . . . , x r Z ) = E ψ ( x r 1 , . . . , x r Z )
Th is eq u ation h as n ot b een s olv ed (y et) ex cep t for th e cas e Z= 1 of th e Hy d r ogen atom w e s a w ear lier . W h at w e can d o is to mak e a v er y cr u d e ap p r o x imation an d ign or e th e Cou lom b r ep u ls ion amon g electr on s . M ath ematically th is s imp lifi es tr emen d ou s ly th e eq u ation , s in ce n o w w e can s imp ly u s e s ep ar ation of v ar iab les to wr ite man y eq u ation s for eac h in d ep en d en t electr on . Ph y s ically , th is is often a go o d en ou gh ap p r o x imation b ecau s e m u tu al r ep u ls ion of electr on is n ot as s tr on g as th e attr action fr om all th e p r oton s . Th en th e S c h r ¨ od in ger eq u ation b ecomes :
2
− 2 m ∇ j − 4 π ǫ
L Z [ k 2
j = 1
0
r j
ψ ( x r 1 , . . . , x r Z ) = E ψ ( x r 1 , . . . , x r Z )
an d w e can wr ite ψ ( x r 1 , . . . , x r Z ) = ψ ( x r 1 ) ψ ( x r 2 ) . . . ψ ( x r Z )
Th en , w e can s olv e for eac h electr on s ep ar ately , as w e d id for th e Hy d r ogen atom eq u ation , an d fi n d for eac h electr on
th e s ame lev el s tr u ctu r e as for th e Hy d r ogen , ex cep t th at th e s in ce th e p oten tial en er gy is n o w
4 π ǫ 0 r j
en er gy (Boh r ’s for m u la) is n o w m u ltip lied b y Z . Th e s olu tion s to th e time-in d ep en d en t S c h r ¨ od in ger eq u ation s ar e
th en th e s ame eigen fu n ction s w e fou n d for th e h y d r ogen atom, ψ ( x r j = ψ l m n ( r , ϑ , ϕ ).
Th u s if w e ign or e th e m u tu al r ep u ls ion amon g electr on s , th e in d iv id u al electr on s o ccu p y on e-p ar ticle h y d r ogen ic s tates ( n, l , m ), called orb i tal s , in th e Cou lom b p oten tial of a n u cleu s with c h ar ge Z e .
Th er e ar e 2 n 2 h y d r ogen ic w a v e fu n ction s (all with th e s ame en er gy E n ) for a giv en v alu e of n . Lo ok in g at th e P er io d ic T ab le w e s ee th is p er io d icit y , with t w o elemen ts in th e n = 1 s h ell, 8 in th e n = 2 s h ell, 18 in th e th ir d s h ell. High er s h ells h o w ev er ar e mor e in fl u en ced b y th e electr on -electr on r ep u ls ion th at w e ign or ed , th u s s imp le con s id er ation s fr om th is mo d el ar e n o lon ger v alid .
Ho w ev er , w e w ou ld ex p ect in s tead th e electr on s in th e atoms to o ccu p y th e s tate with lo w es t en er gy . Th e gr ou n d s tate w ou ld th en b e a s itu ation w er e all th e electr on o ccu p y th eir o wn gr ou n d s tate ( n = 0, l = 0, m = 0). Bu t is th is cor r ect? Th is is n ot wh at is ob s er v ed in n atu r e, oth er wis e all th e atom w ou ld s h o w th e s ame c h emical p r op er ties . S o wh at h ap p en s ?
2
T o u n d er s tan d , w e n eed to an aly ze th e s tatis tical p r op er ties of id en tical p ar ticles . Bu t b efor e th at, w e will in tr o
r − V
d u ce th e s olu tion for an oth er cen tr al p oten tial, th e h ar mon ic os cillator p oten tial V ( r ) = V 0 r 2
0
ap p r o x imation for an y p oten tial clos e to its min im u m).
