Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
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ɧɨ, ɫ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɱɢɫɥɨ ɩɨɜɬɨɪɟɧɢɣ ɭɤɚɡɚɧɧɨɣ ɜɵɲɟ ɩɪɨɰɟɞɭɪɵ ɦɨɠɧɨ
ɜɡɹɬɶ ɫɤɨɥɶ ɭɝɨɞɧɨ ɛɨɥɶɲɢɦ. Ɉɬɫɸɞɚ ɦɵ ɡɚɤɥɸɱɚɟɦ, ɱɬɨ
m
O
ɩɪɢɧɢɦɚɟɬ ɬɨɥɶɤɨ
ɰɟɥɨɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ:
n O , ...,2,1,0 n . (10)
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɥɢɲɶ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚ ɧɟɤɨɬɨɪɨɦ ɲɚɝɟ ()ɦɵ ɩɪɢɞɟɦ ɤ
ɧɭɥɟɜɨɦɭ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ,
nm
0 O n , ɩɪɢɱɟɦ, ɤɚɤ ɜɢɞɧɨ ɢɡ (9),
ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ
0,ɨɬɜɟɱɚɸɳɢɣ ɧɭɥɟɜɨɦɭ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ, ɛɭɞɟɬ
ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɪɚɜɟɧɫɬɜɭ:
00
ˆ
a . (11)
ɉɨɜɬɨɪɧɨɟ ɞɟɣɫɬɜɢɟ ɨɩɟɪɚɬɨɪɚ ɧɚ ɨɛɟ ɫɬɨɪɨɧɵ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɭɠɟ ɧɟ ɞɚɟɬ
ɧɨɜɵɯ ɫɨɛɫɬɜɟɧɧɵɯ ɜɟɤɬɨɪɨɜ, ɬɨ ɟɫɬɶ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ (8), ɤɚɤ ɢ ɞɨɥɠɧɨ
ɛɵɬɶ, ɨɛɪɵɜɚɟɬɫɹ.
a
ˆ
Ɋɚɫɫɦɨɬɪɢɦ ɪɟɡɭɥɶɬɚɬ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɚ aa
ˆˆ
ɧɚ ɜɟɤɬɨɪ O
a
ˆ
, ɫɧɨɜɚ
ɩɪɢɧɢɦɚɹ ɜɨ ɜɧɢɦɚɧɢɟ ɫɨɨɬɧɨɲɟɧɢɹ (3) ɢ (4):
aa
ˆˆ
O
a
ˆ
)1
ˆˆ
(
ˆ
aaa O OO
a
ˆ
)1( .
ȼɢɞɧɨ, ɱɬɨ
O
a
ˆ
– ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ ɨɩɟɪɚɬɨɪɚ aa
ˆˆ
, ɩɪɢɧɚɞɥɟɠɚɳɢɣ
ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ 1, ɬɨ ɟɫɬɶO
a
ˆ
– «ɩɨɜɵɲɚɸɳɢɣ» ɨɩɟɪɚɬɨɪ. ɇɚɣɞɟɦ
ɤɜɚɞɪɚɬ ɧɨɪɦɵ ɷɬɨɝɨ ɜɟɤɬɨɪɚ, ɩɪɢɱɟɦ, ɭɱɢɬɵɜɚɹ (10), ɛɭɞɟɦ ɬɟɩɟɪɶ ɨɛɨɡɧɚɱɚɬɶ
ɧɨɪɦɢɪɨɜɚɧɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ
O ɤɚɤ n ; ɢɦɟɟɦ:
1)1
ˆˆ
(
ˆˆˆˆ
nnaannaannana .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
11
ˆ
nnna . (12)
ɗɬɨɣ ɮɨɪɦɭɥɨɣ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɞɟɣɫɬɜɢɟ ɨɩɟɪɚɬɨɪɚ
a
ˆ
ɧɚ ɧɨɪɦɢɪɨɜɚɧɧɵɟ
ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ ɨɩɟɪɚɬɨɪɚ aa
ˆˆ
. Ⱥɧɚɥɨɝɢɱɧɚɹ ɮɨɪɦɭɥɚ ɞɥɹ ɨɩɟɪɚɬɨɪɚ
ɫɥɟɞɭɟɬ ɢɡ (8):
a
ˆ
1
ˆ
nnna . (13)
ɂɬɚɤ, ɦɵ ɧɚɲɥɢ ɫɩɟɤɬɪ (10) ɨɩɟɪɚɬɨɪɚ aa
ˆˆ
ɢ, ɤɪɨɦɟ ɬɨɝɨ, ɭɛɟɞɢɥɢɫɶ, ɱɬɨ
ɧɚɱɚɜ ɫ ɧɨɪɦɢɪɨɜɚɧɧɨɝɨ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
0, ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɝɨ
ɭɪɚɜɧɟɧɢɸ (11), ɦɨɠɧɨ ɫ ɩɨɦɨɳɶɸ ɮɨɪɦɭɥɵ (12) ɩɨɥɭɱɢɬɶ ɜɫɟ ɨɫɬɚɥɶɧɵɟ
ɫɨɛɫɬɜɟɧɧɵɟ ɞɥɹ ɧɨɪɦɢɪɨɜɚɧɧɵɟ ɜɟɤɬɨɪɵaa
ˆˆ
n :
40
0
ˆ
1
a , 0
!2
)
ˆ
(
2
2
a
, … , 0
!
