mk 23


$ = $0 + $2 (t)
$2 $0
$0un ( = Enun(
r) r).
|n'
"
i ( t) = $( t)
r, r,
"t

"
n
( t) = an(t) un( e-i t
r, r)
n
n = En/  an(t) = const
$2 = 0
( )
" "
n n
(i an(t) + Enan(t) ) un( e-i t = an(t) Enun( + $2 un( e-i t
Ł r) r) r)
n n
u" ( d3
r) r
m
un
"
m n
i am(t) e-i t = Ł'm| $2 |n' an(t)e-i t
Ł
n
am
"
i
mn
am(t) = - an(t) Ł'm| $2 |n' ei t
Ł
n
mn = m - n
$2 
$2 $2
a(t)
"
"
am(t) = ka(k)(t).
m
k=0
"
( ) ( )
i
mn
a(0)(t) + a(1)(t) + 2a(2)(t) + . . . = - a(0)(t) + a(1)(t) + . . . Ł'm| $2 |n' ei t,
Ł Ł Łm
m m n n
n
a(0)(t) = 0,
Ł
m
"
i
mn
a(1)(t) = - a(0)(t) Ł'm| $2 (t) |n' ei t,
Ł
m n
n
"
i
mn
a(2)(t) = - a(1)(t) Ł'm| $2 (t) |n' ei t,
Ł
m n
n
. . .
t0 = -" |k'
a(0) = mk a(0) = (m - k).
m m
a(0)
m
+"t
i
mk
a(1)(t) = - dt2 Ł'm| $2 (t2 ) |k' ei t2
m
-"
a(1)(t) H2 t
m
H2
+"t
i
mk
a(1)(t) = - Ł'm| $2 |k' dt2 ei t2 .
m
-"
t "
( )
2Ąi 1
a(1)(t) = - Ł'm| $2 |k'  (Em - Ek) .
m
|m'
|k'
"
+" +"
" "
mn
d3 " = a" anei t d3 u" un = |am(t)|2 = 1.
r r
m m
n,m m
|am(t)|2 t
|m'
|k' |am(t)|2 |m'

T T
+"/2 +"/2
1 1 ()
2() = lim dt eit () = lim dt () = lim T .
T " T " T "
2Ą 2Ą 2Ą
-T /2 -T/2
( )
2
|am(t)|2 2Ą 1
km = lim = Ł'm| $2 |k'  (Em - Ek)
T "
T 2
2
2Ą
= Ł'm| $2 |k'  (Em - Ek) .
+" +"
1
 = lim dEm(Em) |am(t)|2 = dEm(Em)km
T "
T
(Em)
T
am(t) <" sin t
am(t) <" t
t0
ńł
0 t < t0
ł
H2 (t) =
ół
H2 t t0
+"t
( )
i
mk mk mk
dt2 ei t2 = ei t0 - ei t
mk
t0
mk
)
ei t0 (
mk
= i 1 - ei "t =
mk
"t = t - t0
mk
( )
ei t0
mk
a(1)(t) = Ł'm| $2 |k' 1 - ei "t
m
2
Ł'm| $2 |k' (
) ( )
2
mk mk
a(1)(t) = 1 - e-i "t 1 - ei "t
m
2
2
mk
2
Ł'm| $2 |k'
= 2(1 - cos mk"t)
2
2
mk
2
Ł'm| $2 |k' sin2 mk"t
2
= 4 .
2
2
mk
2
mk
ei t0 a(1)(t)
m
"t
"t
1/"t
mk"t
= Ą
2
2Ą
mk <"
"t
1
0.8
0.6
0.4
0.2
-10 -7.5 -5 -2.5 2.5 5 7.5 10
2
a(1)(t) mk "t = 1 "t = 2
m
"E = Em - Ek
"t "E <" .
H2
"t
"t "
sin2 mk"t
1
2
lim = 2Ą(mk)
2
"t"
"t mk
4
H2
H2 (t) = 2V cos t.
t2
t0 = 0
+"t (
)
i
mk mk
a(1)(t) = - Ł'm| V |k' dt2 ei( +)t2 + ei( -)t2
m
0
( )
mk mk
1 1 - ei( +)t 1 - ei( -)t
= Ł'm| V |k' +
mk +  mk - 
( )
(mk+)t (mk-)t
sin sin
1
2 2
mk mk
= -2i Ł'm| V |k' ei( +)t/2 + ei( -)t/2 .
mk +  mk - 
a(1)(t)
m
2 4
a(1)(t) = |Ł'm| V |k'|2
m
2
( )
(mk+)t (mk-)t
sin2 (mk+)t sin2 (mk-)t sin sin
2 2 2 2
+ + 2 cos  .
2 - 2
(mk + )2 (mk - )2
mk
60 60
50 50
40 40
30 30
20 20
10 10
-2 -1.5 -1 -0.5 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0.5 1 1.5 2
| a(1)(t) |2 mn
m