4.3.3 T he Ha rmoni c Os c i l l ato r P otenti al
0 (wh ic h is an
Th e q u an tu m h .o. is a mo d el th at d es cr ib es s y s tems with a c h ar acter is tic en er gy s p ectr u m, giv en b y a lad d er of ev en ly s p aced en er gy lev els . Th e en er gy d iff er en ce b et w een t w o con s ecu tiv e lev els is ΔE . Th e n u m b er of lev els is in fi n ite, b u t th er e m u s t ex is t a min im u m en er gy , s in ce th e en er gy m u s t alw a y s b e p os itiv e. G iv en th is s p ectr u m, w e ex p ect th e Hamilton ian will h a v e th e for m
2
H | n ) = ( n + 1 )
ω | n ) ,
wh er e eac h lev el in th e lad d er is id en tifi ed b y a n u m b er n . Th e n ame of th e mo d el is d u e to th e an alogy with c h ar acter is tics of clas s ical h .o., wh ic h w e will r ev iew fi r s t.
A . Cla ssica l h a rmon ic oscilla to r a n d h .o. mo d el
2
A clas s ical h .o. is d es cr ib ed b y a p oten tial en er gy V = 1 k x 2 (th e r ad ial p oten tial con s id er ed ab o v e, V ( r ) = V
2
r
r
0 2
0
0
− V ,
±
h as th is for m). If th e s y s tem h as a fi n ite en er gy E , th e motion is b ou n d b y t w o v alu es x 0 , s u c h th at V ( x 0 ) = E . Th e eq u ation of motion is giv en b y
an d th e k in etic en er gy is of cou r s e
d x p ( t )
{ = ,
d t m
d t
d p = − k x
d 2 x
→ m d x 2 = − k x ,
1 2 p 2
T = m x ˙ = .
2 2 m
Th e en er gy is con s tan t s in ce it is a con s er v ativ e s y s tem, with n o d is s ip ation . M os t of th e time th e p ar ticle is in th e p os ition x 0 s in ce th er e th e v elo cit y is zer o, wh ile at x = 0 th e v elo cit y is max im u m.
Th e h .o. os cillator in Q M is an imp or tan t mo d el th at d es cr ib es man y d iff er en t p h y s ical s itu ation s . It d es cr ib es e.g. th e electr omagn etic fi eld , v ib r ation s of s olid -s tate cr y s tals an d (a s imp lifi ed mo d el of ) th e n u clear p oten tial. Th is is b ecau s e an y p oten tial with a lo cal min im u m can b e lo cally d es cr ib ed b y an h .o.. Pr o v id ed th at th e en er gy is lo w en ou gh (or x clos e to x 0 ), an y p oten tial can in fact b e ex p an d ed in s er ies , giv in g: V ( x ) ≈ V ( x 0 ) + b ( x − x 0 ) 2 + . . .
d x 2
x 0
wh er e b = d 2 V | .
It is eas y to s olv e th e eq u ation of motion . In s tead of ju s t s olv in g th e u s u al eq u ation , w e follo w a s ligh tly d iff er en t
r ou te. W e d efi n e d imen s ion les s v ar iab les ,
p
P = √ mω
, X = x √ mω ,
wh er e w e d efi n ed a p ar ameter with u n its of fr eq u en cy : ω = k /m a n d w e in tr o d u ce a comp lex clas s ical v ar iab le (follo win g Ro y J . G lau b er – Ph y s . Rev . 131 , 2766–2788 (1963))
/
1
α = √ 2 ( X + iP ) .
Th e clas s ical eq u ation s of motion for x an d p d efi n e th e ev olu tion of th e v ar iab le α :
→
d t m
d p = − k x
= − iω α ( t )
{ d x = p ( t ) , dα
d t
dt
√ √
Th e ev olu t i o n of α is th er efor e j u s t a r otation in its p h as e s p ace: α ( t ) = α (0) e − iω t .
S in ce X = 2 R e ( α ) an d P = 2 I m ( α ), X an d P os cillate, as u s u al in th e clas s ical cas e:
2
X = √ 1 ( α 0 e − iω t + α 0 ∗ e iω t )
2
P = √ − i ( α 0 e − iω t − α 0 ∗ e iω t )
0
Th e clas s ical en er gy , giv en b y ω / 2( X 2 + P 2 ) = ω α 2 , is con s tan t at all time.
