Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
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Ɉɬɦɟɬɢɦ, ɱɬɨ ɫɩɢɧɨɜɵɣ ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ ɷɥɟɤɬɪɨɧɚ ()ɟɫɬɶ
ɜɟɥɢɱɢɧɚ ɨɬɪɢɰɚɬɟɥɶɧɚɹ. ɉɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɫɩɢɧɨɜɵɣ ɦɚɝɧɢɬɧɵɣ
ɦɨɦɟɧɬ ɷɥɟɤɬɪɨɧɚ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɪɚɜɟɧ (ɫ ɬɨɱɧɨɫɬɶɸ 1)ɦɚɝɧɟɬɨɧɭ Ȼɨɪɚ:
e
P{P
0
.0%
mce
e
2/)00116.1( =|P .
8.6. ɋɩɢɧɨɜɵɟ ɫɨɫɬɨɹɧɢɹ
21
VV ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɜɟɤɬɨɪɚɦɢ
ɞɥɹ ɨɩɟɪɚɬɨɪɚ ɩɪɨɟɤɰɢɢ ɫɭɦɦɚɪɧɨɝɨ ɫɩɢɧɚ ɫ ɫɨɛɫɬɜɟɧɧɵɦɢ
ɡɧɚɱɟɧɢɹɦɢ
21
ˆˆ
ˆ
zzz
ssS
21
VV
z
S :
)
ˆˆ
(
21 zz
ss )(
21
VV VV
211
)
ˆ
(
z
s VV
221
ˆ
z
s
VVV
211
VVV
221
)(
21
VV )(
21
VV . (1)
Ɉɞɧɚɤɨ ɞɥɹ ɨɩɟɪɚɬɨɪɚ ɤɜɚɞɪɚɬɚ ɫɭɦɦɚɪɧɨɝɨ ɫɩɢɧɚ,
21
2
2
2
1
2
21
2
ˆˆ
2
ˆˆ
)
ˆˆ
(
ˆ
ssssssS , (2)
ɧɟ ɜɫɟ ɢɡ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɛɚɡɢɫɧɵɯ ɫɨɫɬɨɹɧɢɣ ɛɭɞɭɬ ɫɨɛɫɬɜɟɧɧɵɦɢ
ɜɟɤɬɨɪɚɦɢ, ɯɨɬɹ ɞɥɹ ɜɫɟ ɨɧɢ ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɜɟɤɬɨɪɚɦɢ ɫ
ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ
2
2
2
1
ˆˆ
ss
2/3)1()1(
2211
ssss .
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɧɚɣɬɢ ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ ɨɩɟɪɚɬɨɪɚ (2), ɩɨɫɬɪɨɢɦ ɢ
ɞɢɚɝɨɧɚɥɢɡɭɟɦ ɦɚɬɪɢɰɭ
21
ˆˆ
ıı
21
ˆˆ
4
ss .
ɉɪɨɧɭɦɟɪɭɟɦ ɛɚɡɢɫɧɵɟ ɫɨɫɬɨɹɧɢɹ, ɧɚɩɪɢɦɟɪ, ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
21
1 pn ,
21
2 np ,
21
3 nn ,
21
4 pp . (3)
ɉɨɥɶɡɭɹɫɶ ɢɡɜɟɫɬɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɷɥɟɦɟɧɬɨɜ ɦɚɬɪɢɰ ɉɚɭɥɢ, ɢɦɟɟɦ:
2
ˆˆ
1
ˆˆ
2121
21
np pVnV VV
xxxx
,
2
ˆˆ
1
ˆˆ
2121
21
np pVnV VV ii
yyyy
,
1)1(
ˆˆ
1
ˆˆ
2121
21
pn pVnV VV
zzzz
.
ɋɭɦɦɢɪɭɹ ɷɬɢ ɪɚɜɟɧɫɬɜɚ ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ
VV
21
ˆˆ
xx
VV
21
ˆˆ
yy
VV
21
ˆˆ
zz 21
ˆˆ
ıı ,
ɩɨɥɭɱɚɟɦ
90
2
ˆ
1
ˆ
ıı 1221 . (4)
Ⱥɧɚɥɨɝɢɱɧɨ ɜɵɱɢɫɥɹɟɬɫɹ ɪɟɡɭɥɶɬɚɬ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɚ ɧɚ ɨɫɬɚɥɶɧɵɟ
ɬɪɢ ɫɨɫɬɨɹɧɢɹ (3):
2
ˆ
1
ˆ
ıı
2
ˆ
1
ˆ
ıı 2122 , (5)
2
ˆ
1
ˆ
ıı 33 , (6)
2
ˆ
1
ˆ
ıı 44 . (7)
ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɢɜ (4)–(7) ɧɚ ɤɚɠɞɨɟ ɢɡ ɛɚɡɢɫɧɵɯ ɫɨɫɬɨɹɧɢɣ (3), ɩɨɥɭɱɚɟɦ
ɢɫɤɨɦɭɸ ɦɚɬɪɢɰɭ:
2
ˆ
1
ˆ
ıı
¸
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
¨
©
§
1000
0100
0012
0021
. (8)
«Ȼɥɨɱɧɨ-ɞɢɚɝɨɧɚɥɶɧɵɣ» ɜɢɞ ɦɚɬɪɢɰɵ (8) ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɨɦɭ
ɨɛɫɬɨɹɬɟɥɶɫɬɜɭ, ɱɬɨ ɫɨɫɬɨɹɧɢɹ
3 ɢ 4,ɤɚɤ ɜɢɞɧɨ ɢɡ (6) ɢ (7), ɭɠɟ ɹɜɥɹɸɬɫɹ
ɫɨɛɫɬɜɟɧɧɵɦɢ ɜɟɤɬɨɪɚɦɢ ɨɩɟɪɚɬɨɪɚ (8) ɫ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ 1.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɫɬɚɥɶɧɵɯ ɞɜɭɯ ɫɨɛɫɬɜɟɧɧɵɯ ɜɟɤɬɨɪɨɜ
ɞɨɫɬɚɬɨɱɧɨ ɞɢɚɝɨɧɚɥɢɡɨɜɚɬɶ ɬɨɥɶɤɨ ɜɟɪɯɧɢɣ ɛɥɨɤ ɜ (8), ɨɬɧɨɫɹɳɢɣɫɹ ɤ
ɞɜɭɦɟɪɧɨɦɭ ɩɨɞɩɪɨɫɬɪɚɧɫɬɜɭ ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɣ ɫ ɛɚɡɢɫɨɦ
1, 2. ɂɡ
ɭɪɚɜɧɟɧɢɹ
0
12
21
det
¸
¸
¹
·
¨
¨
©
§
O
O
(9)
