Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
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¦¦
3
1
3
1
321
2
1
),,(
ij
jiij
xxKxxxU , (4)
ɝɞɟ – ɜɟɳɟɫɬɜɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɨɛɪɚɡɭɸɳɢɟ ɫɢɦɦɟɬɪɢɱɧɵɣ
ɬɟɧɡɨɪ ɜɬɨɪɨɝɨ ɪɚɧɝɚ, – ɤɨɨɪɞɢɧɚɬɵ
jiij
KK
321
,, xxxzy
x
,,.ɉɪɢɜɟɞɟɧɢɟ ɮɨɪɦɵ (4) ɤ
ɫɭɦɦɟ ɤɜɚɞɪɚɬɨɜ ɢɥɢ, ɢɧɵɦɢ ɫɥɨɜɚɦɢ, ɩɪɢɜɟɞɟɧɢɟ ɬɟɧɡɨɪɚ ɤ ɞɢɚɝɨɧɚɥɶɧɨɦɭ
ɜɢɞɭ ɫɜɹɡɚɧɨ ɫ ɡɚɞɚɱɟɣ ɨɬɵɫɤɚɧɢɹ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
ij
K
K
ɢ ɫɨɛɫɬɜɟɧɧɵɯ
ɜɟɤɬɨɪɨɜ ɦɚɬɪɢɰɵ :
j
C
ij
K
¦
G
j
jijij
CKK 0)( . (5)
ɋɢɫɬɟɦɚ ɨɞɧɨɪɨɞɧɵɯ ɭɪɚɜɧɟɧɢɣ (5) ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɥɢɱɢɧ ,
ɩɪɟɞɫɬɚɜɥɹɸɳɢɯ ɫɨɛɨɣ ɤɨɦɩɨɧɟɧɬɵ ɨɪɬɨɜ ɧɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ
, ɢɦɟɟɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɩɪɢ ɭɫɥɨɜɢɢ:
j
C
321
,, XXX
0)(det G
ijij
KK . (6)
ȼ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɬɟɧɡɨɪ ɩɪɢɦɟɬ ɞɢɚɝɨɧɚɥɶɧɵɣ ɜɢɞ, ɩɪɢɱɟɦ
ɟɝɨ ɞɢɚɝɨɧɚɥɶɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ ɛɭɞɭɬ ɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ (6). Ɍɚɤɢɦ
ɨɛɪɚɡɨɦ ɜ ɧɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
321
,, XXX
ij
K
321
,, KKK
2
33
2
22
2
11321
2
1
),,( XKXKXKXXXU . (7)
Ɋɟɲɟɧɢɹɦɢ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (5) ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨɥɨɠɟɧɢɟ ɧɨɜɵɯ
ɤɨɨɪɞɢɧɚɬɧɵɯ ɨɫɟɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɜɨɧɚɱɚɥɶɧɵɯ ɨɫɟɣ ɢ ɬɟɦ ɫɚɦɵɦ
ɭɫɬɚɧɚɜɥɢɜɚɸɬɫɹ ɮɨɪɦɭɥɵ ɩɟɪɟɯɨɞɚ ɨɬ ɤ :
321
,, xxx
321
,, XXX
¦
j
j
n
j
n
xCX
)(
, (8)
ɝɞɟ – ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ, ɩɪɢɧɚɞɥɟɠɚɳɢɣ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ ,
ɧɨɪɦɢɪɨɜɚɧɧɵɣ ɭɫɥɨɜɢɟɦ
)(n
j
C
n
K
2
)(
1
n
C
2
)(
2
n
C
1
2
)(
3
n
C . (9)
Ⱦɥɹ ɩɨɬɟɧɰɢɚɥɚ (1) ɭɤɚɡɚɧɧɵɣ ɦɟɬɨɞ ɞɚɟɬ ,,ɢ ɜɟɞɟɬ ɤ
ɪɟɡɭɥɶɬɚɬɚɦ (2), (3).
