6. Ti m e Evol uti on i n Q uantum M ec hani c s
6.1 Time -de p e nde n t Sc hr o ¨ dinge r e quation
6 .1 .1 S o lu tio n s t o th e S c h r ¨ od in g er equ a tio n
6 .1 .2 U n ita r y E v o lu tio n
6.2 E v olution of w a v e -pac k e ts
6.3 E v olution of op e rators and e xp e c tation v alue s
6 .3 .1 H eis en b er g E qu a t io n
6 .3 .2 E h r en fes t’s th eo r em
Un til n o w w e u s ed q u an tu m mec h an ics to p r ed ict p r op er ties of a toms an d n u clei. S in ce w e w er e in ter es ted mos tly in th e eq u ilib r iu m s tates of n u clei an d in th eir en er gies , w e on ly n eed ed to lo ok at a tim e- indep endent d es cr ip tion of q u an tu m-mec h an ical s y s tems . T o d es cr ib e d y n amical p r o ces s es , s u c h as r ad iation d eca y s , s catter in g an d n u clear r eaction s , w e n eed to s tu d y h o w q u an tu m mec h an ical s y s tems ev olv e in time.
6. 1 Time- dep endent Sc hr o ¨ dinger equation
W h en w e fi r s t in tr o d u ced q u an tu m mec h an ics , w e s a w th at th e fou r th p os tu late of Q M s tates th at:
The evol u tion of a cl os e d s ys tem is u nitar y ( r ever s ibl e) . The evol u tion is given by the tim e- dep endent Schr o ¨ dinger e qu ation
∂ t
i I ∂ | ψ ) = H | ψ )
H
w her e is the Ham il tonian of the s ys tem ( the ener gy op er ator ) and I is the r e du c e d Pl anck c ons tant
( I = h/ 2 π with h th e Plan c k con s tan t, allo win g con v er s ion fr om en er gy to fr eq u en cy u n its ).
| ) | )
W e will fo cu s main ly on th e S c h r ¨ od in ger eq u ation to d es cr ib e th e ev olu tion of a q u an tu m-mec h an ical s y s tem. Th e s tatemen t th at th e ev olu tion of a clos ed q u an tu m s y s tem is u n itar y is h o w ev er mor e gen er al. It mean s th at th e s tate of a s y s tem at a later time t is giv en b y ψ ( t ) = U ( t ) ψ (0) , wh er e U ( t ) is a u n itar y op er ator . An op er ator is u n itar y if its ad join t U † (ob tain ed b y tak in g th e tr an s p os e an d th e comp lex con ju gate of th e op er ator , U † = ( U ∗ ) T ) is eq u al to its in v er s e: U † = U − 1 or U U † = 1 1.
| ) | )
Note th at th e ex p r es s ion ψ ( t ) = U ( t ) ψ (0) is an in tegr al eq u ation r elatin g th e s tate at time zer o with th e s tate at time t . F or ex amp le, clas s ically w e cou ld wr ite th at x ( t ) = x (0) + v t (wh er e v is th e s p eed , for con s tan t s p eed ). W e can as w ell wr ite a d iff er en tial eq u ation th at p r o v id es th e s ame in for mation : th e S c h r ¨ od in ger eq u ation . Clas s ically
d t
for ex amp le, (in th e ex amp le ab o v e) th e eq u iv alen t d iff er en tial eq u ation w ou ld b e d x = v (mor e gen er ally w e w ou ld h a v e Newton ’s eq u ation lin k in g th e acceler ation to th e for ce). In Q M w e h a v e a d iff er en tial eq u ation th at con tr ol th e ev olu tion of clos ed s y s tems . Th is is th e S c h r ¨ od in ger eq u ation :
i I
∂ ψ ( x , t )
∂ t
= H ψ ( x , t )
H
wh er e is th e s y s tem’s Hamilton ian . Th e s olu tion to th is p ar tial d iff er en tial eq u ation giv es th e w a v efu n ction ψ ( x , t ) at an y later time, wh en ψ ( x , 0) is k n o wn .
6.1.1 Sol uti ons to the Sc hr o ¨ di nger equati on
W e fi r s t tr y to fi n d a s olu tion in th e cas e wh er e th e Hamilton ian H =
p ˆ 2
2 m
+ V ( x , t ) is s u c h th at th e p oten tial V ( x , t )
is time in d ep en d en t (w e can th en wr ite V ( x )). In th is cas e w e can u s e s ep ar ation of v ar iab les to lo ok for s olu tion s .
Th at is , w e lo ok for s olu tion s th at ar e a p r o d u ct of a fu n ction of p os ition on ly an d a fu n ction of time on ly :
ψ ( x , t ) = ϕ ( x ) f ( t )
Th en , wh en w e tak e th e p ar tial d er iv ativ es w e h a v e th at
∂ ψ ( x , t ) d f ( t ) ∂ ψ ( x , t ) dϕ ( x ) ∂ 2 ψ ( x , t ) d 2 ϕ ( x )
=
∂ t dt
ϕ ( x ) ,
=
∂ x dx
f ( t ) an d
∂ x 2
= dx 2
f ( t )
Th e S c h r ¨ od in ger eq u ation s imp lifi es to
i I d f ( t ) I 2 d 2 ϕ ( x )
D iv id in g b y ψ ( x , t ) w e h a v e:
dt ϕ ( x ) = − 2 m x 2
f ( t ) + V ( x ) ϕ ( x ) f ( t )
i I
d f ( t ) 1 I 2 d 2 ϕ ( x ) 1
dt f ( t ) = − 2 m x 2 ϕ ( x ) + V ( x )
No w th e LHS is a fu n ction of time on ly , wh ile th e RHS is a fu n ction of p os ition on ly . F or th e eq u ation to h old , b oth s id es h a v e th en to b e eq u al to a con s tan t (s ep ar ation con s tan t):
i I
d f ( t ) 1 I 2 d 2 ϕ ( x ) 1
dt f ( t ) = E , − 2 m x 2 ϕ ( x ) + V ( x ) = E
Th e t w o eq u ation s w e fi n d ar e a s imp le eq u ation in th e time v ar iab le:
d f ( t )
i
E t
= − E f ( t ) , → f ( t ) = f (0) e
− i i
an d
dt I
I 2 d 2 ϕ ( x ) 1
− 2 m x 2 ϕ ( x ) + V ( x ) = E
th at w e h a v e alr ead y s een as th e time-in d ep en d en t S c h r ¨ od in ger eq u ation . W e h a v e ex ten s iv ely s tu d ied th e s olu tion s of th e th is las t eq u ation , as th ey ar e th e eigen fu n ction s of th e en er gy -eigen v alu e p r ob lem, giv in g th e s tation ar y (eq u i lib r iu m) s tates of q u an tu m s y s tems . Note th at for th es e s tation ar y s olu tion s ϕ ( x ) w e can s till fi n d th e cor r es p on d in g total w a v efu n ction , giv en as s tated ab o v e b y ψ ( x , t ) = ϕ ( x ) f ( t ), wh ic h d o es d es cr ib e als o th e time ev olu tion of th e s y s tem:
ψ ( x , t ) = ϕ ( x ) e
− i
E t
i
D o es th is mean th at th e s tates th at u p to n o w w e called s tation a r y ar e in s tead ev olv in g in time?