)
ˆ
(
n
a
n
n
, … .(14)
ɉɨɫɤɨɥɶɤɭ ɫɩɟɤɬɪ (10) ɧɟ ɜɵɪɨɠɞɟɧ (ɤɚɠɞɨɦɭ ɡɧɚɱɟɧɢɸ ɩɪɢɧɚɞɥɟɠɢɬ ɬɨɥɶɤɨ
ɨɞɢɧ ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ), ɫɨɫɬɨɹɧɢɹ (14) ɜɡɚɢɦɧɨ ɨɪɬɨɝɨɧɚɥɶɧɵ:
n
nn
nn
c
G
c
. (15)
ɉɨɤɚɠɟɦ, ɤɚɤ ɧɚɣɬɢ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ )(x
n
\ ɫɨɫɬɨɹɧɢɣ n ɜ
ɤɨɨɪɞɢɧɚɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ.
ɍɱɢɬɵɜɚɹ, ɱɬɨ ɜ ɤɨɨɪɞɢɧɚɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ
x
x
ˆ
,,ɢ
ɩɟɪɟɯɨɞɹ ɤ ɛɟɡɪɚɡɦɟɪɧɨɣ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɟɪɟɦɟɧɧɨɣ
xddip /
ˆ
=
=/
0
Z [ mx , (16)
ɡɚɩɢɲɟɦ ɨɩɟɪɚɬɨɪɵ (1) ɜ ɜɢɞɟ
¸
¸
¹
·
¨
¨
©
§
[
[
d
d
a
2
1
ˆ
,
¸
¸
¹
·
¨
¨
©
§
[
[
d
d
a
2
1
ˆ
. (17)
Ɋɚɜɟɧɫɬɜɨ (11) ɩɪɢɨɛɪɟɬɚɟɬ ɫɦɵɫɥ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ )(
0
x\
ɫɨɫɬɨɹɧɢɹ
0:
a
ˆ
0)(
0
\ x ,
ɬɨ ɟɫɬɶ
0)(
)(
0
0
[\[
[
[\
d
d
. (18)
ɇɟɩɨɫɪɟɞɫɬɜɟɧɧɨɣ ɩɪɨɜɟɪɤɨɣ ɥɟɝɤɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɪɟɲɟɧɢɟɦ ɷɬɨɝɨ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɹ
2/
0
2
)(
[
\ eCx , (19)
ɝɞɟ
C
– ɩɨɫɬɨɹɧɧɚɹ. ɑɬɨɛɵ ɨɩɪɟɞɟɥɢɬɶ
C
, ɜɵɱɢɫɥɢɦ ɧɨɪɦɢɪɨɜɨɱɧɵɣ ɢɧɬɟɝɪɚɥ:
³³
f
f
[
f
f
[
Z
\
2/
0
22
0
2
00 ed
m
Cxd
=
S
Z
0
2
m
C
=
.
ɉɨɥɚɝɚɹ
100 , ɧɚɯɨɞɢɦ:
4/1
0
¸
¸
¹
·
¨
¨
©
§
S
Z
=
m
C . (20)
41
Ɍɟɩɟɪɶ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ, ɜɵɬɟɤɚɸɳɟɣ ɢɡ (12):
1
ˆ
n
n
a
n , ...,3,2,1 n , (21)
ɤɨɬɨɪɚɹ ɜ ɤɨɨɪɞɢɧɚɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɨɡɧɚɱɚɟɬ, ɱɬɨ
)(
2
1
)(
1
[\
¸
¸
¹
·
¨
¨
©
§
[
[ [\
nn
d
d
n
. (22)
Ɋɟɤɭɪɪɟɧɬɧɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (22) ɩɨɡɜɨɥɹɟɬ ɲɚɝ ɡɚ ɲɚɝɨɦ ɨɩɪɟɞɟɥɢɬɶ ɜɨɥɧɨɜɵɟ
ɮɭɧɤɰɢɢ ), ɧɚɱɚɜ ɫ ɭɠɟ ɢɡɜɟɫɬɧɨɣ ɮɭɧɤɰɢɢ )(x
n
\ (
0
x\ . Ɇɨɠɧɨ ɩɨɥɭɱɢɬɶ ɢ
ɹɜɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ). ɋ ɷɬɨɣ ɰɟɥɶɸ ɡɚɦɟɬɢɦ, ɱɬɨ ɜ ɫɢɥɭ ɬɨɠɞɟɫɬɜɚ,
ɫɩɪɚɜɟɞɥɢɜɨɝɨ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɮɭɧɤɰɢɢ
(x
n
\
)([
f
,
¸
¹
·
¨
©
§
[
[
[
¸
¸
¹
·
¨
¨
©
§
[
[
[[
)()1()(
2/2/
22
fe
d
d
ef
d
d
, (23)
ɫɨɨɬɧɨɲɟɧɢɟ (22) ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
)([\
n
¸
¹
·
¨
©
§
[\
[
[[
)(
2
)1(
1
2/2/
22
n
e
d
d
e
n
.