B . Oscilla to r Ha milton ia n : P osition a n d momen tu m op era to rs
Us in g th e op er ator s as s o ciated with p os ition an d momen tu m, th e Hamilton ian of th e q u an tu m h .o. is wr itten as :
p 2 k x 2 p 2
1 2 2
H = 2 m + 2
= + mω x .
2 m 2
2
H
In ter ms of th e d imen s ion les s v ar iab les , P an d X , th e Hamilton ian is = ω ( X 2 + P 2 ).
In an alogy with th e clas s ical v ar iab le a ( t ) [an d its comp lex con ju gate a ∗ ( t ), wh ic h s imp lifi ed th e eq u ation of motion , w e in tr o d u ce t w o op er ator s , a , a † , h op in g to s imp lify th e eigen v alu e eq u ation (time-in d ep en d en t S c h r ¨ od in ger eq u ation ):
a = √ 1 ( X + iP ) = √ 1 ( √ mω x + √ i p )
2 1 2 1 √ m ω
2 1 2 1 m ω
a † = √ 1 ( X − iP ) = √ 1 ( mω x − √ i p ) ,
2 2
Als o, w e d efi n e th e n u m b er op er ator as N = a † a , with eigen v alu es n an d eigen fu n ction s | n ) . Th e Hamilton ian can b e wr itten in ter ms of th es e op er ator s . W e s u b s titu te a , a † at th e p lace of X an d P , y ield in g H = k ω ( a † a + 1 ) = k ω ( N + 1 )
[ ] [ ]
an d th e min im u m en er gy ω / 2 is called th e zer o p oin t en er gy .
Th e comm u tation p r op er ties ar e: a, a † = 1 an d [ N , a ] = − a, N , a † = a † . Als o w e h a v e:
p = i I m ω 1 ( a † − a )
2 m ω
x = I 1 ( a † + a )
2
� P r o v e th e c o mm u ta tio n r ela tio n s h ip s o f th e r a is in g a n d lo w er in g o p er a to r s .
[ a, a † ] = 1 [ X + iP , X − iP ] = 1 i i
2 2
S o w e a ls o h a v e aa † = [ a, a † ] + a † a = 1 + a † a = 1 + N .
( [ X , − iP ] + [ iP , X ]) = − k [ X , P ] = − k [ x, p ] = 1
[ N , a ] = [ a † a, a ] = [ a † , a ] a = − a a n d [ N , a † ] = [ a † a, a † ] = a † [ a, a † ] = a †
�
F r om th e comm u tation r elation s h ip s w e h a v e:
a | n ) = [ a, N ] | n ) = an | n ) − N a | n ) → N ( a | n ) ) = ( n − 1)( a | n ) ) ,
th at is , a | n ) is als o an eigen v ector of th e N op er ator , with eigen v alu e ( n − 1). Th u s w e con fi r m th at th is is th e lo w er in g op er ator : a | n ) = c n | n − 1 ) . S imilar ly , a † | n ) is an eigen v ector of N with eigen v alu e n + 1:
a † | n ) = [ N , a † ] | n ) = N a † | n ) − a † n | n ) → N ( a † | n ) ) = ( n + 1)( a | n ) ) .
W e th u s h a v e a | n ) = c n | n − 1 ) an d a † | n ) = d n | n + 1 ) . W h at ar e th e co efficien ts c n , d n ?
S in ce
an d
( n | N | n ) = ( n | a † a | n ) = n
n
( n | a † a | n ) = ( ( an | )( a | n ) ) = ( n − 1 | n − 1 ) c 2 ,
w e m u s t h a v e c n = √ n . An alogou s ly , s in ce aa † = N + 1, as s een fr om th e comm u tation r elation s h ip :
n
d 2 ( n + 1 | n + 1 ) = ( a † n | a † n ) = ( n | aa † | n ) ( n | ( N + 1) | n ) = n + 1
a | n ) = √ n | n − 1 ) ; a † | n ) = √ n + 1 | n + 1 ) .