| | ( ) J
Th e an s w er is y es , b u t with a ca v eat. Alth ou gh th e s tates th ems elv es ev olv e as s tated ab o v e, an y meas u r ab le q u an tit y (s u c h as th e p r ob ab ilit y d en s it y ψ ( x , t ) 2 or th e ex p ectation v alu es of ob s er v ab le, A = ψ ( x , t ) ∗ A [ ψ ( x , t )]) ar e s till time-in d ep en d en t. (Ch ec k it!)
Th u s w e w er e cor r ect in callin g th es e s tates s tation ar y an d n eglectin g in p r actice th eir time-ev olu tion wh en s tu d y i n g th e p r op er ties of s y s tems th ey d es cr ib e.
L
{ }
Notice th at th e w a v efu n ction b u ilt fr om on e en er gy eigen fu n ction , ψ ( x , t ) = ϕ ( x ) f ( t ), is on ly a p ar ticu lar s olu tion of th e S c h r ¨ od in ger eq u ation , b u t man y oth er ar e p os s ib le. Th es e will b e comp licated fu n ction s of s p ace an d time, wh os e s h ap e will d ep en d on th e p ar ticu lar for m of th e p oten tial V ( x ). Ho w can w e d es cr ib e th es e gen er al s olu tion s ? W e k n o w th at in gen er al w e can wr ite a b as is giv en b y th e eigen fu n ction of th e Hamilton ian . Th es e ar e th e fu n ction s ϕ ( x ) (as d efi n ed ab o v e b y th e time-in d ep en d en t S c h r ¨ od in ger eq u ation ). Th e eigen s tate of th e Hamilton ian d o n ot ev olv e. Ho w ev er w e can wr ite an y w a v efu n ction as
ψ ( x , t ) = c k ( t ) ϕ k ( x )
k
( | )
Th is ju s t cor r es p on d s to ex p r es s th e w a v efu n ction in th e b as is giv en b y th e en er gy eigen fu n ction s . As u s u al, th e co efficien ts c k ( t ) can b e ob tain ed at an y in s tan t in time b y tak in g th e in n er p r o d u ct: ϕ k ψ ( x , t ) .
W h at is th e ev olu tion of s u c h a fu n ction ? S u b s titu tin g in th e S c h r ¨ od in ger eq u ation w e h a v e
|
k c k ( t ) ϕ k ( ∂ t |
c k ( t ) H ϕ k ( x ) |
||
|
L |
L k |
th at b ecomes
i I ∂ (
x ) )
=
∂ t
k
k
k
k
i I L ∂ ( c k ( t )) ϕ ( x ) = L c ( t ) E ϕ ( x )
k k
F or eac h ϕ k w e th en h a v e th e eq u ation in th e co efficien ts on ly
dc k − i E k t
i
i I dt = E k c k ( t ) → c k ( t ) = c k (0) e
A gen er al s olu tion of th e S c h r ¨ od in ger eq u ation is th en
ψ ( x , t ) = c k (0) e
k
i
ϕ k ( x )
L − i E k t
O b s . W e can d efi n e th e eigen -fr eq u en cies I ω k = E k fr om th e eigen -en er gies . Th u s w e s ee th at th e w a v efu n ction is a s u p er p os ition of w a v es ϕ k p r op agatin g in time eac h with a d iff er en t fr eq u en cy ω k .
Th e b eh a v ior of q u an tu m s y s tems –ev en p ar ticles – th u s often is s imilar to th e p r op agation of w a v es . O n e ex amp le is th e d iff r action p atter n for electr on s (an d ev en h ea v ier ob jects ) wh en s catter in g fr om a s lit. W e s a w an ex amp le in th e electr on d iff r action v id eo at th e b egin n in g of th e clas s .
O b s . W h at is th e p r ob ab ilit y of meas u r in g a cer tain en er gy E k at a time t ? It is giv en b y th e co efficien t of th e ϕ k
2 − i E k t 2 2
eigen fu n ction , | c k ( t ) | = | c k (0) e i | = | c k (0) | . Th is mean s th at th e p r ob ab ilit y for th e giv en en er gy is con s tan t,
d o es n ot c h an ge in time. En er gy is th en a s o-called con s tan t of th e motion . Th is is tr u e on ly for th e en er gy eigen v alu es ,
n ot for oth er ob s er v ab les ‘.
| | | | | | | |
| |
Ex amp le: Con s id er in s tead th e p r ob ab ilit y of fi n d in g th e s y s tem at a cer tain p os ition , p ( x ) = ψ ( x , t ) 2 . Th is of cou r s e c h an ges in time. F or ex amp le, let ψ ( x , 0) = c 1 (0) ϕ 1 ( x ) + c 2 (0) ϕ 2 ( x ), with c 1 (0) 2 + c 2 (0) 2 = c 1 2 + c 2 2 = 1 (an d ϕ 1 , 2 n or malized en er gy eigen fu n ction s . Th en at a later time w e h a v e ψ ( x , 0) = c 1 (0) e − iω 1 t ϕ 1 ( x ) + c 2 (0) e − iω 2 t ϕ 2 ( x ). W h at is p ( x , t )?