ɉɪɢɦɟɧɢɦ ɜ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɚɧɚɥɨɝɢɱɧɨɟ ɪɚɜɟɧɫɬɜɨ ɞɥɹ
,)(
1
[\
n
)(
1
[\
n
¸
¹
·
¨
©
§
[\
[
[[
)(
)1(2
)1(
2
2/2/
22
n
e
d
d
e
n
.
ɉɨɜɬɨɪɢɦ ɩɨɞɨɛɧɭɸ ɩɪɨɰɟɞɭɪɭ ɟɳɟ 2n ɪɚɡ. ɇɚ ɩɨɫɥɟɞɧɟɦ ɲɚɝɟ ɜ ɩɪɚɜɨɣ
ɫɬɨɪɨɧɟ ɩɨɹɜɢɬɫɹ ɮɭɧɤɰɢɹ (19), ɢ ɦɵ ɩɪɢɞɟɦ ɤ ɮɨɪɦɭɥɟ:
)([\
n
)(
!2
)1(
22
2/ [[
[
e
d
d
e
n
C
n
n
n
n
.
ɋ ɩɨɦɨɳɶɸ ɩɨɥɢɧɨɦɨɜ ɗɪɦɢɬɚ )([
n
H ,
)([
n
H )()1(
22
[[
[
e
d
d
e
n
n
n
, (24)
ɨɤɨɧɱɚɬɟɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ ɫɨɫɬɨɹɧɢɣ
n ɡɚɩɢɲɟɬɫɹ ɜ
ɜɢɞɟ
42
)([\
n
)(
!2
1
2/
4/1
0
2
[
¸
¸
¹
·
¨
¨
©
§
S
Z
[
n
n
He
n
m
=
. (25)
Ɏɭɧɤɰɢɢ (25) ɫɨɜɩɚɞɚɸɬ ɫ ɜɨɥɧɨɜɵɦɢ ɮɭɧɤɰɢɹɦɢ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ
ɥɢɧɟɣɧɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ (ɫɦ. (5) ɜ ɡɚɞɚɱɟ 6.1).
ɜ) Ɉɛɪɚɳɚɹ ɪɚɜɟɧɫɬɜɚ (1), ɜɵɪɚɡɢɦ ɨɩɟɪɚɬɨɪɵ
x
ˆ
, ɱɟɪɟɡ ,:p
ˆ
a
ˆ
a
ˆ
Z
aa
m
x
ˆˆ
2
ˆ
0
=
, (26)
Z
aa
m
ip
ˆˆ
2
ˆ
0
=
. (27)
ɉɨɞɫɬɚɧɨɜɤɚ (26) ɢ (27) ɜ ɝɚɦɢɥɶɬɨɧɢɚɧ ɥɢɧɟɣɧɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ
22
0
2
ˆ
2
1
2
ˆ
ˆ
xm
m
p
H
ɨɫɰ
Z (28)
ɩɪɢɜɨɞɢɬ ɟɝɨ ɤ ɜɢɞɭ
)
ˆˆˆˆ
(
2
ˆ
0
aaaaH
ɨɫɰ
Z
=
. (29)
ɉɨɫɤɨɥɶɤɭ ɢɡ ɤɨɦɦɭɬɚɰɢɨɧɧɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ (3) ɫɥɟɞɭɟɬ ɪɚɜɟɧɫɬɜɨ
, ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɝɚɦɢɥɶɬɨɧɢɚɧ (29) ɜɵɪɚɡɢɬɶ ɩɨɥɧɨɫɬɶɸ ɱɟɪɟɡ
ɪɚɫɫɦɚɬɪɢɜɚɜɲɢɣɫɹ ɜɵɲɟ ɨɩɟɪɚɬɨɪ
1
ˆˆˆˆ
aaaa
aa
ˆˆ
, ɞɥɹ ɤɨɬɨɪɨɝɨ ɫɩɟɤɬɪɨɦ ɫɨɛɫɬɜɟɧɧɵɯ
ɡɧɚɱɟɧɢɣ ɫɥɭɠɚɬ ɱɢɫɥɚ (ɞɥɹ ɨɩɟɪɚɬɨɪɚ...,2,1,0
n
aa
ˆˆ
ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ
ɪɚɜɧɵ ):1
n
2
ˆˆ
ˆ
0
0
Z
Z
=
=
aaH
ɨɫɰ
. (30)
Ƚɚɦɢɥɶɬɨɧɢɚɧ ɹɜɥɹɟɬɫɹ ɨɞɧɢɦ ɢɡ ɨɫɧɨɜɧɵɯ ɫɪɟɞɫɬɜ ɨɩɢɫɚɧɢɹ ɤɚɧɬɨɜɨɣ
ɫɢɫɬɟɦɵ. Ɋɟɡɭɥɶɬɚɬ (30) ɩɪɢɦɟɱɚɬɟɥɟɧ ɬɟɦ, ɱɬɨ ɡɞɟɫɶ ɨɩɢɫɚɧɢɟ ɫɢɫɬɟɦɵ ɧɟ
ɜɤɥɸɱɚɟɬ ɬɚɤɢɯ ɜɟɥɢɱɢɧ, ɤɚɤ ɦɚɫɫɚ ɢ ɤɨɨɪɞɢɧɚɬɚ ɨɫɰɢɥɥɹɬɨɪɚ (ɜ ɝɚɦɢɥɶɬɨɧɢɚɧ
(30), ɜ ɨɬɥɢɱɢɟ ɨɬ (28), ɜɯɨɞɢɬ ɬɨɥɶɤɨ ɫɨɛɫɬɜɟɧɧɚɹ ɱɚɫɬɨɬɚ ɫɢɫɬɟɦɵ). ɗɬɨ