S o in th e en d w e h a v e :
| )
| ) | )
( | | ) ( | ) ≥
All th e n eigen v alu es of N h a v e to b e n on -n egativ e s in ce n = n N n = ψ n 1 ψ n 1 0 (th is follo ws fr om th e p r op er ties of th e in n er p r o d u ct an d th e fact th at ψ n 1 = a n is ju s t a r egu lar s tate v ector ). Ho w ev er , if w e ap p ly o v er an d o v er th e a (lo w er in g) op er ator , w e cou ld ar r iv e at n egativ e n u m b er s n : w e th er efor e r eq u ir e th at a 0 = 0 to tr u n cate th is p r o ces s . Th e action of th e r ais in g op er ator a † can th en p r o d u ce an y eigen s tate, s tar tin g fr om th e 0 eigen s tate:
( a † ) n
| n ) = √ n !
| 0 ) .
√ √
Th e matr ix r ep r es en tati o n o f th es e op er ator in th e | n ) b as i s (with in fi n ite-d imen s ion al matr ices ) is p ar ticu lar ly s imp le, s in ce ( n | a | n ′ ) = δ n ′ ,n − 1 n an d ( n | a † | n ′ ) = δ n ′ ,n + 1 n + 1:
0 √ 1
√ 0 . . .
√ 0 0 0 . . .
0 0 0 . . . 0 2 0 . . .
a = 0 0 2 . . . a † = 1
√ 0 0 . . .
C. P osition rep resen ta tion
0.5
– 4
– 2
2
4
– 0.5
0.6
0.5
0.4
0.3
0.2
0.1
– 4
– 2
2
4
F ig . 2 9 : Left: H a r mo n ic o s c illa to r w a v efu n c tio n . R ig h t: c o r r es p o n d in g p r o b a b ilit y d is tr ib u tio n fu n c tio n fo r n = 2 (b lu e) a n d
n = 3 (R ed , d o t ted ).
W e h a v e n o w s tar ted fr om a (p h y s ical) d es cr ip tion of th e h .o. Hamilton ian an d mad e a c h an ge of b as is in or d er to ar r iv e at a s imp le d iagon al for m of it. No w th at w e k n o w its eigen k ets , w e w ou ld lik e to go b ac k to a mor e in tu itiv e p ictu r e of p os ition an d momen tu m. W e th u s w an t to ex p r es s th e eigen k ets | n ) in ter ms of th e p os ition r ep r es en tation .
0.4
0.3
0.2
0.1
0.6
0.4
0.2
– 10
– 5
5
10
– 0.2
– 0.4
– 10 – 5 0 5 10
F ig . 3 0 : Left: H a r mo n ic o s c illa to r w a v efu n c t io n . R ig h t : c o r r es p o n d in g p r o b a b ilit y d is t r ib u tio n fu n c tio n fo r n = 4 0 . I n R ed , th e c la s s ic a l p r o b a b ilit y .
J J
| ) | ) ( | ) | )
T h e p os ition r ep r es en tation cor r es p on d s to ex p r es s in g a s tate v ector ψ in th e p os ition b as is : ψ = dx x ψ x =
( | )
| ) | )
dx ψ ( x ) x (wh er e x is th e eigen s tate of th e p os ition op er ator th at is a con tin u ou s v ar iab le, h en ce th e in tegr al). Th is d efi n es th e w a v efu n ction ψ ( x ) = x ψ .