1
1
2
2
c (0) e − iω 1 t ϕ ( x ) + c (0) e − iω 2 t ϕ ( x ) 2
[ ]
= | c 1 (0) | 2 | ϕ 1 ( x ) | 2 + | c 2 (0) | 2 | ϕ 2 ( x ) | 2 + c 1 ∗ c 2 ϕ 1 ∗ ϕ 2 e − i ( ω 2 − ω 1 ) t + c 1 c ∗ 2 ϕ 1 ϕ 2 ∗ e i ( ω 2 − ω 1 ) t
= | c 1 | 2 + | c 2 | 2 + 2Re c 1 ∗ c 2 ϕ 1 ∗ ϕ 2 e − i ( ω 2 − ω 1 ) t
Th e las t ter m d es cr ib es a w a v e in ter fer en ce b et w een d iff er en t comp on en ts of th e in itial w a v efu n ction .
O b s .: Th e ex p r es s ion s fou n d ab o v e for th e time-d ep en d en t w a v efu n c tion ar e on ly v alid if th e p oten tial is its elf time-in d ep en d en t. If th is is n ot th e cas e, th e s olu tion s ar e ev en mor e d ifficu lt to ob tain .
6.1.2 U ni ta ry Evol uti on
W e s a w t w o eq u iv alen t for m u lation of th e q u an tu m mec h an ical ev olu tion , th e S c h r ¨ od in ger eq u ation an d th e Heis en b er g eq u ation . W e n o w p r es en t a th ir d p os s ib le for m u lation : follo win g th e 4 th p os tu late w e ex p r es s th e ev olu tion of a s tate in ter ms of a u n itar y op er ator , called th e p rop agator :
ψ ( x , t ) = U ˆ ( t ) ψ ( x , 0)
with U ˆ † U ˆ = 1 1. (Notice th at a p r ior i th e u n itar y op er ator U ˆ cou ld als o b e a fu n ction of s p ace). W e can s h o w th at th is is eq u iv alen t to th e S c h r ¨ od in ger eq u ation , b y v er ify in g th at ψ ( x , t ) ab o v e is a s olu tion :
H →
i I ∂ U ˆ ψ ( x , 0) = U ˆ ψ ( x , 0) i I ∂ U ˆ
∂ t ∂ t
= H U ˆ
wh er e in th e s econ d s tep w e u s ed th e fact th at s in ce th e eq u ation h old s for an y w a v efu n ction ψ it m u s t h old for th e op er ator th ems elv es . If th e Hamilton ian is time in d ep en d en t, th e s econ d eq u ation can b e s olv ed eas ily , ob tain in g:
i I ∂ U ˆ
∂ t
= H U ˆ → U ˆ ( t ) = e − i H t/ n
wh er e w e s et U ˆ ( t = 0) = 1 1. Notice th at as d es ir ed U ˆ is u n itar y , U ˆ † U ˆ = e i H t/ n e − i H t/ n = 1 1.
6. 2 Evolution of w ave- pac k ets
In S ection 6.1.1 w e lo ok ed at th e ev olu tion of a gen er al w a v efu n ction u n d er a ti me-in d ep en d en t Hamilton ian . Th e s olu tion to th e S c h r ¨ od in ger eq u ation w as giv en in ter ms of a lin ear s u p er p os ition of en er gy eigen fu n ction s , eac h acq u ir in g a time-d ep en d en t p h as e factor . Th e s olu tion w as th en th e s u p er p os ition of w a v es eac h with a d iff er en t fr eq u en cy .
H
{ } → { }
No w w e w an t to s tu d y th e cas e wh er e th e eigen fu n ction s for m for m a con tin u ou s b as is , ϕ k ϕ ( k ) . M or e p r ecis ely , w e w an t to d es cr ib e h o w a fr ee p ar ticle ev olv es in time. W e alr ead y fou n d th e eigen fu n ction s of th e fr ee p ar ticle Hamilton ian ( = p ˆ 2 / 2 m ): th ey w er e giv en b y th e momen tu m eigen fu n ction s e ik x an d d es cr ib e mor e p r op er ly a tr a v elin g w a v e. A p ar ticle lo calized in s p ace in s tead can b e d es cr ib ed b y w a v ep ac k et ψ ( x , 0) in itially w ell lo calized in x -s p ace (for ex amp le, a G au s s ian w a v ep ac k et).
Ho w d o es th is w a v e-fu n ction ev olv e in time? F ir s t, follo win g S ection 2.2.1 , w e ex p r es s th e w a v efu n ction in ter ms of momen tu m (an d en er gy ) eigen fu n ction s :
J
1 ∞
2 π
ψ ( x , 0) = √
− ∞
ψ ¯ ( k ) e
ik x
dk ,
W e s a w th at th is is eq u iv alen t to th e F ou r ier tr an s for m of ψ ¯ ( k ), th en ψ ( x , 0) an d ψ ¯ ( k ) ar e a F ou r ier p air (can b e ob tain ed fr om eac h oth er v ia a F ou r ier tr an s for m).
Th u s th e fu n ction ψ ¯ ( k ) is ob tain ed b y F ou r ier tr an s for min g th e w a v e-fu n ction at t = 0. Notice again th at th e fu n ction
ψ ¯ ( k ) is th e con tin u ou s -v ar iab le eq u iv alen t of th e co efficien ts c k (0).
Th e s econ d s tep is to ev olv e in time th e s u p er p os ition . F r om th e p r ev iou s s ection w e k n o w th at eac h en er gy eigen fu n ction ev olv es b y acq u ir in g a p h as e e − iω ( k ) t , wh er e ω ( k ) = E k / I is th e en er gy eigen v alu e. Th en th e time ev olu tion
J
of th e w a v efu n ction is
wh er e
F or th e fr ee p ar ticle w e h a v e ω k
ψ ( x , t ) = ∞ ψ ¯ ( k ) e i ϕ ( k ) dk ,
− ∞
ϕ ( k ) = k x − ω ( k ) t.