2
021
Z mKK 0
3
K
ɉɨɬɟɧɰɢɚɥ (3) ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɜɭɦɟɪɧɨɦɭ ɨɫɰɢɥɥɹɬɨɪɧɨɦɭ ɞɜɢɠɟɧɢɸ ɜ
ɧɚɩɪɚɜɥɟɧɢɹɯ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɨɫɢ , ɢ ɫɜɨɛɨɞɧɨɦɭ ɞɜɢɠɟɧɢɸ ɱɚɫɬɢɰɵ
z
30
ɜɞɨɥɶ ɨɫɢ . ɉɪɢɦɟɧɢɜ ɤ ɭɪɚɜɧɟɧɢɸ ɒɪɟɞɢɧɝɟɪɚ ɫ ɩɨɬɟɧɰɢɚɥɨɦ (3) ɩɪɨɰɟɞɭɪɭ
ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ, ɩɨɥɭɱɢɦ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ
ɱɚɫɬɢɰɵ ɜ ɜɢɞɟ
z
\ ),,(
21
ZYX
z
pnn
)(
1
X
ɨɫɰ
n
\ )(
2
Y
ɨɫɰ
n
\ )(Z
z
p
\ , (10)
ɝɞɟ ɮɭɧɤɰɢɢ ɞɚɸɬɫɹ ɮɨɪɦɭɥɨɣ (5) ɡɚɞɚɱɢ 6.1,
ɨɫɰ
n
\
z
p
\ – ɩɥɨɫɤɚɹ ɜɨɥɧɚ:
=/
)(
Zpi
p
z
z
eCZ \ . (11)
ɗɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ ɫɨɫɬɨɹɧɢɢ (10) ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɞɢɫɤɪɟɬɧɵɯ
ɭɪɨɜɧɟɣ ɷɧɟɪɝɢɢ ɞɜɭɦɟɪɧɨɝɨ ɨɫɰɢɥɥɹɬɨɪɧɨɝɨ ɞɜɢɠɟɧɢɹ ɢ ɷɧɟɪɝɢɢ ɨɞɧɨɦɟɪɧɨɝɨ
ɫɜɨɛɨɞɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫ ɢɦɩɭɥɶɫɨɦ ff
z
p :
mpnn
zpnn
z
2/)1(
2
210
21
Z H = . (12)
6.4. Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɢ ɡɧɚɱɟɧɢɣ ɢɦɩɭɥɶɫɚ,
dp
pa
pdW
n
=S
2
)(
)(
2
, (1)
ɜ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɹɯ )ɥɢɧɟɣɧɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ,(x
n
\
\ )(x
n
¸
¹
·
¨
©
§
\
¸
¹
·
¨
©
§
b
x
HeCx
n
b
x
n
ɨɫɰ
n
2
2
1
)( , (2)
ɝɞɟ
!2
11
4/1
2
n
b
C
n
n
¸
¸
¹
·
¨
¨
©
§
S
,
0
Z
m
b
=
, (3)
ɜɵɱɢɫɥɢɦ ɤɨɦɩɨɧɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ )(x
n
\ , ɬɨ ɟɫɬɶ – ɤɨɷɮɮɢɰɢɟɧɬɵ
ɪɚɡɥɨɠɟɧɢɹ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ (2) ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ ɢɦɩɭɥɶɫɚ:
)( pa
n
)()(
/
xedxpa
n
xpi
n
\
³
f
f
=
. (4)
Ɉɛɨɡɧɚɱɢɜ
[ b
x
/
, qb
p
=
/
, (5)
ɢɦɟɟɦ:
)(
2/
2
[[
³
f
f
[[
n
qi
nn
HeedbCa . (6)
31
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɢɧɬɟɝɪɚɥɚ (6) ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ, ɧɚɩɪɢɦɟɪ,
ɫɥɟɞɭɸɳɢɦ ɬɨɠɞɟɫɬɜɨɦ, ɩɪɢɦɟɧɢɦɵɦ ɤ ɩɪɨɢɡɜɟɞɟɧɢɸ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɣ
ɮɭɧɤɰɢɢ ɢ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɩɨɥɢɧɨɦɚ :
[ qi
e
n
n
AAA [[ ...
10
[ qi
e ()
n
n
AAA [[ ...
10
)...(
10
n
n
dq
d
iA
dq
d
iAA
¸
¸
¹
·
¨
¨
©
§
[ qi
e .
Ɇɟɧɹɹ ɨɱɟɪɟɞɧɨɫɬɶ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɨ [ ɢ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɣ ɩɨ , ɢ, ɤɪɨɦɟ
ɬɨɝɨ, ɩɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɨɣ (3) ɢɡ ɡɚɞɚɱɢ 4.1, ɩɨɥɭɱɢɦ:
q
[
¸
¸
¹
·
¨
¨
©
§
³
f
f
[[
qi
nnn
eed
dq
d
iHbCa
2/
2
ˆ
2/
2
ˆ
2
q
nn
e
dq
d
iHbC
¸
¸
¹
·
¨
¨
©
§
S . (7)
ɇɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɫ ɭɱɟɬɨɦ ɹɜɧɨɝɨ ɜɢɞɚ ɩɨɥɢɧɨɦɨɜ ɗɪɦɢɬɚ
ɞɚɟɬ:
n
H
¸
¸
¹
·
¨
¨
©
§
2/
0
2
ˆ
q
e
dq
d
iH
2/
2
q
e )(
0
2/
2
qHe
q
,
¸
¸
¹
·
¨
¨
©
§
2/
1
2
ˆ
q
e
dq
d
iH
2/
2
2
q
eiq )(
1
2/
2
qHei
q
,
¸
¸
¹
·
¨
¨
©
§
2/
2
2
ˆ
q
e
dq
d
iH
2/2
2
)42(
q
eq )(
2
2/
2
qHe
q
,
ɢ ɬɚɤ ɞɚɥɟɟ. Ɇɵ ɩɪɢɯɨɞɢɦ ɤ ɡɚɤɥɸɱɟɧɢɸ, ɱɬɨ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ
p
(ɢɥɢ, ɱɬɨ ɬɨ ɠɟ
ɫɚɦɨɟ, ɨɬ =
/
b
p
q ) ɤɨɦɩɨɧɟɧɬ Ɏɭɪɶɟ (4) ɜ ɫɥɭɱɚɟ ɮɭɧɤɰɢɣ
ɚɧɚɥɨɝɢɱɧɚ ɩɨ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɮɨɪɦɟ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
