Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
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ɫɮɟɪɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ,
mmllmllm
YYd
cccc
GG :
³
. (9)
ɉɨɥɭɱɚɟɦ ɪɚɜɟɧɫɬɜɨ:
TTTS
S
³
S
T
)(cossin2
4
12
0
cos
l
ikr
Ped
l
)(
1
2
2/1
2
rkJ
rk
kA
ll
S
. (10)
ɂɧɬɟɝɪɚɥ ɜ (10) ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɦɟɬɨɞɨɦ, ɚɧɚɥɨɝɢɱɧɵɦ ɩɪɢɦɟɧɟɧɧɨɦɭ ɧɚɦɢ ɜ
ɡɚɞɚɱɟ 6.4. ɉɟɪɟɣɞɹ ɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɸ ɩɨ
T
K
cos ɢ ɨɛɨɡɧɚɱɢɜ
[ k
r
, (11)
ɢɦɟɟɦ:
TTT
³
S
T
)(cossin
0
cos
l
ikr
Ped KK
³
[K
)(
1
1
l
i
Ped K
¸
¸
¹
·
¨
¨
©
§
[
³
[K
1
1
ˆ
i
l
ed
d
d
iP
[
[
¸
¸
¹
·
¨
¨
©
§
[
sin
ˆ
2
d
d
iP
l
. (12)
Ɂɞɟɫɶ ) – ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɣ ɨɩɟɪɚɬɨɪ, ɩɨɥɭɱɚɸɳɢɣɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ
ɡɚɦɟɧɵ ɩɟɪɟɦɟɧɧɨɣ
/(
ˆ
[ didP
l
K ɜ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ɩɨɥɢɧɨɦɚ Ʌɟɠɚɧɞɪɚ ,)(K
l
P
l
ll
l
d
d
l
P )1(
!2
1
)(
2
K
K
K
,
ɨɩɟɪɚɬɨɪɨɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ [ did
/
. ɇɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɞɥɹ
ɧɟɫɤɨɥɶɤɢɯ ɩɟɪɜɵɯ ɡɧɚɱɟɧɢɣ
l
ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ
[
[
¸
¸
¹
·
¨
¨
©
§
[
sin
ˆ
d
d
iP
l
)(
2
2/1
[
[
S
l
l
Ji . (13)
ɋ ɭɱɟɬɨɦ (11)–(13) ɧɚɯɨɞɢɦ ɢɡ (10) ɢɫɤɨɦɵɟ ɚɦɩɥɢɬɭɞɵ :
l
A
kliA
l
l
/)12( S . (14)
Ⱦɥɹ ɜɟɥɢɱɢɧɵ (6), ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɣ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ
l
ɢ k ɜɟɪɨɹɬɧɨɫɬɢ ɬɨɝɨ,
ɱɬɨ ɭ ɱɚɫɬɢɰɵ ɜ ɫɨɫɬɨɹɧɢɢ (1) ɛɭɞɟɬ ɨɛɧɚɪɭɠɟɧ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ
l
, ɢɦɟɟɦ:
)12(
2
S
V l
k
l
. (15)
70
Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɢ 1!!
l
ɜɵɪɚɠɟɧɢɟ (15) ɩɟɪɟɯɨɞɢɬ ɜ ɢ
ɦɨɠɟɬ ɛɵɬɶ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɧɨ ɤɚɤ ɩɥɨɳɚɞɶ
2
/2 kl
l
S|V
U'SU2 ɤɨɥɶɰɚ ɪɚɞɢɭɫɚ
p
l
/
= U
ɲɢɪɢɧɨɣ
p
/
= U' . ȼɟɥɢɱɢɧɚ U ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɪɢɰɟɥɶɧɨɟ ɪɚɫɫɬɨɹɧɢɟ,
ɤɨɬɨɪɵɦ ɨɩɪɟɞɟɥɹɥɨɫɶ ɛɵ ɡɧɚɱɟɧɢɟ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
l
= ɱɚɫɬɢɰɵ ɫ ɢɦɩɭɥɶɫɨɦ
p
k = ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ:
p
l
U = .