2 m
= n k 2 . If th e p ar ticle en cou n ter s in s tead a p oten tial (s u c h as in th e p oten tial b ar r ier
or p oten tial w ell p r ob lems w e alr ead y s a w) ω k cou ld h a v e a mor e comp lex for m. W e will th u s con s id er th is mor e
gen er al cas e.
No w, if ψ ¯ ( k ) is stron gl y p eak ed ar ou n d k = k 0 , it is a r eas on ab le ap p r o x imation to T a y lor ex p an d ϕ ( k ) ab ou t k 0 .
— ( k − k 0 ) 2
J
— 0 � {
1
W e can th en ap p r o x imate ψ ¯ ( k ) b y ψ ¯ ( k ) ≈ e 4 ( Δ k ) 2 an d k eep in g ter ms u p to s econ d -or d er in k − k 0 , w e ob tain
2
wh er e
ψ ( x , t ) ∝ ∞ e
− ∞
( k − k ) 2
4 ( Δ k ) 2 ex p − i k x + i ϕ 0 + ϕ ′ 0 ( k − k 0 ) +
ϕ 0 = ϕ ( k 0 ) = k 0 x − ω 0 t,
ϕ 0 ′′ ( k − k 0 ) 2 }� ,
0 d k
ϕ ′ = d ϕ ( k 0 ) = x − v g t,
2
0 d k 2
ϕ ′′ = d ϕ ( k 0 ) = − α t,
− i k x + i { k
x − ω t + ( x − v t ) ( k − k ) + ϕ ′′ ( k − k ) 2 }
1
with
0 0 g
0 2 0 0
dω ( k 0 ) d 2 ω ( k 0 )
ω 0 = ω ( k 0 ) , v g =
, α = .
dk dk 2
As u s u al, th e v ar ian ce of th e in itial w a v efu n ction an d of its F ou r ier tr an s for m ar e r elates : Δk = 1 / (2 Δx ), wh er e Δx
y = ( k − k 0 ) / (2 Δk ), w e get J ∞
is th e in itial wid th of th e w a v e-p ac k et an d Δk th e s p r ead in th e momen tu m. Ch an gin g th e v ar iab le of in tegr ation to
wh er e
ψ ( x , t ) ∝ e i ( k 0 x − ω 0 t )
e i β 1 y − ( 1+ i β 2 ) y 2 dy ,
− ∞
β 1 = 2 Δk ( x − x 0 − v g t ) ,
β 2 = 2 α ( Δk ) 2 t,
J
Th e ab o v e ex p r es s ion can b e r ear r an ged to giv e
ψ ( x , t ) ∝ e i ( k 0 x − ω 0 t ) − ( 1+ iβ 2 ) β 2 / 4 ∞ e − ( 1+ iβ 2 ) ( y − y 0 ) 2 dy ,
− ∞
wh er e y 0 = i β / 2 an d β = β 1 / (1 + i β 2 ).
Again c h an gin g th e v ar iab le of in tegr ation to z = (1 + i β 2 ) 1 / 2 ( y − y 0 ) , w e get
2
ψ ( x , t ) ∝ (1 + i β ) − 1 / 2 e i ( k 0 x − ω 0 t ) − ( 1+ i β 2 ) β 2 / 4 J ∞ e − z 2 dz .
− ∞
Th e in tegr al n o w ju s t r ed u ces to a n u m b er . Hen ce, w e ob tain
wh er e
ψ ( x , t ) ∝
( x − x 0 − v g t ) 2 [ 1 − i 2 α Δ k 2 t ]
e i ( k 0 x − ω 0 t ) e − 4 σ ( t ) 2
� 1 + i 2 α ( Δk ) 2 t ,
σ 2 ( t ) = ( Δx ) 2
α 2 t 2
+ 4 ( Δx ) 2 .
Note th at ev en if w e mad e an ap p r o x imation ear lier b y T a y lor ex p an d in g th e p h as e factor ϕ ( k ) ab ou t k = k 0 , th e ab o v e w a v e-fu n ction is s till id en tical to ou r or igin al w a v e-fu n ction at t = 0.
Th e p r ob ab ilit y d en s it y of ou r p ar ticle as a fu n ction of times is wr itten
2
| ψ ( x , t ) |
∝ σ − 1
( t ) ex p −
( x − x 0 − v g t ) 2
2 σ ( t )
2 .
Hen ce, th e p r ob ab ilit y d is tr ib u tion is a G au s s ian , of c h ar acter is tic wid th σ ( t ) (in cr eas in g in time), wh ic h p eak s at x = x 0 + v g t . No w, th e mos t lik ely p os ition of ou r p ar ticle ob v iou s ly coin cid es with th e p eak of th e d is tr ib u tion fu n ction . Th u s , th e p ar ticle’s mos t lik ely p os ition is giv en b y
x = x 0 + v g t.
It can b e s een th at th e p ar ticle eff ectiv ely mo v es at th e u n ifor m v elo cit y
dω v g = dk ,
wh ic h is k n o wn as th e grou p -v el o ci t y . In oth er w or d s , a p lan e-w a v e tr a v els at th e p h as e-v elo cit y , v p = ω /k , wh er eas a w a v e-p ac k et tr a v els at th e gr ou p -v elo cit y , v g = dω /dt v g = dω /dt . F r om th e d is p er s ion r elation for p ar ticle w a v es th e gr ou p v elo cit y is
d ( I ω ) d E p
v g = d ( I k ) = d p = m .
wh ic h is id en tical to th e clas s ical p ar ticle v elo cit y . Hen ce, th e d is p er s ion r elation tu r n s ou t to b e con s is ten t with clas s ical p h y s ics , after all, as s o on as w e r ealize th at p ar ticles m u s t b e id en tifi ed with w a v e-p ac k ets r ath er th an p lan e-w a v es .