ɩɨɡɜɨɥɹɟɬ ɩɪɢɦɟɧɹɬɶ ɝɚɦɢɥɶɬɨɧɢɚɧɵ ɜɢɞɚ (30) ɞɥɹ «ɤɜɚɧɬɨɜɚɧɢɹ» ɫɢɫɬɟɦ,
ɨɬɥɢɱɧɵɯ ɨɬ ɨɞɢɧɨɱɧɨɣ ɱɚɫɬɢɰɵ, ɞɜɢɠɭɳɟɣɫɹ ɜ ɩɨɥɟ ɭɩɪɭɝɢɯ ɫɢɥ. ɉɨ
ɫɭɳɟɫɬɜɭ, ɥɸɛɚɹ ɫɢɫɬɟɦɚ, ɜ ɤɨɬɨɪɨɣ ɫ ɤɥɚɫɫɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɜɨɡɦɨɠɧɨ
ɜɨɡɛɭɠɞɟɧɢɟ ɤɚɤɨɝɨ-ɥɢɛɨ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ ɩɪɨɰɟɫɫɚ (ɦɨɞɵ ɫ
ɨɩɪɟɞɟɥɟɧɧɨɣ ɱɚɫɬɨɬɨɣ ) ɜ ɤɜɚɧɬɨɜɨɣ ɬɟɨɪɢɢ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ
ɨɫɰɢɥɥɹɬɨɪ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (30). ɗɧɟɪɝɢɹ ɫɢɫɬɟɦɵ, ɨɬɫɱɢɬɵɜɚɟɦɚɹ ɨɬ ɭɪɨɜɧɹ
0
Z
43
ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ 2, ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ:/
00
Z H =
n
n 00
Z HH = , ...,2,1,0 n . (31)
ɉɪɢ ɬɚɤɨɦ ɩɨɞɯɨɞɟ ɜɟɥɢɱɢɧɚ
0
Z= ɬɪɚɤɬɭɟɬɫɹ ɤɚɤ «ɤɜɚɧɬ ɷɧɟɪɝɢɢ» ɦɨɞɵ ɫ
ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɨɣ , ɱɢɫɥɨ ɢɧɬɟɪɩɪɟɬɢɪɭɟɬɫɹ ɤɚɤ ɱɢɫɥɨ ɧɟ
ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɤɜɚɧɬɨɜ ɷɧɟɪɝɢɢ ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ ɫɨɫɬɨɹɧɢɢ
ɫɢɫɬɟɦɵ, ɫɥɭɠɢɬ ɨɩɟɪɚɬɨɪɨɦ ɱɢɫɥɚ ɤɜɚɧɬɨɜ.
0
Z n
aa
ˆˆ
ɋ ɬɨɱɤɢ ɡɪɟɧɢɹ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ ɭɤɚɡɚɧɧɵɟ ɤɜɚɧɬɵ ɷɧɟɪɝɢɢ ɦɨɠɧɨ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɛɨɡɨɧɵ – ɱɚɫɬɢɰɵ, ɩɨɞɱɢɧɹɸɳɢɟɫɹ ɫɬɚɬɢɫɬɢɤɟ Ȼɨɡɟ–
ɗɣɧɲɬɟɣɧɚ. Ⱦɥɹ ɷɬɨɝɨ ɬɢɩɚ ɫɬɚɬɢɫɬɢɤɢ ɯɚɪɚɤɬɟɪɧɨ ɨɬɫɭɬɫɬɜɢɟ ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ
ɦɚɤɫɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɨɟ ɱɢɫɥɨ ɱɚɫɬɢɰ (ɜ ɫɥɭɱɚɟ ɠɟ ɮɟɪɦɢɨɧɨɜ, ɬɨ ɟɫɬɶ –
ɱɚɫɬɢɰ, ɩɨɞɱɢɧɹɸɳɢɯɫɹ ɫɬɚɬɢɫɬɢɤɟ Ɏɟɪɦɢ–Ⱦɢɪɚɤɚ, ɜ ɮɨɪɦɭɥɟ (31)
ɞɨɩɭɫɤɚɥɢɫɶ ɛɵ ɥɢɲɶ ɡɧɚɱɟɧɢɹ 1
n
,0
n ). Ɉɩɟɪɚɬɨɪɵ ɢ ,
ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟɫɹ ɤɨɦɦɭɬɚɰɢɨɧɧɵɦ ɫɨɨɬɧɨɲɟɧɢɟɦ (3), ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ
ɛɨɡɨɧɧɵɦɢ ɨɩɟɪɚɬɨɪɚɦɢ ɪɨɠɞɟɧɢɹ ɢ ɭɧɢɱɬɨɠɟɧɢɹ.
