$
= $
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 ei�t �(�) = 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 = ei�t + e-i�t
 ą�
H2 (t) = V eąi�t
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 + 2�I + . . .
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