ɧɚɯɨɞɢɦ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ:1 O ɢ 3 O . ɂɦ ɨɬɜɟɱɚɸɬ ɧɨɪɦɢɪɨɜɚɧɧɵɟ
ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ
21
2
1
,
21
2
1
. (10)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɫɢɫɬɟɦɵ ɢɡ ɞɜɭɯ ɱɚɫɬɢɰ ɫɨ ɫɩɢɧɨɦ 2
/
1 ɨɩɟɪɚɬɨɪ (2)
ɢɦɟɟɬ ɬɪɟɯɤɪɚɬɧɨ ɜɵɪɨɠɞɟɧɧɨɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ 2)1(
S
S
ɢ
ɧɟɜɵɪɨɠɞɟɧɧɨɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ 0)1(
S
S
. ɗɬɢɦ ɫɨɛɫɬɜɟɧɧɵɦ
ɡɧɚɱɟɧɢɹɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɬɪɢ ɫɨɫɬɨɹɧɢɹ ɫ ɫɭɦɦɚɪɧɵɦ ɫɩɢɧɨɦ 1
S
ɢ
ɫɢɧɝɥɟɬɧɨɟ ɫɨɫɬɨɹɧɢɟ ɫ ɫɭɦɦɚɪɧɵɦ ɫɩɢɧɨɦ 0
S
:
91
°
°
°
°
¯
°
°
°
°
®
pp
nppn
nn
21
2121
21
1,1
2
1
2
1
0,1
1,1
z
z
z
SS
SS
SS
. (11)
2121
2
1
2
1
0,0 nppn
z
SS . (12)
Ɂɧɚɱɢɬɟɥɶɧɨ ɛɨɥɟɟ ɛɵɫɬɪɵɟ ɫɩɨɫɨɛɵ ɪɟɲɟɧɢɹ ɩɨɞɨɛɧɵɯ ɡɚɞɚɱ
ɨɫɧɨɜɵɜɚɸɬɫɹ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɡɚɤɨɧɚ ɫɥɨɠɟɧɢɹ ɦɨɦɟɧɬɨɜ. Ⱦɥɹ ɞɜɭɯ
ɩɪɨɢɡɜɨɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɫɩɢɧɚ ɢ ɡɚɤɨɧ ɫɥɨɠɟɧɢɹ ɦɨɦɟɧɬɨɜ ɝɥɚɫɢɬ, ɱɬɨ
ɫɭɦɦɚɪɧɵɣ ɫɩɢɧ
1
s
2
s
S
ɞɨɥɠɟɧ ɫ ɲɚɝɨɦ, ɪɚɜɧɵɦ 1 , ɩɪɨɛɟɝɚɬɶ ɜɫɟ ɡɧɚɱɟɧɢɹ ɨɬ
ɞɨ
21
ss
21
ss . ɉɪɢɦɟɧɢɬɟɥɶɧɨ ɤ 2/1
21
ss ɦɵ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
ɧɟɦɟɞɥɟɧɧɨ ɧɚɯɨɞɢɦ ɞɨɩɭɫɬɢɦɵɟ ɡɧɚɱɟɧɢɹ ɫɭɦɦɚɪɧɨɝɨ ɫɩɢɧɚ:0,1
S
. ɋɩɟɤɬɪ
ɞɨɩɭɫɬɢɦɵɯ ɡɧɚɱɟɧɢɣ ɜ ɤɚɠɞɨɦ ɢɡ ɞɜɭɯ ɦɭɥɶɬɢɩɥɟɬɨɜ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ
ɡɧɚɱɟɧɢɟɦ
z
S
S
ɡɚɪɚɧɟɟ ɢɡɜɟɫɬɟɧ, ɬɚɤ ɱɬɨ ɦɨɠɧɨ ɫɪɚɡɭ ɠɟ ɭɤɚɡɚɬɶ ɧɚ
ɩɪɢɧɚɞɥɟɠɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ
3 ɢ 4 ɬɪɢɩɥɟɬɭ ɫ 1
S
. ɇɟɞɨɫɬɚɸɳɟɟ ɫɨɫɬɨɹɧɢɟ
ɬɪɢɩɥɟɬɚ, ɩɪɢɧɚɞɥɟɠɚɳɟɟ ɡɧɚɱɟɧɢɸ 0
z
S , ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ, ɩɪɢɦɟɧɹɹ
ɩɨɜɵɲɚɸɳɢɣ ɨɩɟɪɚɬɨɪ
21
ˆˆ
ˆ
ssS (13)
ɤ ɜɟɤɬɨɪɭ ɫɨɫɬɨɹɧɢɹ
1,14
z
SS :
2121
ˆˆ
1,1
ˆ
pppp
ssS . (14)
Ⱦɟɣɫɬɜɢɟ ɩɨɜɵɲɚɸɳɟɝɨ ɨɩɟɪɚɬɨɪɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɮɨɪɦɭɥɨɣ
1,)1()(,
ˆ
zzzz
sssssssss . (15)
ɋ ɭɱɟɬɨɦ ɷɬɨɣ ɮɨɪɦɭɥɵ ɥɟɜɚɹ ɫɬɨɪɨɧɚ ɫɨɨɬɧɨɲɟɧɢɹ (14) ɪɚɜɧɚ
0,12,ɚ
ɩɪɚɜɚɹ ɫɬɨɪɨɧɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦɦɭ
21 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɵ
ɩɪɢɯɨɞɢɦ ɤ ɪɟɡɭɥɶɬɚɬɭ (11). əɜɧɵɣ ɜɢɞ ɫɨɫɬɨɹɧɢɹ (12) ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɭɫɥɨɜɢɹ
ɟɝɨ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɫɨɫɬɨɹɧɢɸ
0,1.
92
8.7. ɚ) ɋɨɝɥɚɫɧɨ ɤɜɚɧɬɨɜɨ-ɦɟɯɚɧɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ ɫɥɨɠɟɧɢɹ ɦɨɦɟɧɬɨɜ
ɫɭɦɦɚ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚj
l
ɢ ɫɩɢɧɚ
s
ɫ ɲɚɝɨɦ 1 ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ ɨɬ
s
l
ɞɨ sl . ɉɪɢ 2
/
1
s
ɷɬɨ ɩɪɚɜɢɥɨ ɞɚɟɬ ɞɜɚ ɞɨɩɭɫɬɢɦɵɯ ɡɧɚɱɟɧɢɹ ɩɨɥɧɨɝɨ
ɦɨɦɟɧɬɚ ɞɥɹ ɤɚɠɞɨɝɨ ɡɚɞɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹj 1t
l
:
2
/
1r
l
j . (1)
Ⱦɥɹ 0
l
ɢɦɟɟɬɫɹ ɨɞɧɨ ɡɧɚɱɟɧɢɟ ɩɨɥɧɨɝɨ ɦɨɦɟɧɬɚ:2
/
1 j .
ɑɢɫɥɨ ɫɨɫɬɨɹɧɢɣ ɜɢɞɚ
zz
ssll ,2/1, ɫ ɡɚɞɚɧɧɵɦ
l
ɪɚɜɧɨ )12(2
l
.