Note th at th e wid th of ou r w a v e-p ac k et gr o ws as time p r ogr es s es : th e c h ar acter is tic time for a w a v e-p ac k et of or igin al wid th Δx Δx to d ou b le in s p atial ex ten t is
t 2 ∼
m ( Δx ) 2
I .
~
S o, if an electr on is or iginally lo calized in a r egion of atomic s cale (i.e., Δx 10 − 10 m ) th en th e d ou b lin g time is on ly ab ou t 10 − 16 s . Clear ly , p ar ticle w a v e-p ac k ets (for fr eely mo v in g p ar ticles ) s p r ead v er y r ap id ly .
Th e r ate of s p r ead in g of a w a v e-p ac k et is u ltimately go v er n ed b y th e s econ d d er iv ativ e of ω ( k ) with r es p ect to k ,
∂ k 2
∂ 2 ω . Th is is wh y th e r elation s h ip b et w een ω an d k is gen er ally k n o wn as a d i sp ersi on rel ati on , b ecau s e it go v er n s h o w w a v e-p ac k ets d is p er s e as time p r ogr es s es .
∝
×
If w e con s id er ligh t-w a v es , th en ω is a l ine ar fu n ction of k an d th e s econ d d er iv ativ e of ω with r es p ect to k is zer o. Th is imp lies th at th er e is n o d is p er s ion of w a v e-p ac k ets , w a v e-p ac k ets p r op agate with ou t c h an gin g s h ap e. Th is is of cou r s e tr u e for an y oth er w a v e for wh ic h ω ( k ) k . An oth er p r op er t y of lin ear d is p er s ion r elation s is th at th e p h as e-v elo cit y , v p = ω /k , an d th e gr ou p -v elo cit y , v g = dω /dk ar e id en tical. Th u s a ligh t p u ls e p r op agates at th e s ame s p eed of a p lan e ligh t-w a v e; b oth p r op agate th r ou gh a v acu u m at th e c h ar acter is tic s p eed c = 3 10 8 m / s .
O f cou r s e, th e d is p er s ion r elation for p ar ticle w a v es is not lin ear in k (for ex amp le for fr ee p ar ticles is q u ad r atic).
Hen ce, p ar ticle p lan e-w a v es an d p ar ticle w a v e-p ac k ets p r o p agate at d iff er en t v elo cities , an d p ar ticle w a v e-p ac k ets als o gr ad u ally d is p er s e as time p r ogr es s es .
6. 3 Evolution of op erato rs and exp ec tation values
Th e S c h r ¨ od in ger eq u ation d es cr ib es h o w th e s tate of a s y s tem ev olv es . S in ce v ia ex p er imen ts w e h a v e acces s to ob s er v ab les an d th eir ou tcomes , it is in ter es tin g to fi n d a d iff er en tial eq u ation th at d ir ectly giv es th e ev olu tion of ex p ectation v alu es .
6.3.1 Hei s enb erg Equati on
W e s tar t fr om th e d efi n ition of ex p ectation v alu e an d tak e its d er iv ativ e wr t time
d ( A ˆ ) = d d t dt
J
d 3 x ψ ( x , t ) ∗ A ˆ [ ψ ( x , t )]
= J d 3 x ∂ ψ ( x , t ) ∗ A ˆ ψ ( x , t ) + J d 3 x ψ ( x , t ) ∗ ∂ A ˆ ψ ( x , t ) + J d 3 x ψ ( x , t ) ∗ A ˆ ∂ ψ ( x , t )
∂ t ∂ t ∂ t
W e th en u s e th e S c h r ¨ od in ger eq u ation :
∂ ψ ( x , t ) i
= − H ψ ( x , t ) ,
∂ ψ ∗ ( x , t ) = i ( H ψ ( x , t )) ∗
∂ t I ∂ t I
an d th e fact ( H ψ ( x , t )) ∗ = ψ ( x , t ) ∗ H ∗ = ψ ( x , t ) ∗ H (s in ce th e Hamilton ian is her m itian H ∗ = H ). W ith th is , w e h a v e
dt I ∂ t I
d \ A ˆ )
= i J d 3 x ψ ( x , t ) ∗ H A ˆ ψ ( x , t ) + J d 3 x ψ ( x , t ) ∗ ∂ A ˆ ψ ( x , t ) − i J d 3 x ψ ( x , t ) ∗ A ˆ H ψ ( x , t )
= i J d 3 x ψ ( x , t ) ∗ [ H A ˆ − A ˆ H ] ψ ( x , t ) + J d 3 x ψ ( x , t ) ∗ ∂ A ˆ ψ ( x , t )
I ∂ t
W e n o w r ewr ite [ H A ˆ − A ˆ H ] = [ H , A ˆ ] as a comm u tator an d th e in tegr als as ex p ectation v alu es :
d A
\ )
ˆ
i
\
)
� �
∂ A
ˆ
dt
= [ H , A ˆ ] +
I
∂ t
d t n
\ )
O b s . Notice th at if th e ob s er v ab le its elf is time in d ep en d en t, th e n th e eq u ation r edu ces to d ( A ˆ ) = i [ H , A ˆ ] . Th en if
th e ob s er v ab le A ˆ comm u tes with th e Hamilton ian , w e h a v e n o ev olu tion at all of th e ex p ectation v alu e. An ob s er v ab le th at comm u tes with th e Hamilton ian is a con s tan t of th e motio n . F or ex amp le, w e s ee again wh y en er gy is a con s tan t of th e motion (as s een b efor e).
Notice th at s in ce w e can tak e th e ex p ectation v alu e with r es p ect to an y w a v efu n ction , th e eq u ation ab o v e m u s t h old als o for th e op er ator s th ems elv es . Th en w e h a v e th e Hei sen b erg eq u ati on :
d A ˆ i
= [ H , A ˆ ] +
dt I ∂ t
∂ A ˆ
Th is is an eq u iv alen t for m u lation of th e s y s tem’s ev olu tion ( eq u iv alen t to th e S c h r ¨ od in ger eq u ation ).
H
O b s . Notice th at if th e op er ator A is time in d ep en d en t an d it comm u tes with th e Hamilton ian th en th e op er ator is con s er v ed , it is a con s tan t of th e motion (n ot on ly its ex p ectation v alu e).