a
ˆ
a
ˆ
6.8. ɚ) ɇɚɢɛɨɥɟɟ ɷɤɨɧɨɦɧɨ ɜɵɱɢɫɥɟɧɢɟ ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ
c
nn
x nxn
ˆ
c
(1)
ɩɪɨɢɡɜɨɞɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɨɩɟɪɚɬɨɪɨɜ ɢa
ˆ
a
ˆ
, ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɜ ɡɚɞɚɱɟ 6.7.
ɉɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɚɦɢ (26), (12) ɢ (13) ɡɚɞɚɱɢ 6.7, ɢɦɟɟɦ:
c
nn
x
c
Z
nan
m
ˆ
2
0
=
c
Z
nan
m
ˆ
2
0
=
c
Z
1
2
0
nn
m
n
=
1
2
)1(
0
c
Z
nn
m
n
=
.
ɋ ɭɱɟɬɨɦ ɪɚɜɟɧɫɬɜɚ
nn
nn
ccc
G
ccc
, ɜɵɪɚɠɚɸɳɟɝɨ ɫɜɨɣɫɬɜɨ
ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɧɨɫɬɢ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɩɨɥɭɱɚɟɦ:
c
nn
x G
Z
c
1,
0
2
nn
m
n
=
1,
0
2
)1(
c
G
Z
nn
m
n
=
. (2)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɬɥɢɱɧɵɟ ɨɬ ɧɭɥɹ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɤɨɨɪɞɢɧɚɬɵ ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
nn
x
,1
1, nn
x
0
2 Zm
n
=
. (3)
44
ɛ) ɋɨɨɬɧɨɲɟɧɢɟ (2) ɩɨɡɜɨɥɹɟɬ ɥɟɝɤɨ ɨɩɪɟɞɟɥɹɬɶ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ
ɨɩɟɪɚɬɨɪɨɜ
m
x
ˆ
. ȼ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ ɫɪɟɞɧɟɝɨ ɡɧɚɱɟɧɢɹ ɤɜɚɞɪɚɬɚ ɤɨɨɪɞɢɧɚɬɵ
ɧɚɯɨɞɢɦ, ɱɬɨ
nxnx
22
ˆ
cc
¦
c
nxnnxn
n
ˆˆ
1
ˆ
nxn nxn
ˆ
1 1
ˆ
nxn
¸
¹
·
¨
©
§
Z
2
1
ˆ
1
0
n
m
nxn
=
. (4)
ɇɢɠɟ ɦɵ ɜɨɫɩɪɨɢɡɜɟɞɟɦ ɪɟɡɭɥɶɬɚɬ (4) ɞɪɭɝɢɦ ɫɩɨɫɨɛɨɦ. ɉɪɟɞɜɚɪɢɬɟɥɶɧɨ
ɞɨɤɚɠɟɦ ɨɞɧɭ ɩɨɥɟɡɧɭɸ ɜ ɜɵɱɢɫɥɟɧɢɹɯ ɮɨɪɦɭɥɭ ɞɥɹ ɫɪɟɞɧɢɯ ɡɧɚɱɟɧɢɣ
ɷɪɦɢɬɨɜɚ ɨɩɟɪɚɬɨɪɚ ɫ ɞɢɫɤɪɟɬɧɵɦ ɫɩɟɤɬɪɨɦ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ.