ɂɡ ɥɢɧɟɣɧɵɯ ɤɨɦɛɢɧɚɰɢɣ ɬɚɤɢɯ ɫɨɫɬɨɹɧɢɣ ɨɛɪɚɡɭɸɬɫɹ ɦɭɥɶɬɢɩɥɟɬɵ ɫɨɫɬɨɹɧɢɣ
mj, ɫ ɨɩɪɟɞɟɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ
l
mj ,, (ɢ 2
/
1
s
). Ɇɭɥɶɬɢɩɥɟɬ ɫ ɡɚɞɚɧɧɵɦ
ɜɤɥɸɱɚɟɬ 1ɱɥɟɧɨɜ. ɍɤɚɡɚɧɧɵɟ ɦɭɥɶɬɢɩɥɟɬɵ ɭɫɥɨɜɢɦɫɹ ɨɛɨɡɧɚɱɚɬɶ
ɫɢɦɜɨɥɨɦ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɷɥɟɤɬɪɨɧɚ (
j 2 j
...,,, d
p
s
) ɫ ɧɢɠɧɢɦ ɢɧɞɟɤɫɨɦ,
ɪɚɜɧɵɦ ɩɨɥɧɨɦɭ ɦɨɦɟɧɬɭ . Ɍɨɝɞɚ ɩɟɪɟɱɢɫɥɟɧɢɟ ɷɥɟɤɬɪɨɧɧɵɯ ɫɨɫɬɨɹɧɢɣ
ɩɪɟɞɫɬɚɜɢɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ.
j
Ⱦɜɚ
s
-ɫɨɫɬɨɹɧɢɹ (0
l
) ɫ ɪɚɡɥɢɱɚɸɳɢɦɢɫɹ ɡɧɚɱɟɧɢɹɦɢ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ
ɨɛɪɚɡɭɸɬ ɞɭɛɥɟɬ ɫ 2
2/1
s
/
1 j . ɒɟɫɬɶ
p
-ɫɨɫɬɨɹɧɢɣ (1
l
) ɨɛɪɚɡɭɸɬ ɞɭɛɥɟɬ
ɫ 2
2/1
p
/
1 j ɢ ɤɜɚɪɬɟɬ ɫ 2
2/3
p
/
3 j . Ⱦɟɫɹɬɶ -ɫɨɫɬɨɹɧɢɣ ɚɧɚɥɨɝɢɱɧɵɦ
ɨɛɪɚɡɨɦ «ɪɚɫɳɟɩɥɹɸɬɫɹ» ɧɚ ɱɟɬɵɪɟ ɫɨɫɬɨɹɧɢɹ ɢ ɲɟɫɬɶ ɫɨɫɬɨɹɧɢɣ . ɂ
ɬɚɤ ɞɚɥɟɟ.
d
2/3
d
2/5
d
ɛ) ȼ ɫɢɥɭ ɫɨɨɬɧɨɲɟɧɢɹ
slslslJ
ˆ
ˆ
2
ˆ
ˆ
)
ˆ
ˆ
(
ˆ
2222
(2)
ɞɥɹ ɨɩɟɪɚɬɨɪɚ
222
ˆ
ˆˆ
ˆ
ˆ
2
slJsl (3)
ɫɨɛɫɬɜɟɧɧɵɦɢ ɜɟɤɬɨɪɚɦɢ ɹɜɥɹɸɬɫɹ ɫɨɫɬɨɹɧɢɹ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ
l
j,,
s
. Ʉɚɤ ɪɚɡ ɬɚɤɢɟ ɫɨɫɬɨɹɧɢɹ ɛɵɥɢ ɩɟɪɟɱɢɫɥɟɧɵ ɜɵɲɟ. ɋɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ
ɨɩɟɪɚɬɨɪɚ (3) ɩɪɢ 2
/
1
s
ɢ 2
/
1r
l
j ɪɚɜɧɵ ɜɟɥɢɱɢɧɟ
¯
®
.2/1,1
;2/1,
)1()1()1(
ljl
ljl
sslljj (4)
ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɩɪɢ 2
/
1
s
ɨɩɟɪɚɬɨɪ «ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ»
ɢɦɟɟɬ ɞɜɚ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɹ
sl
ˆ
ˆ
A
j
GH , ɤɨɬɨɪɵɟ ɨɬɧɨɫɹɬɫɹ ɤ ɦɭɥɶɬɢɩɥɟɬɚɦ
ɫɨɫɬɨɹɧɢɣ ɫ ɪɚɡɥɢɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɩɨɥɧɨɝɨ ɦɨɦɟɧɬɚ :j
2/
2/1
Al
lj
GH
, 2/)1(
2/1
G
H
lA
lj
. (5)
93
ɉɪɢ 0
l
ɢɦɟɟɬɫɹ ɥɢɲɶ ɨɞɢɧ ɦɭɥɶɬɢɩɥɟɬ – ɞɭɛɥɟɬ ɫɨɫɬɨɹɧɢɣ ɫ
2/1
s 2
/
1 j ;
ɨɛɚ ɫɨɫɬɨɹɧɢɹ ɜ ɧɟɦ ɨɛɥɚɞɚɸɬ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɷɧɟɪɝɢɟɣ.
ɗɬɢ ɪɟɡɭɥɶɬɚɬɵ ɨɬɪɚɠɚɸɬ ɧɟɤɨɬɨɪɵɟ ɱɟɪɬɵ ɞɨɫɬɚɬɨɱɧɨ ɫɥɨɠɧɨɣ ɬɟɨɪɢɢ
ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ, ɨɫɧɨɜɚɧɧɨɣ ɧɚ ɪɟɥɹɬɢɜɢɫɬɫɤɨɦ
ɨɩɢɫɚɧɢɢ ɞɜɢɠɟɧɢɹ ɷɥɟɤɬɪɨɧɚ ɫ ɩɨɦɨɳɶɸ ɭɪɚɜɧɟɧɢɹ Ⱦɢɪɚɤɚ. ɗɧɟɪɝɢɹ ɫɩɢɧ-
ɨɪɛɢɬɚɥɶɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɩɨɹɜɥɹɟɬɫɹ ɜ ɬɚɤɨɣ ɬɟɨɪɢɢ ɤɚɤ ɨɞɧɚ ɢɡ
ɪɟɥɹɬɢɜɢɫɬɫɤɢɯ ɩɨɩɪɚɜɨɤ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɭɱɟɬɚ ɪɟɥɹɬɢɜɢɫɬɫɤɢɯ ɩɨɩɪɚɜɨɤ ɜ
ɷɧɟɪɝɟɬɢɱɟɫɤɨɦ ɫɩɟɤɬɪɟ ɚɬɨɦɚ ɜɨɡɧɢɤɚɟɬ ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɬɨɧɤɚɹ ɫɬɪɭɤɬɭɪɚ
ɭɪɨɜɧɟɣ – ɪɚɫɳɟɩɥɟɧɢɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ
n
H ɫ ɪɚɡɥɢɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ
ɩɨɥɧɨɝɨ ɦɨɦɟɧɬɚ . ɉɨɪɹɞɨɤ ɜɟɥɢɱɢɧɵ ɷɬɨɝɨ ɪɚɫɳɟɩɥɟɧɢɹ ɨɰɟɧɢɜɚɟɬɫɹ
ɜɵɪɚɠɟɧɢɟɦ , ɝɞɟ
j
RyA
2
~ D ce =
/
2
D – ɩɨɫɬɨɹɧɧɚɹ ɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ.