H
Con s id er for ex amp le th e an gu lar momen tu m op er ator L ˆ 2 for a cen tr al p oten tial s y s tem (i.e. with p oten tial th at on ly d ep en d s on th e d is tan ce, V ( r )). W e h a v e s een wh en s olv in g th e 3D time-in d ep en d en t eq u ation th at [ , L ˆ 2 ] = 0. Th u s th e an gu lar momen tu m is a con s tan t of th e motion .
6.3.2 Ehrenfes t’s theo rem
W e n o w ap p ly th is r es u lt to calcu late th e ev olu tion of th e ex p ectation v alu es for p os ition an d momen tu m.
= I ( [ H , x ˆ ] ) = I
[ + V ( x ) , x ˆ ] 2 m
d ( x ˆ ) i i ( p ˆ 2 )
dt
No w w e k n o w th at [ V ( x ) , x ˆ ] = 0 an d w e alr ead y calcu lated [ p ˆ 2 , x ˆ ] = − 2 i I p ˆ . S o w e h a v e:
d
dt m
( x ˆ )
= ( p ˆ
1
)
Notice th at th is is th e s ame eq u ation th at lin k s th e clas s ical p os ition with momen tu m (r emem b er p/m = v v elo cit y ). No w w e tu r n to th e eq u ation for th e momen tu m:
d ⟨ p ˆ ⟩ i
i p ˆ 2
2 m
dt = k ⟨ [ H , p ˆ ] ⟩ = k
[ + V ( x ) , p ˆ ] 2 m
Her e of cou r s e [ p ˆ 2 , p ˆ ] = 0, s o w e on ly n eed to calcu late [ V ( x ) , p ˆ ]. W e s u b s titu te th e ex p licit ex p r es s ion for th e
momen tu m:
[ V ( x ) , p ˆ ] f ( x ) = V ( x ) − i k ∂ f ( x ) − − i k ∂ ( V ( x ) f ( x ))
∂ f ( x )
∂ V ( x )
∂ x
∂ f ( x )
∂ x
∂ V ( x )
Th en ,
= − V ( x ) i k
+ i k
∂ x
f ( x ) + i k
∂ x
V ( x ) = i k
∂ x
f ( x )
∂ x
d
dt
⟨ p ˆ
⟩
= −
∂ V ( x )
∂ x
O b s . Notice th at in th es e t w o eq u ation s k h as b een can celed ou t. Als o th e eq u ation in v olv e on ly r eal v ar iab les (as in clas s ical mec h an ics ).
∂ x
O b s . Us u ally , th e d er iv ativ e of a p oten tial fu n ction is a for ce, s o w e can wr ite − ∂ V ( x ) = F ( x )
If w e cou ld ap p r o x imate ⟨ F ( x ) ⟩ ≈ F ( ⟨ x ⟩ ), th en th e t w o eq u ation s ar e r ewr itten :
dt
m
dt
d ⟨ x ˆ ⟩ = 1 ⟨ p ˆ ⟩ d ⟨ p ˆ ⟩ = F ( ⟨ x ⟩ )
⟨ ⟩ →
⟨ ⟩ →
Th es e ar e t w o eq u ation s in th e ex p ectation v alu es on ly . Th en w e cou ld ju s t mak e th e s u b s titu tion s p ˆ p an d x ˆ x (i.e. id en tify th e ex p ectation v alu es of Q M op er ator s with th e cor r es p on d in g clas s ical v ar iab les ). W e ob tain in th is w a y th e u s u al clas s ical eq u ation of motion s . Th is is Eh ren fes t’ s th eorem .
|ψ(x)| 2
F(<x>)
F(x)
|ψ( x)| 2
∆x
F(x)
F(<x>) ≈ <F(x)>
<x>
F(<x>) ≠ <F(x)>
<x>
F ig . 4 0 : Lo c a liz ed (left) a n d s p rea d -o u t (r ig h t) w a v efu n c t io n . I n th e p lo t th e a b s o lu te v a lu e s qu a r e o f th e w a v efu n c t io n is s h o w n in b lu e (c o r r es p o n d in g t o th e p o s itio n p r o b a b ilit y d en s it y) fo r a s ys tem a p p r o a c h in g th e c la s s ic a l limit (left) o r s h o w in g mo r e qu a n tu m b eh a vio r . T h e fo r c e a c t in g o n th e s ys tem is s h o w n in b la c k (s a me in th e t w o p lo ts ) . T h e s h a d ed a r ea s in d ic a te th e r eg io n o v er w h ic h | ψ ( x ) | 2 is n o n -n eg lig ib le, t h u s g ivin g a n id ea o f t h e r eg io n o v er w h ic h th e fo r c e is a v er a g ed . T h e w a v efu n c tio n s g iv e th e s a me a v er a g e p o s itio n ⟨ x ⟩ . H o w ev er , w h ile fo r t h e left o n e F ( ⟨ x ⟩ ) ≈ ⟨ F ( x ) ⟩ , fo r th e r ig h t w a v efu n c t io n F ( ⟨ x ⟩ ) / = ⟨ F ( x ) ⟩
∂ x
∂ ( x ⟩
W h en is th e ap p r o x imation ab o v e v alid ? W e w an t D ∂ V ( x ) E ≈ ∂ V ( ( x ⟩ ) . Th is mean s th at th e w a v efu n ction is lo calized
— ⟨ ⟩ ∫ −
en ou gh s u c h th at th e wid th of th e p os ition p r ob ab ilit y d is tr ib u tion is s mall comp ar ed to th e t y p ical len gth s cale o v er wh ic h th e p oten tial v ar ies . W h en th is con d ition is s atis fi ed , th en th e ex p ectation v alu es of q u an tu m-mec h an ical p r ob ab ilit y ob s er v ab le will follo w a clas s ical tr a jector y .