ɉɭɫɬɶ ) – ɷɪɦɢɬɨɜ ɨɩɟɪɚɬɨɪ ɮɢɡɢɱɟɫɤɨɣ ɜɟɥɢɱɢɧɵ, ɡɚɜɢɫɹɳɟɣ ɨɬ
ɜɟɳɟɫɬɜɟɧɧɨɝɨ ɩɚɪɚɦɟɬɪɚ , – ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɷɬɨɝɨ ɨɩɟɪɚɬɨɪɚ. Ɍɟ
ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ
(
ˆ
Of
O
k
f
k , ɤɨɬɨɪɵɟ ɩɪɢɧɚɞɥɟɠɚɬ ɞɢɫɤɪɟɬɧɵɦ ɫɨɛɫɬɜɟɧɧɵɦ
ɡɧɚɱɟɧɢɹɦ ɨɩɟɪɚɬɨɪɚ , ɨɛɥɚɞɚɸɬ ɫɯɨɞɹɳɢɦɫɹ ɧɨɪɦɢɪɨɜɨɱɧɵɦ ɢɧɬɟɝɪɚɥɨɦ ɢ
ɩɨɷɬɨɦɭ ɦɨɝɭɬ ɛɵɬɶ ɧɨɪɦɢɪɨɜɚɧɵ ɧɚ ɟɞɢɧɢɰɭ:
f
ˆ
1 kk . (5)
Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɨɛɟ ɫɬɨɪɨɧɵ ɪɚɜɟɧɫɬɜɚ (5) ɩɨ
O
ɢ ɨɛɨɡɧɚɱɚɹ kk
~
/ Oww ,
ɩɨɥɭɱɚɟɦ:
0
~~
kkkk . (6)
ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɢɜ ɧɚ
k ɨɛɟ ɫɬɨɪɨɧɵ ɪɚɜɟɧɫɬɜɚ
kfkf
k
ˆ
(7)
ɢ ɩɪɨɞɢɮɮɟɪɟɧɰɢɪɨɜɚɜ ɩɨ ɩɨɥɭɱɢɜɲɟɟɫɹ ɫ ɭɱɟɬɨɦ (5) ɫɨɨɬɧɨɲɟɧɢɟO
k
fkfk
ˆ
, (8)
ɢɦɟɟɦ:
Ow
w
Ow
w
k
f
kfkk
f
kkfk
~
ˆ
ˆ
ˆ
~
. (9)
Ɂɞɟɫɶ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɜ ɫɢɥɭ (7) ɪɚɜɧɨ
kkf
k
~
; ɬɪɟɬɶɟ ɫɥɚɝɚɟɦɨɟ ɫ ɭɱɟɬɨɦ
ɷɪɦɢɬɨɜɨɫɬɢ ɨɩɟɪɚɬɨɪɚ ɢ ɜɟɳɟɫɬɜɟɧɧɨɫɬɢ ɟɝɨ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
ɩɪɢɧɢɦɚɟɬ ɜɢɞ
f
ˆ
45
kkfkkfkfk
k
~~
ˆ
~
ˆ
. (10)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫ ɭɱɟɬɨɦ (6) ɞɜɚ ɭɩɨɦɹɧɭɬɵɯ ɱɥɟɧɚ ɜ ɥɟɜɨɣ ɫɬɨɪɨɧɟ (9) ɜɡɚɢɦɧɨ
ɭɧɢɱɬɨɠɚɸɬɫɹ, ɢ ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɩɨɥɟɡɧɨɦɭ ɫɨɨɬɧɨɲɟɧɢɸ:
Ow
w
Ow
w
k
f
k
f
k
ˆ
. (11)
ɉɪɨɜɟɪɢɦ ɫ ɩɨɦɨɳɶɸ ɷɬɨɣ ɮɨɪɦɭɥɵ ɪɟɡɭɥɶɬɚɬ (4), ɩɪɟɞɫɬɚɜɢɜ
ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ )ɝɚɦɢɥɶɬɨɧɢɚɧɚ ɤɚɤ ɟɝɨ ɦɚɬɪɢɱɧɵɟ
ɷɥɟɦɟɧɬɵ ɜ ɛɚɡɢɫɟ, ɨɛɪɚɡɨɜɚɧɧɨɦ ɫɬɚɰɢɨɧɚɪɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ:
2/1(
0
Z n=
ɨɫɰ
H
ˆ
¸
¸
¹
·
¨
¨
©
§
Z nxm
m
p
n
22
0
2
ˆ
2
1
2
ˆ
¸
¹
·
¨
©
§
Z
2
1
0
n= . (12)
Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɥɟɜɭɸ ɢ ɩɪɚɜɭɸ ɫɬɨɪɨɧɵ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɩɨ , ɩɨɥɭɱɚɟɦ
ɫɨɨɬɧɨɲɟɧɢɟ
0
Z
Z nxnm
2
0
ˆ
¸
¹
·
¨
©
§
2
1
n
= ,
ɩɨɞɬɜɟɪɠɞɚɸɳɟɟ ɫɩɪɚɜɟɞɥɢɜɨɫɬɶ ɮɨɪɦɭɥɵ (4):
2
x
¸
¹
·
¨
©
§
Z
2
1
0
n
m
=
. (13)
Ɉɬɦɟɬɢɦ, ɱɬɨ
0 x (ɬɚɤ ɤɚɤ ɞɢɚɝɨɧɚɥɶɧɵɟ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ (2)
ɪɚɜɧɵ ɧɭɥɸ), ɢ ɩɨɷɬɨɦɭ ɧɚɣɞɟɧɧɚɹ ɜɟɥɢɱɢɧɚ
2
x ɫɨɜɩɚɞɚɟɬ ɫɨ
ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɧɨɣ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶɸ ɤɨɨɪɞɢɧɚɬɵ ɨɫɰɢɥɥɹɬɨɪɚ.
ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɤɨɨɪɞɢɧɚɬɵ ɪɚɜɧɚ
2
)( xG
2/1
0
)
2
1
(
¸
¸
¹
·
¨
¨
©
§
Z
G n
m
x
=
. (14)
ɋ ɭɱɟɬɨɦ ɮɨɪɦɭɥ (12)–(13) ɢ ɪɚɜɟɧɫɬɜɚ
0 p (ɨɛɨɫɧɨɜɚɧɢɟ ɤɨɬɨɪɨɦɭ
ɱɢɬɚɬɟɥɶ ɥɟɝɤɨ ɧɚɣɞɟɬ ɜ ɤɚɱɟɫɬɜɟ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɭɩɪɚɠɧɟɧɢɹ), ɢɦɟɟɦ:
2/1
0
2
)
2
1
(
¸
¹
·
¨
©
§
Z G nmpp
=
. (15)
ȼɢɞɧɨ, ɱɬɨ
46
2
/
)2
/
1( == t GG n
p
x
, (16)
ɜ ɫɨɝɥɚɫɢɢ ɫ ɫɨɨɬɧɨɲɟɧɢɟɦ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ Ƚɟɣɡɟɧɛɟɪɝɚ.
6.9. ɚ) Ɋɚɫɫɦɚɬɪɢɜɚɟɦɨɟ ɫɨɫɬɨɹɧɢɟ ɨɫɰɢɥɥɹɬɨɪɚ ɢɦɟɟɬ ɜɢɞ ɫɭɩɟɪɩɨɡɢɰɢɢ
ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ
n ,
¦
f
D
0
0
n
n
t
nA , (1)
ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
!
2/
2
n
eA
n
n
D
D
. (2)
ɗɤɫɩɨɧɟɧɰɢɚɥɶɧɵɣ ɦɧɨɠɢɬɟɥɶ ɜ (2) ɨɛɟɫɩɟɱɢɜɚɟɬ ɧɨɪɦɢɪɨɜɤɭ ɜɟɤɬɨɪɚ
ɫɨɫɬɨɹɧɢɹ (1) ɧɚ ɟɞɢɧɢɰɭ. Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɩɨɥɭɱɢɬɶ ɫɨɫɬɨɹɧɢɟ
D , ɨɬɧɟɫɟɧɧɨɟ
ɤ ɩɪɨɢɡɜɨɥɶɧɨɦɭ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ
t
, ɞɨɫɬɚɬɨɱɧɨ ɭɦɧɨɠɢɬɶ ɧɚ , ɝɞɟ
– ɷɧɟɪɝɢɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
n
A
=/ti
n
e
H
)2/1(
0
Z H n
n
= n :
¦
f
D
0
)(
n
n
ntA , (3)
!
)(
2/
2
n
etA
n
n
D
D
=/ti
n
e
H
. (4)
ɉɪɢɫɬɭɩɢɦ ɤ ɜɵɱɢɫɥɟɧɢɸ
DD xtx
ˆ
)(,ɩɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɨɣ (2)
ɡɚɞɚɱɢ 6.8:
DD x
ˆ
¦¦
c
nn
)()(
*
tAtA
nn
c
c
nxn
ˆ
0
2 Z
m
=
¦
f
0n
ntAtA
nn
)()(
*
1
0
2 Zm
=
¦
f
0n
1)()(
*
1
ntAtA
nn
. (5)
ȼ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ (5) ɜ ɩɟɪɜɨɣ ɢɡ ɞɜɭɯ ɫɭɦɦ ɱɥɟɧ ɫ 0 n ɪɚɜɟɧ ɧɭɥɸ, ɬɚɤ ɱɬɨ
ɡɞɟɫɶ ɦɨɠɧɨ ɡɚɦɟɧɢɬɶ ɧɚ 1; ɷɬɨ ɞɚɟɬ:n n
DD x
ˆ
0
2 Zm
=
¦
f
0n
..1)()(
1
*
ɫɤntAtA
nn
, (6)
ɝɞɟ «ɤ.ɫ.» ɨɡɧɚɱɚɟɬ ɜɵɪɚɠɟɧɢɟ, ɤɨɦɩɥɟɤɫɧɨ ɫɨɩɪɹɠɟɧɧɨɟ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ
ɩɪɟɞɵɞɭɳɟɦɭ ɜɵɪɚɠɟɧɢɸ, ɜɵɩɢɫɚɧɧɨɦɭ ɹɜɧɨ. ɋ ɭɱɟɬɨɦ (4) ɢɦɟɟɦ:
47
¦
f
0n
1)()(
1
*
ntAtA
nn
!
*)(
1
0
2
n
e
nn
n
f
D
DD
¦
HH =/)(
1
ti
nn
e
D
D
¦
f
ZD
!