Ⱦɚɥɶɧɟɣɲɟɟ ɭɬɨɱɧɟɧɢɟ ɤɚɪɬɢɧɵ ɭɪɨɜɧɟɣ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ ɫɜɹɡɚɧɨ ɫ
ɪɚɞɢɚɰɢɨɧɧɵɦɢ ɩɨɩɪɚɜɤɚɦɢ, ɜɨɡɧɢɤɚɸɳɢɦɢ ɜɫɥɟɞɫɬɜɢɟ ɤɜɚɧɬɨɜɚɧɢɹ
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ, ɢ, ɤɪɨɦɟ ɬɨɝɨ, – ɫ ɭɱɟɬɨɦ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ,
ɫɨɡɞɚɜɚɟɦɨɝɨ ɫɩɢɧɨɜɵɦ ɦɚɝɧɢɬɧɵɦ ɦɨɦɟɧɬɨɦ ɹɞɪɚ. ȼɡɚɢɦɨɞɟɣɫɬɜɢɟ ɷɥɟɤɬɪɨɧɚ
ɫ ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ ɹɞɪɚ ɩɪɢɜɨɞɢɬ ɤ ɜɨɡɧɢɤɧɨɜɟɧɢɸ ɬɚɤ ɧɚɡɵɜɚɟɦɨɣ
ɫɜɟɪɯɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ. ȼ ɱɚɫɬɧɨɫɬɢ, ɷɬɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ
ɨɬɜɟɬɫɬɜɟɧɧɨ ɡɚ ɪɚɫɳɟɩɥɟɧɢɟ ɷɥɟɤɬɪɨɧɧɨɝɨ ɭɪɨɜɧɹ ɧɚ ɞɜɚ
ɛɥɢɡɤɢɯ ɩɨ ɷɧɟɪɝɢɢ ɩɨɞɭɪɨɜɧɹ, ɱɬɨ ɫɥɟɞɭɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɫɭɳɟɫɬɜɟɧɧɵɣ
ɷɮɮɟɤɬ ɜ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɪɟɱɶ ɢɞɟɬ ɨ ɬɨɱɧɨɣ ɤɚɪɬɢɧɟ ɨɫɧɨɜɧɨɝɨ ɢ
ɛɥɢɠɚɣɲɟɝɨ ɤ ɧɟɦɭ ɜɨɡɛɭɠɞɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɣ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ (ɫɦ. ɡɚɞɚɱɭ 8.8).
2/1
1s
8.8. ɍɤɚɡɚɧɧɵɣ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɨɩɟɪɚɬɨɪ ɷɧɟɪɝɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧɬɨɜ ɷɥɟɤɬɪɨɧɚ ɢ ɩɪɨɬɨɧɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
C
2
ˆ
1
ˆ
ıı
21
ˆˆ
4
ss
C . (1)
ɋɨɝɥɚɫɧɨ ɡɚɤɨɧɭ ɫɥɨɠɟɧɢɹ ɦɨɦɟɧɬɨɜ ɞɨɩɭɫɬɢɦɵɟ ɡɧɚɱɟɧɢɹ ɫɭɦɦɚɪɧɨɝɨ ɫɩɢɧɚ
ɷɥɟɤɬɪɨɧɚ ɢ ɩɪɨɬɨɧɚ
S
ɪɚɜɧɵ 1 ɢ 0, ɩɪɢɱɟɦ ɫɨɫɬɨɹɧɢɹ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ
ɡɧɚɱɟɧɢɟɦ
S
ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɞɥɹ ɨɩɟɪɚɬɨɪɚ (1), ɬɚɤ ɤɚɤ
2
1
2
1
2
21
ˆˆ
ˆ
ˆˆ
2
ssSss . (2)
ɋɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɨɩɟɪɚɬɨɪɚ (2) ɩɪɢ 2/1
21
ss ɪɚɜɧɵ ɜɟɥɢɱɢɧɟ
¯
®
.0,2/3
;1,2/1
)1()1()1(
2211
S
S
ssssSS (3)
Ɍɪɢɩɥɟɬ ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɣ (ɫɨ ɫɩɢɧɨɦ )1
S
ɷɥɟɤɬɪɨɧɚ ɢ ɹɞɪɚ,
ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɜ ɤɚɱɟɫɬɜɟ ɟɞɢɧɨɣ ɫɩɢɧɨɜɨɣ ɩɨɞɫɢɫɬɟɦɵ ɚɬɨɦɚ, ɩɪɢɧɚɞɥɟɠɢɬ
ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ ɷɧɟɪɝɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ C , ɚ ɫɢɧɝɥɟɬ (0
S
)
94
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɨɛɫɬɜɟɧɧɨɦɭ ɡɧɚɱɟɧɢɸ C3 . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɭɪɨɜɟɧɶ ɷɧɟɪɝɢɢ
s
1-ɫɨɫɬɨɹɧɢɹ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ ɢɦɟɟɬ ɫɜɟɪɯɬɨɧɤɭɸ ɫɬɪɭɤɬɭɪɭ – ɢɫɩɵɬɵɜɚɟɬ
ɪɚɫɳɟɩɥɟɧɢɟ ɧɚ ɞɜɚ ɭɪɨɜɧɹ, ɪɚɡɞɟɥɟɧɧɵɯ ɷɧɟɪɝɟɬɢɱɟɫɤɢɦ ɢɧɬɟɪɜɚɥɨɦ C4.
Ɉɰɟɧɢɦ ɦɚɫɲɬɚɛ ɜɟɥɢɱɢɧɵ C ɤɚɤ ɡɧɚɱɟɧɢɟ ɷɧɟɪɝɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
ɦɚɝɧɢɬɧɨɝɨ ɦɨɦɟɧɬɚ ɷɥɟɤɬɪɨɧɚ
Be
P P ɫ ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ
, ɫɨɡɞɚɜɚɟɦɵɦ ɦɚɝɧɢɬɧɵɦ ɩɨɥɟɦ ɩɪɨɬɨɧɚ ɧɚ ɪɚɫɫɬɨɹɧɢɹɯ ɨɬ
ɹɞɪɚ ɩɨɪɹɞɤɚ ɪɚɡɦɟɪɚ ɷɥɟɤɬɪɨɧɧɨɣ «ɨɪɛɢɬɵ»; ɩɨɥɭɱɚɟɦ:
33
/~/
Bpp
arB PP
p
e
p
e
B
B
B
pB
B
m
m
Ry
m
m
aa
BC
2
3
2
3
~~~~ D
P
PP
P , (4)
ɝɞɟ 137
/
1
/
2
| D ce = – ɩɨɫɬɨɹɧɧɚɹ ɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ,
4
105~/
pe
mm –
ɨɬɧɨɲɟɧɢɟ ɦɚɫɫɵ ɷɥɟɤɬɪɨɧɚ ɤ ɦɚɫɫɟ ɩɪɨɬɨɧɚ.