As s u me for ex amp le ψ ( x ) is an eigen s tate of th e p os ition op er ator ψ ( x ) = δ ( x x ¯ ). Th en x ˆ = dx x δ ( x x ¯ ) 2 = x ¯ an d
∂ V ( x ) = ∫ ∂ V ( x ) δ ( x − ⟨ x ⟩ ) dx = ∂ V ( ⟨ x ⟩ )
∂ x ∂ x
∂ ⟨ x ⟩
n o lon ger h a v e an eq u alit y b u t on ly an ap p r o x imation if ∆x ≪ L = ∂ V ( x ) ( or lo calized w a v efu n ction ).
If in s tead th e w a v efu n ction is a p ac k et cen ter ed ar ou n d ⟨ x ⟩ b u t with a fi n ite wid th ∆x (i.e. a G au s s ian fu n ction ) w e
1 — 1
V ∂ x
6. 4 F ermi’ s Golden R ule
W e con s id er n o w a s y s tem with an Hamilton ian H 0 , of wh ic h w e k n o w th e eigen v alu es an d eigen fu n ction s :
H 0 u k ( x ) = E k u k ( x ) = I ω k u k ( x )
L
Her e I ju s t ex p r es s ed th e en er gy eigen v alu es in ter ms of th e fr eq u en cies ω k = E k / I . Th en , a gen er al s tate will ev olv e as :
ψ ( x , t ) = c k (0) e − iω k t u k ( x )
k
If th e s y s tem is in its eq u ilib r iu m s tate, w e ex p ect it to b e s tation ar y , th u s th e w a v efu n ction will b e on e of th e eigen fu n ction s of th e Hamilton ian . F or ex amp le, if w e con s id er an atom or a n u cleu s , w e u s u ally ex p ect to fi n d it in its gr ou n d s tate (th e s tate with th e lo w es t en er gy ). W e con s id er th is to b e th e in itial s tate of th e s y s tem:
ψ ( x , 0) = u i ( x )
(wh er e i s tan d s for initial ). No w w e as s u me th at a p er tu r b ation is ap p lied to th e s y s tem. F or ex amp le, w e cou ld h a v e a las er illu min atin g th e atom, or a n eu tr on s catter in g with th e n u cleu s . Th is p er tu r b ation in tr o d u ces an ex tr a p oten tial V ˆ in th e s y s tem’s Hamilton ian (a p r ior i V ˆ can b e a fu n ction of b oth p os ition an d time V ˆ ( x , t ), b u t w e will con s id er th e s imp ler cas e of time-in d ep en d en t p oten tial V ˆ ( x )). No w th e h amilton ian r ead s :
H = H 0 + V ˆ ( x )
h
W h at w e s h ou ld d o, is to fi n d th e eigen v alu es { E v } an d eigen fu n ction s { v h ( x ) } of th is n ew Hamilton ian an d ex p r es s
u i ( x ) in th is n ew b as is an d s ee h o w it ev olv es :
u i ( x ) = d h (0) v h → ψ ( x , t ) = d h (0) e
h h
h
v h ( x ) .
L ′ L − iE v t/ n
M os t of th e time h o w ev er , th e n ew Hamilton ian is a comp lex on e, an d w e can n ot calcu late its eigen v alu es an d eigen fu n ction s . Th en w e follo w an oth er s tr ategy .
Con s id er th e ex amp les ab o v e (atom+ las er or n u cleu s + n eu tr on ): W h at w e w an t to calcu late is th e p r ob ab ilit y of mak in g a tr an s ition fr om an atom/n u cleu s en er gy lev el to an oth er en er gy lev el, as in d u ced b y th e in ter action . S in ce
H
L
0 is th e or igin al Hamilton ian d es cr ib in g th e s y s tem, it mak es s en s e to alw a y s d es cr ib e th e s tate in ter ms of its en er gy lev els (i.e. in ter ms of its eigen fu n ction s ). Th en , w e gu es s a s olu tion for th e s tate of th e for m:
ψ ′ ( x , t ) = c k ( t ) e − iω k t u k ( x )
k
Th is is v er y s imilar to th e ex p r es s ion for ψ ( x , t ) ab o v e, ex cep t th at n o w th e co efficien t c k ar e time d ep en d en t. Th e time-d ep en d en cy d er iv es fr om th e fact th at w e ad d ed an ex tr a p oten tial in ter action to th e Hamilton ian .
∂ t
0
Let u s n o w in s er t th is gu es s in to th e S c h r ¨ od in ger eq u ation , i I ∂ ψ ′ = H ψ ′ + V ˆ ψ ′ :
i I L [ c ˙ k ( t ) e − iω k t u k ( x ) − iω c k ( t ) e − iω k t u k ( x ) ] = L c k ( t ) e − iω k t ( H 0 u k ( x ) + V ˆ [ u k ( x )] )
k k
(wh er e c ˙ is th e time d er iv ativ e). Us in g th e eigen v alu e eq u ation to s imp lify th e RHS w e fi n d
u k ( x )
=
, k
I ω k u k ( x ) + c k ( t ) e
k
, , , ,
L [ i I c ˙ ( t ) e − iω k t u ( x ) + , I ω , c , ( t ) , e − , i ω , k t , ] L [ c , ( t ) , e − , iω , k t , ,
− iω k t V ˆ [ u ( x )] ]
L L
k
k
k
k k
i I c ˙ k ( t ) e − iω k t u k ( x ) = c k ( t ) e − iω k t V ˆ [ u k ( x )]
k k
No w let u s tak e th e in n er p r o d u ct of eac h s id e with u h ( x ):
L i I c ˙
k
( t ) e − iω k t J ∞ u ∗ ( x ) u ( x ) dx = L c ( t ) e − iω k t J ∞ u ∗ ( x ) V ˆ [ u ( x )] dx
− ∞
k
− ∞
k
h
k
k
h
k
Th en in th e s u m o v er k t h − e ∞ on ly ter m th at s u r v iv es is th e on e k = h :
h
k
h
In th e LHS w e fi n d th at J ∞
u ∗ h ( x ) u k ( x ) dx = 0 for h / = k an d it is 1 for h = k (th e eigen fu n ction s ar e or th on or mal).