)(x
ɨɫɰ
n
\
ɨɫɰ
n
\
x
:
)(2)(
2/
2
qHeCbia
n
q
n
n
n
S . (8)
ɉɨɞɫɬɚɜɥɹɹ (8) ɜ (1) ɢ ɭɱɢɬɵɜɚɹ (3), ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ
ɜɟɪɨɹɬɧɨɫɬɢ:
dp
m
p
He
mn
pdW
n
mp
n
¸
¸
¹
·
¨
¨
©
§
Z
ZS
Z
=
=
=
0
2
/
0
0
2
!2
1
)( . (9)
6.5. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɛɟɫɫɩɢɧɨɜɨɣ ɱɚɫɬɢɰɵ ɫ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɡɚɪɹɞɨɦ ɜ
ɩɪɢɫɭɬɫɬɜɢɢ ɜɟɤɬɨɪɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ
A ɢɦɟɟɬ ɜɢɞ:
q
2
ˆ
2
1
ˆ
¸
¸
¹
·
¨
¨
©
§
Ap
c
q
m
H , (1)
32
ɝɞɟ . Ɋɚɫɤɪɵɜ ɫɤɨɛɤɢ ɫ ɭɱɟɬɨɦ ɡɚɞɚɧɧɨɝɨ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɹɜɧɨɝɨ ɜɢɞɚ
ɤɨɦɩɨɧɟɧɬ ɜɟɤɬɨɪɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ,
=ip
ˆ
ByA
x
, 0
zy
AA , (2)
ɩɨɥɭɱɢɦ
m
p
H
x
2
ˆ
ˆ
2
2
2
2
2
ˆˆ
2
xxxxx
A
mc
q
pAAp
mc
q
m
p
y
2
ˆ
2
m
p
z
2
ˆ
2
m
p
x
2
ˆ
2
m
p
z
2
ˆ
2
2
2
2
2
ˆ
2
ˆ
y
mc
qBm
py
mc
qB
m
p
x
y
¸
¹
·
¨
©
§
. (3)
ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɤɨɦɦɭɬɚɬɢɜɧɨ ɫ ɨɩɟɪɚɬɨɪɚɦɢ
ɢ , ɬɚɤ ɤɚɤ ɨɧɨ ɧɟ ɫɨɞɟɪɠɢɬ
ɤɨɨɪɞɢɧɚɬ
x
p
ˆ
z
p
ˆ
x
ɢ . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ
ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɢɫɤɚɬɶ ɤɚɤ ɬɚɤɢɟ ɮɭɧɤɰɢɢ, ɤɨɬɨɪɵɟ
ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɞɥɹ , ɢ
z
x
p
ˆ
z
p
ˆ
H
ˆ
:
)(),,(
//
yeCzyx
zpixpi
zx
) \
==
. (4)
Ɂɞɟɫɶ ) – ɧɟɢɡɜɟɫɬɧɚɹ ɮɭɧɤɰɢɹ.( y)
ɉɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ (4) ɜ ɭɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ
\H \H
ˆ
(5)
ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (3) ɨɩɟɪɚɬɨɪɧɵɟ ɱɥɟɧɵ ɜ
H
ˆ
, ɫɨɞɟɪɠɚɳɢɟ ɢ ,
ɡɚɦɟɧɹɸɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɱɢɫɥɨɜɵɦɢ ɜɟɥɢɱɢɧɚɦɢ. ɋ ɩɨɦɨɳɶɸ ɬɚɤɨɣ
ɡɚɦɟɧɵ ɢ ɩɪɢɦɟɧɟɧɢɹ ɨɛɨɡɧɚɱɟɧɢɣ
x
p
ˆ
z
p
ˆ
mc
qB
Z
0
, (6)
qB
pc
y
x
0
, (7)
m
p
E
z
2
2
H (8)
ɭɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ (5) ɩɪɢɜɨɞɢɬɫɹ ɤ ɭɪɚɜɧɟɧɢɸ ɞɥɹ ɥɢɧɟɣɧɨɝɨ
ɨɫɰɢɥɥɹɬɨɪɚ, ɫɨɜɟɪɲɚɸɳɟɝɨ ɤɨɥɟɛɚɧɢɹ ɜɞɨɥɶ ɨɫɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɶɧɨɣ
ɬɨɱɤɢ :
y
0
yy
)()()(
2
1
2
ˆ
2
0
2
0
2
yEyyym
m
p
y
) )
¸
¸
¹
·
¨
¨
©
§
Z . (9)
33
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɨɩɭɫɬɢɦɵɟ ɡɧɚɱɟɧɢɹ ɷɧɟɪɝɢɢ (8) ɢ ɨɬɜɟɱɚɸɳɢɟ ɢɦ ɮɭɧɤɰɢɢ
ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:)( y)
)2/1(
0
Z nE
n
= , ...,2,1,0 n , (10)
)()(
0
yyy
ɨɫɰ
nn
\ ) . (11)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɬɟ ɪɟɲɟɧɢɹ ɧɚɲɟɣ ɢɫɯɨɞɧɨɣ ɡɚɞɚɱɢ, ɤɨɬɨɪɵɟ ɞɨɩɭɫɤɚɸɬ
ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɜ ɮɨɪɦɟ (4), ɧɚɣɞɟɧɵ:
)/(),,(
//
,,
qBcpyeCzyx
x
ɨɫɰ
n
zpixpi
npp
zx
zx
\ \
==
. (12)
Ʉɜɚɧɬɨɜɵɦɢ ɱɢɫɥɚɦɢ ɧɚɣɞɟɧɧɵɯ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ ɫɥɭɠɚɬ
ɜɟɥɢɱɢɧɵ , ɫ ɧɟɩɪɟɪɵɜɧɵɦ ɫɩɟɤɬɪɨɦ ɡɧɚɱɟɧɢɣ ɢ ɱɢɫɥɨ , ɩɪɢɧɢɦɚɸɳɟɟ
ɬɨɥɶɤɨ ɞɢɫɤɪɟɬɧɵɟ ɡɧɚɱɟɧɢɹ 0, 1, 2, … . ɗɧɟɪɝɢɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ,
x
p
z
pn
)2/1(
2
0
2
,
Z H n
m
p
z
np
z
= , (13)
ɢɦɟɟɬ ɜɢɞ ɫɭɦɦɵ ɷɧɟɪɝɢɢ ɫɜɨɛɨɞɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜɞɨɥɶ ɨɫɢ ɫ
ɢɦɩɭɥɶɫɨɦ ɢ ɞɢɫɤɪɟɬɧɵɯ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ (10), ɧɚɡɵɜɚɟɦɵɯ
ɭɪɨɜɧɹɦɢ Ʌɚɧɞɚɭ.