ɍɞɚɟɬɫɹ ɬɚɤɠɟ ɩɪɨɜɟɫɬɢ ɬɨɱɧɨɟ ɤɜɚɧɬɨɜɨ-ɦɟɯɚɧɢɱɟɫɤɨɟ ɜɵɱɢɫɥɟɧɢɟ
ɷɧɟɪɝɢɢ ɫɜɟɪɯɬɨɧɤɨɝɨ ɪɚɫɳɟɩɥɟɧɢɹ
s
1-ɭɪɨɜɧɹ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ (ɫɦ., ɧɚɩɪɢɦɟɪ,
[8]). Ɋɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɚ ɯɨɪɨɲɨ ɫɨɝɥɚɫɭɸɬɫɹ ɫ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦɢ ɞɚɧɧɵɦɢ.
ȼ ɟɞɢɧɢɰɚɯ ɱɚɫɬɨɬɵ ɷɬɨ ɪɚɫɳɟɩɥɟɧɢɟ ɫɨɫɬɚɜɥɹɟɬ 1420 ɆȽɰ, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɦɭ ɢɡɥɭɱɟɧɢɸ ɫ ɞɥɢɧɨɣ ɜɨɥɧɵ 21 ɫɦ. ɇɚɛɥɸɞɟɧɢɟ ɬɚɤɨɝɨ
ɢɡɥɭɱɟɧɢɹ ɫɵɝɪɚɥɨ ɜɚɠɧɭɸ ɪɨɥɶ ɜ ɫɬɚɧɨɜɥɟɧɢɢ ɪɚɞɢɨɚɫɬɪɨɧɨɦɢɢ; ɩɨ
ɧɚɛɥɸɞɚɟɦɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɩɪɢɯɨɞɹɳɟɝɨ ɢɡ ɤɨɫɦɨɫɚ ɢɡɥɭɱɟɧɢɹ ɧɚ ɞɥɢɧɟ
ɜɨɥɧɵ 21 ɫɦ ɢ ɟɝɨ ɞɨɩɥɟɪɨɜɫɤɨɦɭ
ɫɞɜɢɝɭ ɛɵɥɢ ɢɡɭɱɟɧɵ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ɩɥɨɬɧɨɫɬɢ ɢ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɜɨɞɨɪɨɞɚ ɜ Ƚɚɥɚɤɬɢɤɟ.
8.9. ɋɢɫɬɟɦɚ ɞɜɭɯ ɷɥɟɤɬɪɨɧɨɜ, ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɫɢɧɝɥɟɬɧɨɦ ɫɩɢɧɨɜɨɦ
ɫɨɫɬɨɹɧɢɢ
0,0
z
SS , ɜ ɤɨɨɪɞɢɧɚɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɞɥɹ ɨɪɛɢɬɚɥɶɧɨɣ
ɱɚɫɬɢ ɩɨɥɧɨɝɨ ɜɟɤɬɨɪɚ ɫɨɫɬɨɹɧɢɹ ɨɩɢɫɵɜɚɟɬɫɹ ɫɢɦɜɨɥɢɱɟɫɤɢɦ ɪɚɜɟɧɫɬɜɨɦ
),(
2112
rr\ ) 0,0
z
SS , (1)
ɝɞɟ ɞɜɭɯɷɥɟɤɬɪɨɧɧɚɹ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ),(
21
rr\ ɞɨɥɠɧɚ ɛɵɬɶ ɜɵɪɚɠɟɧɚ ɱɟɪɟɡ
ɨɞɧɨɷɥɟɤɬɪɨɧɧɵɟ ɮɭɧɤɰɢɢ ,)(
r
p
M )(r
q
M , ɚ ɜɟɤɬɨɪ ɫɩɢɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ ɢɦɟɟɬ
ɜɢɞ:
2121
2
1
2
1
0,0 nppn
z
SS . (2)
ɗɥɟɤɬɪɨɧɵ ɹɜɥɹɸɬɫɹ ɮɟɪɦɢɨɧɚɦɢ ɢ ɩɨɬɨɦɭ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɪɢɧɰɢɩɨɦ
ɧɟɪɚɡɥɢɱɢɦɨɫɬɢ ɨɞɢɧɚɤɨɜɵɯ ɱɚɫɬɢɰ, ɫɨɫɬɨɹɧɢɟ (1) ɞɨɥɠɧɨ ɛɵɬɶ
ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɵɦ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɟɪɟɫɬɚɧɨɜɤɟ ɧɨɦɟɪɨɜ ɱɚɫɬɢɰ 1 ɢ 2 (ɢɥɢ –
ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɟɪɟɫɬɚɧɨɜɤɟ ɫɨɜɨɤɭɩɧɨɫɬɟɣ ɤɜɚɧɬɨɜɵɯ ɱɢɫɟɥ ɤɚɠɞɨɣ ɢɡ
95
ɱɚɫɬɢɰ, ɧɚɩɪɢɦɟɪ ɢ )). ɉɨɫɤɨɥɶɤɭ ɫɩɢɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ (2)
ɭɠɟ ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɨ ɩɨ ɧɨɦɟɪɚɦ 1 ɢ 2, ɬɨ ɨɪɛɢɬɚɥɶɧɚɹ ɱɚɫɬɶ ɜ (1) ɞɨɥɠɧɚ
ɛɵɬɶ ɫɢɦɦɟɬɪɢɱɧɨɣ:
),( n pa ,( p qb
),(
21
rr\ ),(
12
rr\ . (3)
ȼ ɫɥɭɱɚɟ ɪɚɡɥɢɱɢɦɵɯ ɱɚɫɬɢɰ 1 ɢ 2 (ɧɚɩɪɢɦɟɪ, ɷɥɟɤɬɪɨɧɚ ɢ ɩɪɨɬɨɧɚ),
ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɫɨɫɬɨɹɧɢɹɯ ɢ ))(
1
r
p
M (
2
r
q
M , ɨɪɛɢɬɚɥɶɧɚɹ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ
ɫɢɫɬɟɦɵ ɢɦɟɥɚ ɛɵ ɜɢɞ ɩɪɨɢɡɜɟɞɟɧɢɹ )(
1
r
p
M )(
2
r
q
M . Ⱦɥɹ ɫɢɫɬɟɦɵ ɠɟ
ɨɞɢɧɚɤɨɜɵɯ ɱɚɫɬɢɰ ɩɪɢ ɭɫɥɨɜɢɢ (3) ɨɪɛɢɬɚɥɶɧɚɹ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɞɨɥɠɧɚ
ɛɵɬɶ ɪɚɜɧɚ ɫɢɦɦɟɬɪɢɡɨɜɚɧɧɨɦɭ «ɩɪɨɢɡɜɟɞɟɧɢɸ» ɨɞɧɨɱɚɫɬɢɱɧɵɯ ɮɭɧɤɰɢɣ:
)()()()(
2
1
),(
122121
rrrrrr
qpqp
MMMM \
. (4)
Ɇɧɨɠɢɬɟɥɶ
2
/
1 ɡɞɟɫɶ ɜɜɟɞɟɧ ɞɥɹ ɫɨɯɪɚɧɟɧɢɹ ɧɨɪɦɢɪɨɜɤɢ – ɟɫɥɢ ɮɭɧɤɰɢɢ
p
M
ɢ ɧɨɪɦɢɪɨɜɚɧɵ ɧɚ ɟɞɢɧɢɰɭ, ɬɨ ɮɭɧɤɰɢɹ (4) ɬɚɤɠɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɸ
ɧɨɪɦɢɪɨɜɤɢ ɧɚ ɟɞɢɧɢɰɭ:
q
M
1),(
2
212
3
1
3
\
³³
rrrr dd . (5)
8.10. ɚ) ɋɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɷɧɟɪɝɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɱɚɫɬɢɰ
ɜ ɫɨɫɬɨɹɧɢɢ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ ɫɭɦɦɚɪɧɵɦ ɫɩɢɧɨɦ
)()(
1221
rrrr UU
S
,
),(
2112
rr\ ) S , (1)
ɫ ɭɱɟɬɨɦ ɧɨɪɦɢɪɨɜɤɢ ɫɩɢɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ ɧɚ ɟɞɢɧɢɰɭ ɢɦɟɟɬ ɜɢɞ ɢɧɬɟɝɪɚɥɚ ɩɨ
ɤɨɨɪɞɢɧɚɬɚɦ ɱɚɫɬɢɰ:
))
1212
ˆ
UU
\\
³³
SSUdd *
2
3
1
3
rr
³³
2
3
1
3
rr dd ),(*
21
rr\ ),(
21
rr
\
)(
21
rr U . (2)
ɋɭɦɦɚɪɧɨɦɭ ɫɩɢɧɭ 0
S
ɨɬɜɟɱɚɟɬ ɫɢɦɦɟɬɪɢɱɧɚɹ ɩɨ ɧɨɦɟɪɚɦ ɱɚɫɬɢɰ 1,2
ɨɪɛɢɬɚɥɶɧɚɹ ɱɚɫɬɶ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɫɢɫɬɟɦɵ (ɬɚɤ ɤɚɤ ɷɥɟɤɬɪɨɧɵ ɹɜɥɹɸɬɫɹ
ɮɟɪɦɢɨɧɚɦɢ, ɚ ɫɩɢɧɨɜɚɹ ɱɚɫɬɶ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ ɩɪɢ 0
S
ɭɠɟ
ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɚ):
)()()()(
2
1
),(
122121
rrrrrr
qpqp
MMMM \
. (3)
ɉɨɞɫɬɚɜɢɜ (3) ɜ (2) ɢ ɩɪɨɢɡɜɟɞɹ ɜ ɧɟɤɨɬɨɪɵɯ ɫɥɚɝɚɟɦɵɯ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ
96
ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ , ɩɨɥɭɱɢɦ
21
rr l
JAU , (4)
ɝɞɟ
A
2
12
3
1
3
)(
³³
M rrr
p
dd
2
2
)(r
q
M )(
21
rr
U , (5)
J
³³
2
3
1
3
rr dd *)(
1
r
p
M )(
1
r
q
M *)(
2
r
q
M
)(
2
r
p
M )(
21
rr U . (6)
ȼɟɥɢɱɢɧɚ )ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɷɧɟɪɝɢɸ ɤɭɥɨɧɨɜɫɤɨɝɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɷɥɟɤɬɪɨɧɨɜ:
(
21
rr U
21
2
rr
e
U .
ɛ) ɋɩɢɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ
S ɫɢɫɬɟɦɵ ɞɜɭɯ ɷɥɟɤɬɪɨɧɨɜ ɩɪɢ ɫɭɦɦɚɪɧɨɦ
ɫɩɢɧɟ 1
S
ɦɨɠɟɬ ɫɨɜɩɚɞɚɬɶ ɫ ɨɞɧɢɦ ɢɡ ɫɨɫɬɨɹɧɢɣ ɬɪɢɩɥɟɬɚ
z
SS ,1 ɢɥɢ
ɢɦɟɬɶ ɜɢɞ ɢɯ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ. ȼ ɥɸɛɨɦ ɢɡ ɷɬɢɯ ɫɥɭɱɚɟɜ ɫɨɫɬɨɹɧɢɟ
1 S
ɫɢɦɦɟɬɪɢɱɧɨ ɩɨ ɧɨɦɟɪɚɦ ɱɚɫɬɢɰ 1 ɢ 2, ɬɚɤ ɤɚɤ ɬɚɤɨɣ ɫɢɦɦɟɬɪɢɟɣ ɨɛɥɚɞɚɟɬ
ɤɚɠɞɵɣ ɢɡ ɱɥɟɧɨɜ ɬɪɢɩɥɟɬɚ (ɫɦ. ɮɨɪɦɭɥɵ (11) ɜ ɡɚɞɚɱɟ 8.6). ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
ɬɪɟɛɨɜɚɧɢɸ ɚɧɬɢɫɢɦɦɟɬɪɢɢ ɞɜɭɯɮɟɪɦɢɨɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
12
) ɬɟɩɟɪɶ ɨɬɜɟɱɚɟɬ
ɚɧɬɢɫɢɦɦɟɬɪɢɡɨɜɚɧɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɨɪɛɢɬɚɥɶɧɵɯ ɮɭɧɤɰɢɣ ɢ :
p
M
q
M
)()()()(
2
1
),(
122121
rrrrrr
qpqp
MMMM \
. (7)
ɉɨɞɫɬɚɜɢɜ (7) ɜ (2), ɩɨɥɭɱɢɦ
JAU , (8)
ɝɞɟ ɜɟɥɢɱɢɧɵ
A
ɢ
J
ɞɚɸɬɫɹ ɮɨɪɦɭɥɚɦɢ (5), (6). Ʉɚɤ ɜɢɞɢɦ, ɫɥɚɝɚɟɦɨɟ
J
ɜɯɨɞɢɬ ɜ (4) ɢɜ (8) ɫ ɪɚɡɥɢɱɧɵɦ ɡɧɚɤɨɦ.
ȼɟɥɢɱɢɧɭ
J
ɧɚɡɵɜɚɸɬ ɨɛɦɟɧɧɵɦ ɢɧɬɟɝɪɚɥɨɦ, ɚ ɨ ɬɨɦ ɨɛɫɬɨɹɬɟɥɶɫɬɜɟ, ɱɬɨ
ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɨɞɢɧɚɤɨɜɵɯ ɱɚɫɬɢɰ
U ɡɚɜɢɫɢɬ ɨɬ ɡɧɚɱɟɧɢɹ ɫɭɦɦɚɪɧɨɝɨ
ɫɩɢɧɚ (ɞɚɠɟ ɩɪɢ ɧɟ ɡɚɜɢɫɹɳɟɦ ɨɬ ɫɩɢɧɚ ɱɚɫɬɢɰ ɨɩɟɪɚɬɨɪɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
U
),
ɝɨɜɨɪɹɬ ɤɚɤ ɨɛ ɨɛɦɟɧɧɨɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ. ȼ ɫɥɭɱɚɟ ɪɚɡɥɢɱɢɦɵɯ ɱɚɫɬɢɰ
ɨɪɛɢɬɚɥɶɧɚɹ ɱɚɫɬɶ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɫɢɫɬɟɦɵ ɛɵɥɚ ɛɵ ɪɚɜɧɚ ɨɛɵɱɧɨɦɭ
ɩɪɨɢɡɜɟɞɟɧɢɸ , ɢ ɜɦɟɫɬɨ (4) ɢɥɢ (8) ɦɵ ɢɦɟɥɢ ɛɵ ɫɨɨɬɧɨɲɟɧɢɟ)(
1
r
p
M )(
2
r
q
M
AU , ɧɟ ɫɨɞɟɪɠɚɳɟɟ ɨɛɦɟɧɧɨɝɨ ɢɧɬɟɝɪɚɥɚ.
97
Ƚɥɚɜɚ 9. ȼɈɁɆɍɓȿɇɂȿ, ɉȿɊȿɏɈȾɕ.
Ⱦɥɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɫɥɨɠɧɵɯ ɤɜɚɧɬɨɜɨ-ɦɟɯɚɧɢɱɟɫɤɢɯ ɡɚɞɚɱ
ɩɪɢɦɟɧɹɸɬ ɦɟɬɨɞɵ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ. ȼ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɝɚɦɢɥɶɬɨɧɢɚɧ
ɫɢɫɬɟɦɵ
H
ˆ
ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɮɨɪɦɟ
VHH
ˆˆˆ
0
, (I)
ɝɞɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɪɚɜɧɟɧɢɸ ɒɪɟɞɢɧɝɟɪɚ, ɞɨɩɭɫɤɚɸɳɟɦɭ ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ,
ɚ ɨɩɟɪɚɬɨɪ , ɧɚɡɵɜɚɟɦɵɣ
ɜɨɡɦɭɳɟɧɢɟɦ, ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɦɚɥɚɹ ɜɟɥɢɱɢɧɚ
ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ .
0
ˆ
H
V
ˆ
0
ˆ
H
Ɋɚɡɥɢɱɚɸɬ ɫɬɚɰɢɨɧɚɪɧɵɟ ɢ ɧɟɫɬɚɰɢɨɧɚɪɧɵɟ ɡɚɞɚɱɢ. ȼ ɫɬɚɰɢɨɧɚɪɧɨɦ
ɫɥɭɱɚɟ ɨɩɟɪɚɬɨɪɵ ɢ ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɜɪɟɦɟɧɢ, ɬɚɤ ɱɬɨ ɤɜɚɧɬɨɜɚɹ ɫɢɫɬɟɦɚ ɫ
ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (I) ɨɛɥɚɞɚɟɬ ɫɬɚɰɢɨɧɚɪɧɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ
0
ˆ
HV
ˆ
n
H ɢ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɷɧɟɪɝɟɬɢɱɟɫɤɢɦ ɫɩɟɤɬɪɨɦ
n
H . ɗɬɢ ɜɟɤɬɨɪɵ ɫɨɫɬɨɹɧɢɣ ɢ
ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ (ɜ ɞɢɫɤɪɟɬɧɨɣ ɱɚɫɬɢ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɫɩɟɤɬɪɚ) ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ
ɜɢɞɟ ɪɹɞɨɜ ɩɨ ɫɬɟɩɟɧɹɦ ɦɚɥɨɝɨ ɩɚɪɚɦɟɬɪɚ, ɨɩɪɟɞɟɥɹɸɳɟɝɨ ɦɚɥɨɫɬɶ ɨɩɟɪɚɬɨɪɚ
ɜɨɡɦɭɳɟɧɢɹ
ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɧɟɜɨɡɦɭɳɟɧɧɨɦɭ ɝɚɦɢɥɶɬɨɧɢɚɧɭ ; ɫɬɟɩɟɧɶ
ɦɚɥɨɫɬɢ ɱɥɟɧɨɜ ɪɹɞɚ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɩɪɢɧɹɬɨ ɭɤɚɡɵɜɚɬɶ ɜɜɟɪɯɭ ɜ ɫɤɨɛɤɚɯ:
V
ˆ
0
ˆ
H
¦
H
k
nkn
kC
)0(
,
nk
C
)0(
nk
C
)1(
nk
C ...
)2(
nk
C ; (II)
n
H
)0(
n
H
)1(
n
H ...
)2(
H
n
. (III)
Ʉɚɤ ɩɪɚɜɢɥɨ, ɜ ɜɚɠɧɵɯ ɞɥɹ ɩɪɚɤɬɢɤɢ ɡɚɞɚɱɚɯ ɪɹɞɵ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ
ɩɪɨɫɭɦɦɢɪɨɜɚɬɶ ɧɟ ɭɞɚɟɬɫɹ. ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɨɫɧɨɜɧɨɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɟɬ
ɨɬɵɫɤɚɧɢɟ ɥɢɲɶ ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɜɵɯ ɱɥɟɧɨɜ ɪɹɞɚ. ȿɫɥɢ ɷɬɢ ɱɥɟɧɵ ɦɚɥɵ ɩɨ
ɫɪɚɜɧɟɧɢɸ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɜɟɥɢɱɢɧɚɦɢ ɧɭɥɟɜɨɝɨ ɩɨɪɹɞɤɚ ɢ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɭɛɵɜɚɸɬ, ɬɨ ɢɯ ɦɨɠɧɨ ɬɪɚɤɬɨɜɚɬɶ ɤɚɤ ɩɨɩɪɚɜɤɢ,
ɨɛɭɫɥɨɜɥɟɧɧɵɟ ɨɩɟɪɚɬɨɪɨɦ ɜɨɡɦɭɳɟɧɢɹ. ɂɧɨɝɞɚ ɩɪɢɛɟɝɚɸɬ ɤ ɜɵɛɨɪɨɱɧɨɦɭ
ɫɭɦɦɢɪɨɜɚɧɢɸ ɪɹɞɨɜ ɢɥɢ ɤ ɩɟɪɟɫɬɪɨɣɤɟ ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ ɧɨɜɨɝɨ ɦɚɥɨɝɨ
ɩɚɪɚɦɟɬɪɚ, ɧɨ ɜ ɧɚɲɟɦ ɢɡɥɨɠɟɧɢɢ ɩɨɞɨɛɧɵɟ ɜɨɩɪɨɫɵ ɧɟ ɛɭɞɭɬ ɡɚɬɪɨɧɭɬɵ.
ɍɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (I), ɨɩɪɟɞɟɥɹɸɳɟɟ
ɫɬɚɰɢɨɧɚɪɧɵɟ ɫɨɫɬɨɹɧɢɹ (II) ɢ ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ (III), ɫɜɨɞɢɬɫɹ ɤ ɫɢɫɬɟɦɟ
ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɪɚɡɥɨɠɟɧɢɹ ɢɫɤɨɦɵɯ ɬɨɱɧɵɯ ɜɟɤɬɨɪɨɜ
ɫɨɫɬɨɹɧɢɹ
nk
C
n
H ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɜɟɤɬɨɪɚɦ
)0(
k
ɧɟɜɨɡɦɭɳɟɧɧɨɝɨ
ɝɚɦɢɥɶɬɨɧɢɚɧɚ:
¦
c
cc
HH
k
nkkknk
k
n
CVC)(
)0(
, (IV)
98
ɝɞɟ – ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ ɜɨɡɦɭɳɟɧɢɹ, ɜɵɱɢɫɥɟɧɧɵɟ ɜ ɛɚɡɢɫɟ
kk
V
c
)0(
k
. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɞɫɬɚɧɨɜɤɢ ɜ (IV) ɜɟɥɢɱɢɧ ɢ
nk
C
n
H ɜ ɮɨɪɦɟ ɪɹɞɨɜ ɬɟɨɪɢɢ
ɜɨɡɦɭɳɟɧɢɣ ɢ ɩɪɢɪɚɜɧɢɜɚɧɢɹ ɞɪɭɝ ɞɪɭɝɭ ɜɟɥɢɱɢɧ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɩɨɪɹɞɤɚ
ɦɚɥɨɫɬɢ ɢɡ ɥɟɜɨɣ ɢ ɩɪɚɜɨɣ ɱɚɫɬɟɣ (IV) ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɫɤɨɦɵɟ ɩɨɩɪɚɜɤɢ ɢ
. ɉɪɢ ɷɬɨɦ ɮɨɪɦɭɥɵ ɞɥɹ ɫɥɭɱɚɟɜ, ɜ ɤɨɬɨɪɵɯ ɧɟɜɨɡɦɭɳɟɧɧɵɣ ɭɪɨɜɟɧɶ
ɧɟ ɜɵɪɨɠɞɟɧ, ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɚɸɬɫɹ ɨɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɪɟɡɭɥɶɬɚɬɨɜ ɜ
ɫɥɭɱɚɹɯ ɫ ɜɵɪɨɠɞɟɧɢɟɦ.
)(m
nk
C
)(m
n
H
)0(
n
H
ȼ ɨɬɫɭɬɫɬɜɢɟ ɜɵɪɨɠɞɟɧɢɹ ɭɪɨɜɧɹ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɤɨɬɨɪɵɦɢ
ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨɩɪɚɜɤɢ ɩɟɪɜɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ ɤ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ
ɝɚɦɢɥɶɬɨɧɢɚɧɚ, ɞɚɸɬɫɹ ɮɨɪɦɭɥɚɦɢ:
)0(
n
H
nk
nk
C G
)0(
, ,0
)1(
nn
C
)0(
)0(
)1(
k
n
nk
nk
V
C
HH
ɩɪɢ nk z . (V)
ɉɨɩɪɚɜɤɢ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɤ ɧɟɜɵɪɨɠɞɟɧɧɨɦɭ ɭɪɨɜɧɸ ɷɧɟɪɝɢɢ
ɢɦɟɸɬ ɜɢɞ:
)0(
n
H
nnn
V H
)1(
, (VI)
H
)2(
n
)0(
)0(
2
'
k
n
kn
k
V
HH
¦
, (VII)
ɝɞɟ ɲɬɪɢɯ ɭ ɡɧɚɤɚ ɫɭɦɦɵ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɫɥɚɝɚɟɦɨɟ ɫ
nk ɞɨɥɠɧɨ ɛɵɬɶ
ɢɫɤɥɸɱɟɧɨ ɢɡ ɞɚɧɧɨɝɨ ɜɵɪɚɠɟɧɢɹ (ɫɭɦɦɢɪɨɜɚɧɢɟ ɜɟɞɟɬɫɹ ɩɨ
nk z ). ɍɤɚɡɚɧɧɵɟ
ɮɨɪɦɭɥɵ ɩɪɢɦɟɧɢɦɵ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ
ɜɨɡɦɭɳɟɧɢɹ ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɪɚɡɧɨɫɬɹɦɢ
ɧɟɜɨɡɦɭɳɟɧɧɵɯ ɭɪɨɜɧɟɣ ɷɧɟɪɝɢɢ:
)0(
)0(
k
nnk
V HH . (VIII)
ɉɪɢ ɧɚɥɢɱɢɢ ɜɵɪɨɠɞɟɧɢɹ ɭɪɨɜɧɹ ɫ ɤɚɤɨɣ-ɥɢɛɨ ɤɪɚɬɧɨɫɬɶɸ
)0(
n
H
g
ɞɚɧɧɨɦɭ ɭɪɨɜɧɸ ɩɪɢɧɚɞɥɟɠɚɬ
g
ɧɟɜɨɡɦɭɳɟɧɧɵɯ ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɧɵɯ
ɫɨɫɬɨɹɧɢɣ
)0(
k
(ɝɞɟ ). ȼɫɹɤɚɹ ɥɢɧɟɣɧɚɹ ɤɨɦɛɢɧɚɰɢɹ ɷɬɢɯ
ɫɨɫɬɨɹɧɢɣ, ɫ ɩɪɨɢɡɜɨɥɶɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ , ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ
ɫɨɛɫɬɜɟɧɧɵɦ ɜɟɤɬɨɪɨɦ ɧɟɜɨɡɦɭɳɟɧɧɨɝɨ ɝɚɦɢɥɶɬɨɧɢɚɧɚ, ɩɪɢɧɚɞɥɟɠɚɳɢɦ ɬɨɦɭ
ɠɟ ɫɚɦɨɦɭ ɜɵɪɨɠɞɟɧɧɨɦɭ ɭɪɨɜɧɸ ɷɧɟɪɝɢɢ. ȼɨɡɦɭɳɟɧɢɟ ɩɨɪɨɠɞɚɟɬ
ɩɨɩɪɚɜɤɢ ɤ ɭɪɨɜɧɸ ɢ ɬɟɦ ɫɚɦɵɦ ɱɚɫɬɢɱɧɨ ɢɥɢ ɩɨɥɧɨɫɬɶɸ ɫɧɢɦɚɟɬ
g
nnk ,...,
1
)0(
k
C
V
ˆ
)0(
n
H
99