2
0
0
2
n
e
n
n
ti
ti
e
0
Z
D . (7)
ɉɨɞɫɬɚɜɢɜ (7) ɜ (6), ɩɨɥɭɱɢɦ:
DD x
ˆ
0
2 Zm
=
titi
ee
00
*
ZZ
DD . (8)
Ɂɚɩɢɲɟɦ ɤɨɦɩɥɟɤɫɧɭɸ ɜɟɥɢɱɢɧɭ D ɜ ɮɨɪɦɟ
G
D D
i
e ɢ ɨɛɨɡɧɚɱɢɦ
C
m
D
Z
0
2=
. (9)
ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɩɪɟɞɫɬɚɜɢɬɶ ɪɟɡɭɥɶɬɚɬ (8) ɜ ɜɢɞɟ
)(cos)(
0
GZ tCtx . (10)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɤɨɨɪɞɢɧɚɬɵ ɨɫɰɢɥɥɹɬɨɪɚ ɜ ɤɨɝɟɪɟɧɬɧɨɦ
ɫɨɫɬɨɹɧɢɢ
D ɢɫɩɵɬɵɜɚɟɬ ɝɚɪɦɨɧɢɱɟɫɤɨɟ ɤɨɥɟɛɚɧɢɟ ɫ ɱɚɫɬɨɬɨɣ , ɚɦɩɥɢɬɭɞɚ
0
Z
C ɢ ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ G ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟɥɹɸɬɫɹ ɤɨɦɩɥɟɤɫɧɵɦ ɩɚɪɚɦɟɬɪɨɦ D.
ɛ) ɂɡ (3) ɫɥɟɞɭɟɬ, ɱɬɨ ɞɥɹ ɨɫɰɢɥɥɹɬɨɪɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɤɨɝɟɪɟɧɬɧɨɦ
ɫɨɫɬɨɹɧɢɢ
D , ɜɟɪɨɹɬɧɨɫɬɶ ɨɛɧɚɪɭɠɢɬɶ ɫɨɫɬɨɹɧɢɟ
n
W n ɟɫɬɶ
2
)(tAW
nn
2
!
2
D
D
e
n
n
. (11)
ɇɚɣɞɟɦ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ «ɱɢɫɥɚ ɤɜɚɧɬɨɜ» ɜ ɤɨɝɟɪɟɧɬɧɨɦ ɫɨɫɬɨɹɧɢɢn
D :
¦
f
1n
n
Wnn
D
D
f
¦
2
!)1(
2
1
e
n
n
n
2
2
D
D e
!
2
0
n
n
n
D
¦
f
2
D
.
ɂɬɚɤ,
n
2
D
, (12)
ɢ ɩɨɷɬɨɦɭ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɢ (11) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
n
W
n
n
e
n
n
!
. (13)
48
ȼɵɪɚɠɟɧɢɟ (13) ɧɚɡɵɜɚɸɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɉɭɚɫɫɨɧɚ.
ɇɚɣɞɟɦ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɤɜɚɞɪɚɬɚ ɱɢɫɥɚ ɤɜɚɧɬɨɜ:
¦
f
1
22
n
n
Wnn
f
¦
n
n
n
e
n
nn
!)1(
1
f
¦
n
n
n
e
n
nn
!
)1(
1
0
¦
f
0n
n
Wnn
¦
f
0n
n
Wn
nn
2
.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ,
nnn
2
2
. (14)
Ɉɬɦɟɬɢɦ, ɱɬɨ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ n
G
ɨɤɚɡɵɜɚɟɬɫɹ ɪɚɜɧɨɣ ɤɨɪɧɸ
ɤɜɚɞɪɚɬɧɨɦɭ ɢɡ ɫɪɟɞɧɟɝɨ ɡɧɚɱɟɧɢɹ :n
nnnn G
2
2
. (15)
ɗɬɚ ɜɟɥɢɱɢɧɚ ɪɚɫɬɟɬ ɫ ɭɜɟɥɢɱɟɧɢɟɦ
n , ɨɞɧɚɤɨ ɨɬɧɨɫɢɬɟɥɶɧɚɹ ɮɥɭɤɬɭɚɰɢɹ
ɱɢɫɥɚ ɤɜɚɧɬɨɜ ɭɛɵɜɚɟɬ:
1
1
G
n
n
n
ɩɪɢ
1!!n . (16)
ɋ ɜɟɥɢɱɢɧɨɣ
n ɫɜɹɡɚɧɨ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɷɧɟɪɝɢɢ ɨɫɰɢɥɥɹɬɨɪɚ H .
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɡɚɦɟɬɢɦ, ɱɬɨ
DD H H
ˆ
¦
¦
c
nn
)()(
*
tAtA
nn
c
nHn
ˆ
c
, (17)
ɢ ɭɱɬɟɦ ɪɚɜɟɧɫɬɜɨ
nn
nnHn
c
GZ
c
)2/1(
ˆ
0
= . (18)
ȼ ɢɬɨɝɟ (17) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
2/
00
ZZ H == n . (19)
ɋ ɭɱɟɬɨɦ (9) ɢ (12) ɫɨɨɬɧɨɲɟɧɢɟ (19) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ:
22
1
0
22
0
Z
Z H
=
Cm . (20)
ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɜ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ (20) ɢɦɟɟɬ ɜɢɞ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɷɧɟɪɝɢɢ
ɤɥɚɫɫɢɱɟɫɤɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ, ɤɨɥɟɛɥɸɳɟɝɨɫɹ ɫ ɚɦɩɥɢɬɭɞɨɣ C . ȿɫɥɢ
1!!D , ɬɨ
49