z
z
p
ɍɪɨɜɧɢ (13) ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɜɟɥɢɱɢɧɵ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɟɸɬ
ɛɟɫɤɨɧɟɱɧɭɸ ɤɪɚɬɧɨɫɬɶ ɜɵɪɨɠɞɟɧɢɹ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɪɟɲɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ
ɒɪɟɞɢɧɝɟɪɚ, ɩɪɢɧɚɞɥɟɠɚɳɢɦ ɨɩɪɟɞɟɥɟɧɧɨɣ ɷɧɟɪɝɢɢ (13), ɹɜɥɹɟɬɫɹ ɧɟ ɬɨɥɶɤɨ
ɮɭɧɤɰɢɹ (12), ɧɨ ɢ ɥɸɛɚɹ ɥɢɧɟɣɧɚɹ ɤɨɦɛɢɧɚɰɢɹ ɬɚɤɢɯ ɮɭɧɤɰɢɣ ɫɨ
ɜɫɟɜɨɡɦɨɠɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ . ɉɨɫɤɨɥɶɤɭ ɫɩɟɤɬɪ ɧɟɩɪɟɪɵɜɟɧ, ɭɤɚɡɚɧɧɚɹ
ɥɢɧɟɣɧɚɹ ɤɨɦɛɢɧɚɰɢɹ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɡɚɩɢɲɟɬɫɹ ɜ ɜɢɞɟ ɢɧɬɟɝɪɚɥɚ ɩɨ
x
p
x
p
x
p
x
p
)/()()(
/
/
,
qBcpyepCdpe
x
ɨɫɰ
n
xpi
xx
zpi
np
x
z
z
\ \
³
f
f
=
=
r , (14)
ɝɞɟ ) – ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤɰɢɹ.(
x
pC
Ɉɬɦɟɬɢɦ, ɱɬɨ ɜɟɤɬɨɪɧɵɦ ɩɨɬɟɧɰɢɚɥɨɦ (1) ɨɩɢɫɵɜɚɟɬɫɹ ɦɚɝɧɢɬɧɨɟ ɩɨɥɟ
ɫ ɤɨɦɩɨɧɟɧɬɚɦɢ
AB rot
0
yx
BB , BB
z
. (15)
ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɞɜɢɠɟɧɢɟ ɡɚɪɹɠɟɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɨɞɧɨɪɨɞɧɨɦ
ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ, ɩɚɪɚɥɥɟɥɶɧɨɦ ɨɫɢ , ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɧɚɥɨɠɟɧɢɹ
ɪɚɜɧɨɦɟɪɧɨɝɨ ɩɪɹɦɨɥɢɧɟɣɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫ ɢɦɩɭɥɶɫɨɦ ɜɞɨɥɶ ɨɫɢ ɢ
ɜɪɚɳɟɧɢɹ ɫ ɱɚɫɬɨɬɨɣ ɜ ɩɥɨɫɤɨɫɬɢ
z
z
p z
mcqB /
0
Z y
x
, ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɚ
34
qBcp /
A
U , ɝɞɟ
22
yx
ppp
A
– ɩɨɩɟɪɟɱɧɚɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɩɪɚɜɥɟɧɢɹ
ɫɨɫɬɚɜɥɹɸɳɚɹ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ;
B
U ɧɚɡɵɜɚɸɬ ɪɚɞɢɭɫɨɦ ɥɚɪɦɨɪɨɜɫɤɨɣ ɨɪɛɢɬɵ.