√
2 1 2 1 m ω
Th e w a v e fu n ction d es cr ip tion in th e x r ep r es en tation of th e q u an tu m h .o. can b e fou n d b y s tar tin g with th e gr ou n d s tate w a v efu n ction . S in ce a | 0 ) = 0 w e h a v e √ 1 ( X + iP ) | 0 ) = √ 1 ( mω x + √ i p ) | 0 ) = 0. In th e x r ep res en tation ,
giv en ψ 0 ( x ) = ( x | 0 )
1 √ ip d
2
− m ω x / 2
√ 2 k ( x | ( mω x + √ mω ) | 0 ) = 0 → ( mω x + dx ) ψ 0 ( x ) = 0 → ψ 0 ( x ) ∝ e
Th e oth er eigen s tates ar e b u ilt u s in g Her mite P oly n omials H n
d iff er en tial eq u ation s :
( x ) , u s in g th e for m u la 9 | n ) =
† n
( a )
√ n ! | 0 ) to d er iv e
ψ n ( x ) = ( x | n ) = √ n ! 2 n
mω x − √ mω dx
ψ 0 ( x )
1 [ √ 1 d ] n
I
2 n n !
x
0
F ig. ?? ).
2
0
with s olu tion s ψ n ( x ) = ( x | n ) = √ 1 H n ( x ) ψ 0 ( x ). Th e n = 2 an d n = 3 w a v efu n ction s ar e p lotted in th e follo win g fi gu r e, wh ile th e s econ d fi gu r e d is p la y s th e p r ob ab ilit y d is tr ib u tion fu n ction . Notice th e d iff er en t p ar it y for ev en an d o d d n u m b er an d th e n u m b er of zer os of th es e fu n ction s . Clas s ically , th e p r ob ab ilit y th at th e os cillatin g p ar ticle is at a giv en v alu e of x is s imp ly th e fr action of time th at it s p en d s th er e, wh ic h is in v er s ely p r op or tion al to its v elo cit y v ( x ) = x ω 1 − x 2 at th at p os ition . F or lar ge n , th e p r ob ab ilit y d is tr ib u tion b ecomes clos e to th e clas s ical on e (s ee
9 F o r mo r e d eta ils o n H er mite P o lyn o mia ls a n d th eir g en er a to r fu n c tio n , lo o k o n C o h en -T a n n o u d ji. On lin e in fo rma tio n fr o m: E r ic W. W eis s tein . H er mite P o lyn o mia l . F r o m M a th W o r ld – A W o lfr a m W eb R es o u r c e.
4. 4 Identic al pa rtic les
W e s tar t fi r s t with th e s imp les t cas e of a t w o-p ar ticle s y s tem. Th e w a v efu n ction is th en : ψ ( x r 1 , x r 2 ) an d if w e as s u me th at th er e is n o in ter action b et w een th e t w o p ar ticles , w e will b e ab le to d es cr ib e th e s tates u s in g s ep ar ation of v ar iab les :
ψ ( x r 1 , x r 2 ) = ψ a ( x r 1 ) ψ b ( x r 2 )
wh er e a an d b lab el t w o d iff er en t s in gle-p ar ticle s tates . Imp licit in th is ex p r es s ion is th e as s u mp tion th at I can d is tin gu is h th e t w o p ar ticles b y s ome mean , an d lin k p ar ticle on e to th e p os ition 1 an d th e s tate a . Ho w ev er , if w e con s id er t w o id en tical p ar ticles (2 electr on s , t w o p h oton s , t w o n eu tr on s ) th er e is n o p h y s ical mean to d is tin gu is h th em. Ev en if w e tr y to meas u r e th em in or d er to k eep tr ac k of wh ic h on e is wh ic h , w e k n o w th at in th e p r o ces s w e d es tr o y th e s tate (b y th e w a v efu n ction collap s e) s o n ot ev en th is is a p os s ib ilit y .