mk
 t  > 0
 :
2
a(1)(t)
m
km = lim
t"
t
2Ą
= |Ł'm| V |k'|2 ((Em - Ek + ) + (Em - Ek - )) .
 0
2(Em - Ek) V H2 /2
1/2
 0
Em = Ek ą .
2 cos t = eit + e-it
ą
H2 (t) = V eąit
2Ą
km = |Ł'm| V |k'|2 (Em - Ek ą )
S0
|ąI(t)' = ei$ t/ |ąS(t)' ,
S0 S0
I(t) = ei$ t/ Se-i$ t/ ,
|ąI(t)'
d
2
i |ąI(t)' = $I |ąI(t)' .
dt
$
d
i |nS(t)' = $ |nS(t)' = En |nS(t)' ,
dt
n
|nS(t)' = e-iE t/ |nS(0)' ,
En - Em
Ł'nS(t)| S |mS(t)' = Ł'nS(0)| S |mS(0)' exp(i t)
H0 |nS(t)'
"
n
|S(t)' = |nS(0)' an(t)e-iE t/ .
n=1
n
Ł'nS(0)|S(t)' = an(t)e-iE t/
"
S0 S0
Ł'nS(0)|I(t)' = Ł'nS(0)| ei$ t/ |S(t)' = Ł'nS(0)| ei$ t/ |mS(0)' Ł'mS(0)|S(t)'
m
n n
= eiE t/ an(t)e-iE t/ = an(t).
"
2
S0 S0
Ł'mS(0)| $I |nS(0)' = Ł'mS(0)| ei$ t/ |kS(0)' Ł'kS(0)| $S |lS(0)' Ł'lS(0)| e-i$ t/ |nS(0)'
k,l
2
mn
= Ł'mS(0)| $S |nS(0)' ei t.
Ł' '
"
mn
i am = an m|$2 |n ei t
Ł
n
"
d
i Ł'mS(0)|I(t)' = Ł'mS(0)| $I |nS(0)' Ł'nS(0)|I(t)'
dt
n
= Ł'mS(0)| $ |I(t)' ,
|mS(0)'
|ąI(t)' = I(t, t0) |ąI(t0)'
d
2
i I(t, t0) = $I(t)UI(t, t0),
dt
2
I(t0, t0) = 1 $I(t)
( )
i
2
I(t, t0) = exp - $I(t - t0) ,
2
$I(t)
dt t0 t
+"t
i
2
I(t, t0) = 1 - dt2 $I(t2 )UI(t2 , t0).
t0
2 2
$I(t2 )  $I(t2 )
(1) (2)
I = 1 + I + 2I + . . .
t
HI(t' )
= + +
HI(t' )
HI(t'' )
t0
I(t, t0)
  = 1
2
( )2
+"t +"t +"t
i i
2 2 2
I(t, t0) = 1 - dt2 $I(t2 ) + dt2 dt2 2 $I(t2 )$I(t2 2 )
t0 t0 t0
2 2 2
( )3
+"t +"t +"t
i
2 2 2
+ dt2 dt2 2 dt2 2 2 $I(t2 )$I(t2 2 )$I(t2 2 2 ) + . . .
t0 t0 t0
t0 t
2
HI


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