Ʉɚɤ ɜɢɞɧɨ ɢɡ (13), ɜ ɤɜɚɧɬɨɜɨɣ ɦɟɯɚɧɢɤɟ ɤɥɚɫɫɢɱɟɫɤɨɦɭ ɜɪɚɳɟɧɢɸ ɱɚɫɬɢɰɵ ɩɨ
ɥɚɪɦɨɪɨɜɫɤɢɦ ɨɪɛɢɬɚɦ ɦɨɠɧɨ ɫɨɩɨɫɬɚɜɢɬɶ ɭɪɨɜɧɢ Ʌɚɧɞɚɭ – ɧɚɛɨɪ ɞɢɫɤɪɟɬɧɵɯ
ɡɧɚɱɟɧɢɣ ɷɧɟɪɝɢɢ, ɯɚɪɚɤɬɟɪɧɵɣ ɞɥɹ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ ɫ ɫɨɛɫɬɜɟɧɧɨɣ
ɱɚɫɬɨɬɨɣ .mcqB /
0
Z
6.6. ɉɪɟɠɞɟ ɜɫɟɝɨ ɜɵɩɢɲɟɦ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
ɞɥɹ ɱɚɫɬɢɰɵ ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɜɟɤɬɨɪɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ
BxA
y
~
, . (1) 0
~~
zx
AA
ɗɬɨ ɥɟɝɤɨ ɫɞɟɥɚɬɶ, ɡɚɦɟɬɢɜ, ɱɬɨ (1) ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɤɨɦɩɨɧɟɧɬɵ ɩɨɥɹ
ByA
x
,0
zy
AA (2)
ɜ ɧɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɩɨɥɭɱɚɸɳɟɣɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɜɨɪɨɬɚ ɢɫɯɨɞɧɵɯ
ɨɫɟɣ ɧɚ ɭɝɨɥ 2
/
S ɜɨɤɪɭɝ ɨɫɢ . ɇɟ ɪɟɲɚɹ ɡɚɧɨɜɨ ɭɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ,
ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɢɡɜɟɫɬɢ ɡɚɦɟɧɭ
z
y
x
o ,
x
y o (ɚ ɬɚɤɠɟ ) ɜ
ɜɵɪɚɠɟɧɢɢ (12) ɡɚɞɚɱɢ 6.5. Ɉɩɭɫɬɢɜ ɞɥɹ ɭɩɪɨɳɟɧɢɹ ɡɚɩɢɫɢ ɧɨɪɦɢɪɨɜɨɱɧɵɣ
ɦɧɨɠɢɬɟɥɶ
yx
pp o
C , ɩɨɥɭɱɚɟɦ:
)/(),,(
~
/
/
,,
qBcpxeezyx
y
ɨɫɰ
n
ypi
zpi
npp
y
z
zy
\ \
=
=
, (3)
Ɍɟɩɟɪɶ ɛɭɞɟɦ ɩɪɢɦɟɧɹɬɶ ɮɨɪɦɭɥɵ ɤɚɥɢɛɪɨɜɨɱɧɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ,
ɤɨɬɨɪɵɟ, ɤɚɤ ɢɡɜɟɫɬɧɨ, ɢɦɟɸɬ ɜɢɞ
f
AA
~
,
t
f
c w
w
M M
1
~
, (4)
\ \
~
/ cqfi
e
=
. (5)
ɋɪɚɜɧɢɜ (1) ɢ (2), ɧɚɯɨɞɢɦ
f
:
yB
x
f
w
w
,
xB
y
f
w
w
, 0
w
w
z
f
. (6)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤɚɥɢɛɪɨɜɨɱɧɭɸ ɮɭɧɤɰɢɸ
f
ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɜ ɮɨɪɦɟ:
x
y
B
f
)(r , (7)
ɢ ɩɪɢ ɷɬɨɦ, ɫɨɝɥɚɫɧɨ (5), ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɜɢɞɚ
35
)/()(
/)(
,,
qBcpxe
y
ɨɫɰ
n
ypzpiyx
c
qB
i
npp
yz
zy
\ \
=
=
r (8)
ɞɨɥɠɧɚ ɨɤɚɡɚɬɶɫɹ ɨɞɧɢɦ ɢɡ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ ɫ ɜɟɤɬɨɪɧɵɦ
ɩɨɬɟɧɰɢɚɥɨɦ (2).
Ⱦɨɩɭɫɬɢɦɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɪɟɲɟɧɢɣ ɭɤɚɡɚɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ
ɩɪɢɜɟɞɟɧɵ ɜ ɮɨɪɦɭɥɚɯ (12) ɢ (14) ɡɚɞɚɱɢ 6.5. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚɦ ɧɟɨɛɯɨɞɢɦɨ
ɭɛɟɞɢɬɶɫɹ ɜ ɬɨɦ, ɱɬɨ ɩɨɫɥɟɞɧɹɹ ɢɡ ɭɩɨɦɹɧɭɬɵɯ ɮɨɪɦɭɥ ɜɤɥɸɱɚɟɬ ɜ ɤɚɱɟɫɬɜɟ
ɱɚɫɬɧɨɝɨ ɫɥɭɱɚɹ ɢ ɧɚɣɞɟɧɧɭɸ ɡɞɟɫɶ ɮɭɧɤɰɢɸ (8).
ɉɨɥɶɡɭɹɫɶ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ )
ɜ ɜɢɞɟ ɪɚɡɥɨɠɟɧɢɹ ɜ ɢɧɬɟɝɪɚɥ Ɏɭɪɶɟ
ɩɨ ɩɥɨɫɤɢɦ ɜɨɥɧɚɦ,
(x
ɨɫɰ
n
\
³
f
f
S
\
=
=
/
)(
2
)(
xpi
xn
x
ɨɫɰ
n
x
epa
dp
x , (9)
ɢ ɭɱɢɬɵɜɚɹ ɜ (8) ɜɵɬɟɤɚɸɳɟɟ ɢɡ (9) ɫɨɨɬɧɨɲɟɧɢɟ
³
f
f
S
\
=
=
=
qB
ppc
ixpi
xn
x
y
ɨɫɰ
n
yx
x
epa
dp
qBcpx
/
)(
2
)/( , (10)
ɢɦɟɟɦ:
z
p
i
npp
z
zy
e
=
\ )(
,,
r
³
f
f
S
==
=
qB
ppc
i
ypxp
i
xn
x
yxyx
epa
dp
)(
2
yx
c
qB
i
=
. (11)
Ɂɚɦɟɧɚ (ɫɞɜɢɝ) ɩɟɪɟɦɟɧɧɨɣ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ cqBypp
xx
/o ɩɪɢɜɨɞɢɬ ɤ
ɜɡɚɢɦɧɨɦɭ ɫɨɤɪɚɳɟɧɢɸ ɧɟɤɨɬɨɪɵɯ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɯ ɦɧɨɠɢɬɟɥɟɣ ɩɨɞ ɡɧɚɤɨɦ
ɢɧɬɟɝɪɚɥɚ ɜ (11), ɢ ɦɵ ɩɨɥɭɱɚɟɦ:
=/
,,
)(
zpi
npp
z
zy
e \ r
³
f
f
S
=
=
=
qB
ppc
ixpi
xn
x
yx
x
ecqBypa
dp
/
)/(
2
. (12)
ɉɨ ɯɨɞɭ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ 6.4 ɛɵɥɨ ɩɨɤɚɡɚɧɨ, ɱɬɨ ɮɭɧɤɰɢɢ ɢ
ɚɧɚɥɨɝɢɱɧɵ ɞɪɭɝ ɞɪɭɝɭ. Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɜ ɫɢɥɭ ɫɨɨɬɧɨɲɟɧɢɹ (8) ɡɚɞɚɱɢ 6.4
ɫɩɪɚɜɟɞɥɢɜɨ ɪɚɜɟɧɫɬɜɨ
))(qa
n
([\
ɨɫɰ
n
¸
¹
·
¨
©
§
S
S
b
yqBcp
He
n
b
bicqBypa
x
n
by
qB
cp
n
n
xn
x
/
!2
)(
2)()/(
22
2/)(
4/12
)/(2)( qBcpybi
x
ɨɫɰ
n
n
\S , (13)
ɝɞɟ
36
0
/ Z mb = , mcqB /
0
Z .
ȼɜɟɞɟɦ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɡɚɜɢɫɹɳɭɸ ɨɬ ɢ ɜɟɥɢɱɢɧɭ :
x
p
y
p ),(
yx
ppC
=
=
qBppci
n
yx
yx
e
b
ippC
/
2
)(),(
S
. (14)
ɍɱɢɬɵɜɚɹ ɜ (12) ɪɚɜɟɧɫɬɜɚ (13) ɢ (14), ɦɵ ɧɚɯɨɞɢɦ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɚɹ ɡɞɟɫɶ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ (8) ɢ ɜ ɫɚɦɨɦ ɞɟɥɟ ɞɨɩɭɫɤɚɟɬ
ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɜ ɜɢɞɟ
)/(),()(
/
/
,,
qBcpyeppCdpe
x
ɨɫɰ
n
xpi
yxx
zpi
npp
x
z
zy
\ \
³
f
f
=
=
r , (15)
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɮɨɪɦɭɥɟ (14) ɡɚɞɚɱɢ 6.5. ɗɬɚ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɩɢɫɵɜɚɟɬ ɨɞɧɨ ɢɡ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ ɩɪɢɧɚɞɥɟɠɚɳɢɯ
ɜɵɪɨɠɞɟɧɧɨɦɭ ɫ ɛɟɫɤɨɧɟɱɧɨɣ ɤɪɚɬɧɨɫɬɶɸ ɭɪɨɜɧɸ
¸
¹
·
¨
©
§
H
2
1
2
2
,
n
cm
qB
m
p
z
np
z
=
. (16)
Ⱦɥɹ ɛɨɥɟɟ ɩɨɥɧɨɝɨ ɚɧɚɥɢɡɚ ɫɥɟɞɭɟɬ ɧɚɱɢɧɚɬɶ ɧɟ ɫ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ (3), ɚ
ɫ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ )(
~
,
r
np
z
\ , ɩɨɥɭɱɚɟɦɨɝɨ ɭɦɧɨɠɟɧɢɟɦ (3) ɧɚ ɩɪɨɢɡɜɨɥɶɧɵɣ
ɤɨɷɮɮɢɰɢɟɧɬ )ɫ ɩɨɫɥɟɞɭɸɳɢɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɨ :(
y
pN
y
p
)/()()(
~
/
/
,
qBcpxepNdpe
y
ɨɫɰ
n
ypi
yy
zpi
np
y
z
z
\ \
³
f
f
=
=
r . (17)
ɉɪɨɢɡɜɟɞɹ ɡɚɬɟɦ ɤɚɥɢɛɪɨɜɨɱɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ (5) ɢ ɜɵɩɨɥɧɢɜ ɩɪɢɦɟɧɢɬɟɥɶɧɨ
ɤ ɩɨɫɬɪɨɟɧɧɨɣ ɬɚɤ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ )(
,
r
np
z
\ ɬɟ ɠɟ ɞɟɣɫɬɜɢɹ, ɱɬɨ ɢ ɜɵɲɟ (ɧɚ
ɷɬɨɬ ɪɚɡ – ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɩɨ ), ɦɵ ɫɧɨɜɚ ɩɪɢɞɟɦ ɤ ɮɨɪɦɭɥɟ (14)
ɡɚɞɚɱɢ 6.5. ȼ ɬɚɤɨɣ ɮɨɪɦɭɥɟ ɤɨɷɮɮɢɰɢɟɧɬ )ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ ɮɭɧɤɰɢɢ
, ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɧɧɨɣ ɩɨ .
y
p
(
x
pC
),(
yx
ppC )(
y
pN
y
p
Ɉɛɫɭɞɢɦ ɨɫɧɨɜɧɵɟ ɱɟɪɬɵ ɩɥɨɬɧɨɫɬɢ ɜɟɪɨɹɬɧɨɫɬɢ
2
)(r\
ɜ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɹɯ ɱɚɫɬɢɰɵ. ɂɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɟɬ
ɡɚɜɢɫɢɦɨɫɬɶ ɷɬɨɣ ɜɟɥɢɱɢɧɵ ɨɬ ɤɨɨɪɞɢɧɚɬ
x
ɢ , ɨɫɢ ɤɨɬɨɪɵɯ
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɧɚɩɪɚɜɥɟɧɢɸ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ .
y
B
ɋɨɫɬɨɹɧɢɸ ) (ɮɨɪɦɭɥɚ (12) ɡɚɞɚɱɢ 6.5) ɨɬɜɟɱɚɟɬ ɨɞɧɨɪɨɞɧɨɟ
ɪɚɫɩɪɟɞɟɥɟɧɢɟ «ɷɥɟɤɬɪɨɧɧɨɣ ɩɥɨɬɧɨɫɬɢ» ɜɞɨɥɶ ɨɫɢ
(
,,
r
npp
zx
\
x
ɢ ɯɚɪɚɤɬɟɪɧɨɟ ɞɥɹ
37
ɥɢɧɟɣɧɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɢɞɚ
2
0
)( yy
ɨɫɰ
n
\ ɜɞɨɥɶ . ɉɪɢ
ɜɟɪɨɹɬɧɨɫɬɶ ɨɛɧɚɪɭɠɟɧɢɹ ɱɚɫɬɢɰɵ ɪɟɡɤɨ ɭɛɵɜɚɟɬ; ɷɬɨ ɭɛɵɜɚɧɢɟ
ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɦ ɦɚɫɲɬɚɛɨɦ
y
rfoy
qBcmb //
0
== Z . (18)
ȼɟɥɢɱɢɧɭ
qBc /= ɧɚɡɵɜɚɸɬ ɦɚɝɧɢɬɧɨɣ ɞɥɢɧɨɣ.
ȿɫɥɢ ɛɵ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ )(
,,
r
npp
zx
\ ɹɜɥɹɥɚɫɶ ɟɞɢɧɫɬɜɟɧɧɵɦ
ɪɟɲɟɧɢɟɦ, ɬɨ ɩɨɞɨɛɧɚɹ ɜɵɞɟɥɟɧɧɨɫɬɶ ɨɞɧɨɣ ɢɡ ɞɜɭɯ ɤɨɨɪɞɢɧɚɬ
x
ɢ , ɤɨɧɟɱɧɨ,
ɜɵɡɵɜɚɥɚ ɛɵ ɭɞɢɜɥɟɧɢɟ. ȼ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɤɚɠɞɨɦɭ ɭɪɨɜɧɸ ɷɧɟɪɝɢɢ (16)
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɜɨɥɧɨɜɵɯ ɮɭɧɤɰɢɣ (ɮɨɪɦɭɥɚ (14) ɡɚɞɚɱɢ
6.5), ɫɪɟɞɢ ɤɨɬɨɪɵɯ, ɤɚɤ ɦɵ ɭɠɟ ɭɛɟɞɢɥɢɫɶ, ɢɦɟɟɬɫɹ ɢ ɪɟɲɟɧɢɟ (8). ɗɬɨ
ɪɟɲɟɧɢɟ ɜɨɫɫɬɚɧɚɜɥɢɜɚɟɬ «ɪɚɜɧɨɩɪɚɜɢɟ» ɨɫɟɣ
y
x
ɢ , ɩɨɫɤɨɥɶɤɭ ɨɧɨ ɞɚɟɬ
ɨɞɧɨɪɨɞɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɢ ɨɛɧɚɪɭɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜɞɨɥɶ ɢ
ɯɚɪɚɤɬɟɪɧɨɟ ɞɥɹ ɨɫɰɢɥɥɹɬɨɪɚ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɢɞɚ
y
y
2
0
)( xx
ɨɫɰ
n
\ ɜɞɨɥɶ
x
.
ɍɛɵɜɚɧɢɟ ɷɬɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɜɞɨɥɶ
x
ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɦɚɝɧɢɬɧɨɣ ɞɥɢɧɨɣ (18).
Ɍɚɤɚɹ ɠɟ ɤɚɪɬɢɧɚ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɟɲɟɧɢɸ (3) ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ ɫ
ɜɟɤɬɨɪɧɵɦ ɩɨɬɟɧɰɢɚɥɨɦ (1).
6.7. ɚ) ɉɨɞɫɬɚɜɢɜ ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ aaaaaa
ˆˆˆˆ
]
ˆ
,
ˆ
[
{ ɭɤɚɡɚɧɧɵɟ
ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɨɩɟɪɚɬɨɪɧɵɟ ɜɵɪɚɠɟɧɢɹ
¸
¸
¹
·
¨
¨
©
§
Z
Z
0
0
ˆ
ˆ
2
ˆ
m
pi
x
m
a
=
,
¸
¸
¹
·
¨
¨
©
§
Z
Z
0
0
ˆ
ˆ
2
ˆ
m
pi
x
m
a
=
, (1)
ɢɦɟɟɦ:
]
ˆ
,
ˆ
[)
ˆˆˆˆ
(]
ˆ
,
ˆ
[ px
i
xppx
i
aa
==
.