4.4.1 B os ons , fermi ons
2
1
In q u an tu m mec h an ics id en tical p ar ticle ar e fu n d amen tally in d is tin gu is h ab le. Th en th e ex p r es s ion ab o v e d o es n ot cor r ectly d es cr ib e th e s tate an y mor e. In or d er to faith fu lly d es cr ib e a s tate in wh ic h w e can n ot k n o w if p ar ticle a or b is at r 1 or r 2 , w e can tak e a lin ear com b in ation of th es e t w o p os s ib ilities : ψ ( x r 1 , x r 2 ) = A 1 ψ a ( x r 1 ) ψ b ( x r 2 ) + A 2 ψ b ( x r 1 ) ψ a ( x r 2 ). No w, s in ce th e t w o p os s ib ilities h a v e th e s ame p r ob ab ilit y , w e h a v e | A 1 | = | A 2 | = √ . Th en th er e ar e t w o p os s ib le
com b in ation s :
1
ψ ( x r 1 , x r 2 ) = √ 2 [ ψ a ( x r 1 ) ψ b ( x r 2 ) ± ψ b ( x r 1 ) ψ a ( x r 2 )]
Th es e t w o com b in ation s d es cr ib e t w o t y p es of p ar ticle. Th e com b in ation with th e p lu s s ign d es cr ib es b os on s , p ar ticles th at ar e in v ar ian t u n d er ex c h an ge of a p ar ticle p air . Th e com b in ation with th e min u s s ign d es cr ib es fer mion s :
- all p ar ticles with in teger s p in ar e b oson s
- all p ar ticles with h alf-in teger s p in ar e fermi on s
(Th is can b e p r o v ed in r elativ is tic Q M ).
4.4.2 Exc hange op erato r
W e can d efi n e an op er ator P ˆ th at in ter c h an ges th e t w o p ar ticles :
P ˆ [ ψ ( x r 1 , x r 2 )] = ψ ( x r 2 , x r 1 )
±
±
H
n n
1 2 {
S in ce of cou r s e P ˆ [ P ˆ [ ψ ( x r 1 , x r 2 )]] = ψ ( x r 1 , x r 2 ), w e h a v e th at P ˆ 2 = 1. Th en th e eigen v alu es of P ˆ m u s t b e 1. [ If ϕ n is an eigen fu n ction of P ˆ with eigen v alu e p n , w e h a v e P ˆ 2 ϕ n = p 2 ϕ n = ϕ n , fr om wh ic h p 2 = 1. ] If t w o p ar ticles ar e id en tical, th en the Hamilton ian is in v ar ian t with r es p ect to th eir ex c h an ge an d [ , P ˆ ] = 0. Th en w e can fi n d en er gy eigen fu n ction s th at ar e common eigen fu n ction s of th e ex c h an ge op er ator , or ψ ( x r 1 , x r 2 ) = ψ ( x r 2 , x r 1 ). Th en if th e s y s tem is in itially in s u c h a s tate, it will b e alw a y s b e in a s tate with th e s ame ex c h an ge s y mmetr y . F or th e con s id er ation s ab o v e, h o w ev er , w e h a v e s een th at th e w a v efu n ction is n ot on ly allo w ed , b u t it m u s t b e in a s tate with a d efi n ite s y mmetr y :
ψ ( x r , x r ) = ψ ( x r 2 , x r 1 ) b os on s
− ψ ( x r 2 , x r 1 ) fer mion s
4.4.3 P aul i exc l us i on p ri nc i pl e
F r om th e for m of th e allo w ed w a v efu n ction for fer mion s , it follo ws th at t w o fer mion s can n ot o ccu p y th e s ame s tate. As s u me th at ψ a ( x r ) = ψ b ( x r ), th en w e alw a y s h a v e th at
1
ψ f ( x r 1 , x r 2 ) = √ 2 [ ψ a ( x r 1 ) ψ b ( x r 2 ) − ψ b ( x r 1 ) ψ a ( x r 2 )] = 0 .
Th is is th e w ell-k n o wn P au li ex clu s ion p r in cip le. Notice th at of cou r s e it ap p lies to an y fer mion s . F or ex amp le, it ap p lies to electr on s , an d th is is th e r eas on wh y electr on s d o n ot p ile u p in th e lo w es t en er gy lev el of th e atomic s tr u ctu r e, b u t for m a s h ell mo d el. W e will s ee th at th e s ame ap p lies as w ell to p r oton s an d n eu tr on s , giv in g r is e to th e s h ell mo d el for n u clei.
MIT OpenCourseWare http://ocw.mit.edu
22.02 Introduction to Applied Nuclear Physics
Spring 2012
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .