h 0 h
Ż Ż
 m x
� = �(x, t) x
t �"(x, t)
�(x, t)
p(x, t) = �"(x, t)�(x, t) = |�(x, t)|2 ,
x t p(x, t)
x
x2 x2
P [x1, x2](t) = p(x, t) dx = |�(x, t)|2 dx ,
x1 x1
t [x1, x2]
x1 < x2 t
"
|�(x, t)|2dx = 1 .
-"
x t x
t
t
x
t
x 1
 �(x, t)
x
t {�} x
t
�(x)
�
�
�
"
|�(x)|2dx = S ,
-"
S 0 < S < "
� = N� ,
"
N = ( S)-1 �(x)
�
"
Ć|� = Ć"(x)�(x) dx ,
-"
||�|| = �|� .
||�||2 = �|� = 1 .
= {Ć1, Ć2, . . .} ,
Ćm|Ćn = �mn ,
 m, n = 1, 2, . . .
"
� = cm Ćm ,
m=1
cm
cm = Ćm|� .
�
"
|cm|2 = S ,
m=1
0 < S < "
Ć
Q
�
Ć
Q�
Ć Ć Ć
Q(Ć + �) = QĆ + Q� ,
Ć Ć
Q(c �) = c Q� ,
Ć � c
Ć Ć
Ć|Q� = QĆ|� ,
Ć �
x, y, z
px, py, pz x px
 x px
x = x ,
Ć
d
px = -iŻ .
Ć hdx
x x px
Ć Ć
Ć
V = V (x) V (x)
x
p2
Ćx
Ć
T = m
2m
Ć Ć
$
= T + V .
Ć ĆQ
Ć
R = P
Ć Ć(Q�)
Ć
R� = P ,
�
ĆQ Ć Ć
Ć
P = QP
Ć, Ć ĆQ Ć Ć
Ć
[P Q] = P - QP .
Ć, Ć Ć Ć
[P Q] = 0 0
Ć Ć
P Q
c c
Ć Ć
P Q
Ć, Ć ĆQ Ć Ć
Ć
[P Q] = P - QP = i
,
Ć


= 0

Ć Ć
P Q