ɋ ɭɱɟɬɨɦ ɢɡɜɟɫɬɧɨɝɨ ɤɨɦɦɭɬɚɰɢɨɧɧɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ ɨɩɟɪɚɬɨɪɨɜ
ɤɨɨɪɞɢɧɚɬɵ ɢ ɢɦɩɭɥɶɫɚ,
=ipx ]
ˆ
,
ˆ
[ , (2)
ɧɚɯɨɞɢɦ ɤɨɦɦɭɬɚɬɨɪ ɨɩɟɪɚɬɨɪɨɜ ɢa
ˆ
a
ˆ
:
1]
ˆ
,
ˆ
[
aa . (3)
ɛ) Ɉɩɟɪɚɬɨɪ aa
ˆˆ
ɷɪɦɢɬɨɜ; ɨɧ ɞɨɥɠɟɧ ɢɦɟɬɶ ɜɟɳɟɫɬɜɟɧɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ
ɡɧɚɱɟɧɢɹ (ɦɵ ɢɯ ɜɪɟɦɟɧɧɨ ɨɛɨɡɧɚɱɢɦ ɤɚɤ O) ɢ ɜɡɚɢɦɧɨ ɨɪɬɨɝɨɧɚɥɶɧɵɟ
38
ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ O . ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɞɥɹ ɤɚɤɨɝɨ-ɥɢɛɨ ɫɨɛɫɬɜɟɧɧɨɝɨ
ɡɧɚɱɟɧɢɹ O ɫɭɳɟɫɬɜɭɟɬ ɧɨɪɦɢɪɨɜɚɧɧɵɣ ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ
O :
aa
ˆˆ
O OO , (4)
1 OO . (5)
Ɋɚɫɫɦɨɬɪɢɦ ɪɟɡɭɥɶɬɚɬ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɚ aa
ˆˆ
ɧɚ ɜɟɤɬɨɪ Oa
ˆ
, ɭɱɢɬɵɜɚɹ
ɪɚɜɟɧɫɬɜɚ (3) ɢ (4):
aa
ˆˆ
Oa
ˆ
)1
ˆˆ
(
aa Oa
ˆ
O
)
ˆˆ
(
ˆ
aaa
Oa
ˆ
O
O a
ˆ
)1( . (6)
ȼɢɞɧɨ, ɱɬɨ
Oa
ˆ
– ɬɚɤɠɟ ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ ɨɩɟɪɚɬɨɪɚ (ɟɫɥɢ ɬɨɥɶɤɨaa
ˆˆ
0
ˆ
zOa ), ɩɪɢɱɟɦ ɨɬɜɟɱɚɸɳɟɟ ɷɬɨɦɭ ɜɟɤɬɨɪɭ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɧɚ ɟɞɢɧɢɰɭ
ɦɟɧɶɲɟ ɩɟɪɜɨɧɚɱɚɥɶɧɨɝɨ:.Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, – «ɩɨɧɢɠɚɸɳɢɣ»
ɨɩɟɪɚɬɨɪ.
1O a
ˆ
ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɢɜ ɨɛɟ ɫɬɨɪɨɧɵ ɪɚɜɟɧɫɬɜɚ (4) ɧɚ
O ɢ ɩɪɢɧɹɜ ɜɨ
ɜɧɢɦɚɧɢɟ (5), ɩɨɥɭɱɢɦ:
O OO
aa
ˆˆ
,
ɢɥɢ
O OO aa
ˆˆ
. (7)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɫɥɢ
0
ˆ
zOa , ɬɨ 0!O (ɩɨɫɤɨɥɶɤɭ ɥɟɜɚɹ ɫɬɨɪɨɧɚ ɜ (7)
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɤɜɚɞɪɚɬ ɧɨɪɦɵ ɜɟɤɬɨɪɚ), ɢ ɜɟɤɬɨɪ
Oa
ˆ
ɦɨɠɧɨ ɧɨɪɦɢɪɨɜɚɬɶ
ɧɚ ɟɞɢɧɢɰɭ: ɧɨɪɦɢɪɨɜɚɧɧɵɣ ɫɨɛɫɬɜɟɧɧɵɣ ɜɟɤɬɨɪ, ɩɪɢɧɚɞɥɟɠɚɳɢɣ
ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ 1, ɟɫɬɶO
O
O
O a
ˆ
1
1
. (8)
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɞɟɣɫɬɜɭɹ ɪɚɡ ɨɩɟɪɚɬɨɪɨɦ ɧɚ ɨɩɪɟɞɟɥɹɟɦɵɟ
ɮɨɪɦɭɥɨɣ (8) ɧɨɪɦɢɪɨɜɚɧɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ, ɩɨɥɭɱɚɟɦ ɪɹɞ ɫɨɛɫɬɜɟɧɧɵɯ
ɜɟɤɬɨɪɨɜ, ɨɬɜɟɱɚɸɳɢɣ ɭɛɵɜɚɸɳɟɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
. Ɉɞɧɚɤɨ ɞɚɧɧɵɣ ɩɪɨɰɟɫɫ ɩɨɫɬɪɨɟɧɢɹ ɫɨɛɫɬɜɟɧɧɵɯ ɜɟɤɬɨɪɨɜ
ɡɚɜɟɞɨɦɨ ɧɟ ɦɨɠɟɬ ɩɪɨɞɨɥɠɚɬɶɫɹ ɞɨ ɛɟɫɤɨɧɟɱɧɨɫɬɢ. ȼ ɫɚɦɨɦ ɞɟɥɟ, ɧɚ ɤɚɠɞɨɦ
ɲɚɝɟ ɦɵ ɛɭɞɟɦ ɢɦɟɬɶ ɪɚɜɟɧɫɬɜɨ, ɚɧɚɥɨɝɢɱɧɨɟ (7),
ma
ˆ
mOOO ,...,1,
mmama O OO )(
ˆ
)(
ˆ
, (9)
ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ
0
ˆ
zO ma ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɧɟɪɚɜɟɧɫɬɜɨ ,0!O m
39