k
L i I c ˙
k
( t ) e − iω k t J ∞ u ∗ ( x ) u ( x ) dx = i I c ˙
− ∞
( t ) e − iω h t
O n th e RHS w e d o n ot h a v e an y s imp lifi cation . T o s h or ten th e n ota tion h o w ev er , w e call V h k th e in tegr al:
V = J ∞ u ∗ ( x ) V ˆ [ u ( x )] dx
Th e eq u ation th en s imp lifi es to:
h k h k
− ∞
h
I
k
hk
c ˙ ( t ) = − i L c ( t ) e i ( ω h − ω k ) t V
k
Th is is a d iff er en tial eq u ation for th e co efficien ts c h ( t ). W e can ex p r es s th e s ame r elation u s in g an in tegr al eq u ation :
c h ( t ) = − I
c k ( t ′ ) e i ( ω h − ω k ) t V hk dt ′ + c h (0)
i L J t ′
k
0
W e n o w mak e an imp or tan t ap p ro x i mati on . W e s aid at th e b egin n in g th at th e p oten tial V ˆ is a p er tu r b ation , th u s w e as s u me th at its eff ects ar e s mall (or th e c h an ges h ap p en s lo wly ). Th en w e can ap p r o x imate c k ( t ′ ) in th e in tegr al with its v alu e at time 0, c k ( t = 0):
c ( t ) = − L
J ′
i t
h I c k (0) e i ( ω h − ω k ) t V h k dt ′ + c h (0)
k 0
[Notice: for a b etter ap p r o x imation , an iter ativ e p r o ced u r e can b e u s ed wh ic h r ep laces c k ( t ′ ) with its fi r s t or d er s olu tion , th en s econ d etc.].
No w let’s go b ac k to th e in itial s cen ar io, in wh ic h w e as s u med th at th e s y s tem w as in itially at r es t, in a s tation ar y
i J t ′
s tate ψ ( x , 0) = u i ( x ). Th is mean s th at c k (0) = 0 for all k = /
i . Th e eq u ation th en r ed u ces to:
or , b y callin g Δω h = ω h − ω i ,
c h ( t ) = − I
c h ( t ) = − I V hi e iΔ ω h t dt ′ = − I Δω
i J t
0
e i ( ω h − ω i ) t V hi dt ′
0
1 − e iΔ ω h t
′ V h i � �
h
H
→ | |
W h at w e ar e r eally in ter es ted in is th e p r ob ab ilit y of mak in g a tr an s ition fr om th e in itial s tate u i ( x ) to an oth er s tate u h ( x ): P ( i h ) = c h ( t ) 2 . Th is tr an s ition is cau s ed b y th e ex tr a p oten tial V ˆ b u t w e as s u me th at b oth in itial an d fi n al s tates ar e eigen fu n ction s of th e or igin al Hamilton ian 0 (n otice h o w ev er th at th e fi n al s tate will b e a s u p er p os ition of all p os s ib le s tates to wh ic h th e s y s tem can tr an s ition to).
W e ob tain
P ( i → h ) = I 2 Δω 2 s in
4 | V h i | 2 ( Δω h t ) 2
h
2
z Δ ω / 2
Th e fu n ction s i n z is called a s in c fu n ction (s ee fi gu r e 41 ). T ak e s i n( Δ ω t/ 2) . In th e limit t → ∞ (i.e. as s u min g w e ar e d es cr ib in g th e s tate of th e s y s tem after th e n ew p oten tial h as h ad a lon g time to c h an ge th e s tate of th e q u an tu m s y s tem) th e s in c fu n ction b ecomes v er y n ar r o w, u n til wh en w e can ap p r o x imate it with a d elta fu n ction . Th e ex act limit of th e fu n ction giv es u s :
P ( i → h ) =
2 π V h i 2 t
| |
I 2
δ ( Δω h )
d t
W e can th en fi n d th e tr an s ition r ate fr om i → h as th e p r ob ab ilit y of tr an s ition p er u n it time, W ih = d P ( i → h ) :
W ih = I 2 | V hi | δ ( Δω h )
2 π
2
Th is is th e s o-called F ermi ’ s Gol d en R u l e , d es cr ib in g th e tr an s ition r ate b et w een s tates .
O b s .: Th is tr an s ition r ate d es cr ib es th e tr an s ition fr om u i t o a sin gle lev el u h with a giv en en er gy E h = I ω h . In man y cas es th e fi n al s tate is an u n b ou n d s tate, wh ic h , as w e s a w, can tak e on a con tin u ou s of p os s ib le en er gy a v ailab le. Th en , in s tead of th e p oin t-lik e d elta fu n ction , w e con s id er th e tr an s ition to a s et of s tates with en er gies in a s mall in ter v al E → E + dE . Th e tr an s ition r ate is th en p r op or tion al to th e n u m b er of s tates th at can b e fou n d with th is
Δ ω / 2
F ig . 4 1 : S in c fu n c tio n s i n ( Δ ω t / 2) . Left: S in c fu n c tio n a t s h o r t times . R ig h t: S in c fu n c tio n a t lo n g er times , th e fu n c tio n b ec o min g n a r r o w er a n d c lo s er to a Dir a c d elt a fu n c t io n
en er gy . Th e n u m b er of s tate is giv en b y dn = ρ ( E ) dE , wh er e ρ ( E ) is called th e d en s it y of s tates (w e will s ee h o w to calcu late th is in a later lectu r e). Th en , F er mi’s G old en r u le is mor e gen er ally ex p r es s ed as :
W ih = I | V hi | ρ ( E h ) | E h = E i
2 π
2
[Note, b efor e mak in g th e s u b s titu tion δ ( Δω ) → ρ ( E ) w e n eed to wr ite δ ( Δω ) = I δ ( I Δω ) = I δ ( E h − E i ) →
I ρ ( E h ) | E h = E i . Th is is wh y in th e fi n al for m u lation for th e G old en r u le w e on ly h a v e a factor I an d n ot its s q u ar e.]
MIT OpenCourseWare http://ocw.mit.edu
22.02 Introduction to Applied Nuclear Physics
Spring 2012
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