ȼ ɥɢɬɟɪɚɬɭɪɟ ɜɦɟɫɬɨ ɮɭɧɤɰɢɣ Ȼɟɫɫɟɥɹ ɩɨɥɭɰɟɥɨɝɨ ɩɨɪɹɞɤɚ ɱɚɫɬɨ
ɩɪɢɦɟɧɹɸɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɫɮɟɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ Ȼɟɫɫɟɥɹ :
l
j
)(
2
)(
2/1
[
[
S
[
ll
Jj . (16)
ɋ ɭɱɟɬɨɦ ɨɩɪɟɞɟɥɟɧɢɹ (16) ɪɚɫɫɦɨɬɪɟɧɧɨɟ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɪɚɡɥɨɠɟɧɢɟ ɩɥɨɫɤɨɣ
ɜɨɥɧɵ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɩɪɢɧɢɦɚɟɬ ɫɥɟɞɭɸɳɢɣ
ɜɢɞ:
¦
f
0
)12(
l
lzki
lie )(cos)( T
ll
Prkj . (17)
7.7. ɚ) ɍɪɨɜɟɧɶ ɷɧɟɪɝɢɢ ɫɮɟɪɢɱɟɫɤɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ, ɛɥɢɠɚɣɲɢɣ ɤ
ɨɫɧɨɜɧɨɦɭ ɷɧɟɪɝɟɬɢɱɟɫɤɨɦɭ ɭɪɨɜɧɸ, ɬɪɟɯɤɪɚɬɧɨ ɜɵɪɨɠɞɟɧ. ɉɪɢɧɚɞɥɟɠɚɳɢɟ
ɷɬɨɦɭ ɭɪɨɜɧɸ ɜɨɡɛɭɠɞɟɧɧɵɟ ɫɨɫɬɨɹɧɢɹ ɨɫɰɢɥɥɹɬɨɪɚ ɫɨɝɥɚɫɧɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ
ɜɵɛɟɪɟɦ ɜ ɜɢɞɟ
xrf )()(
100
\ r , yrf )()(
010
\ r , zrf )()(
001
\ r , (1)
ɝɞɟ ɮɭɧɤɰɢɹ )(
r
f
ɡɚɜɢɫɢɬ ɨɬ ɤɨɨɪɞɢɧɚɬ ɬɨɥɶɤɨ ɱɟɪɟɡ ɪɚɞɢɚɥɶɧɭɸ ɩɟɪɟɦɟɧɧɭɸ
. Ɉɛɨɡɧɚɱɢɦ ɫɨɫɬɨɹɧɢɹ, ɨɩɢɫɵɜɚɟɦɵɟ ɬɪɟɦɹ ɜɨɥɧɨɜɵɦɢ
ɮɭɧɤɰɢɹɦɢ (1), ɤɚɤ
2/1222
)( zyxr
1, 2 ɢ 3, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɗɬɢ ɜɟɤɬɨɪɵ ɫɨɫɬɨɹɧɢɣ
ɧɨɪɦɢɪɨɜɚɧɵ ɢ ɜɡɚɢɦɧɨ ɨɪɬɨɝɨɧɚɥɶɧɵ, ɩɨɫɤɨɥɶɤɭ ɩɨɞɨɛɧɵɦɢ ɠɟ ɫɜɨɣɫɬɜɚɦɢ
ɨɛɥɚɞɚɸɬ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ (1).
ɉɨɞɟɣɫɬɜɭɟɦ ɧɚ ɮɭɧɤɰɢɢ (1) ɨɩɟɪɚɬɨɪɨɦ . ȼ ɯɨɞɟ
ɜɵɱɢɫɥɟɧɢɣ ɦɨɠɧɨ ɦɧɨɠɢɬɟɥɶ )
)//(
ˆ
xyyxil
z
wwww
(
r
f
ɜɵɧɨɫɢɬɶ ɢɡ ɩɨɞ ɡɧɚɤɚ , ɬɚɤ ɤɚɤ ɜ
ɪɟɡɭɥɶɬɚɬɟ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɨɜ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɧɚ ɮɭɧɤɰɢɸ, ɡɚɜɢɫɹɳɭɸ
ɨɬ ɤɨɨɪɞɢɧɚɬ ɬɨɥɶɤɨ ɱɟɪɟɡ ɪɚɞɢɚɥɶɧɭɸ ɩɟɪɟɦɟɧɧɭɸ
z
l
ˆ
r
, ɩɨɥɭɱɚɟɬɫɹ ɧɭɥɶ. ɂɦɟɟɦ:
21
ˆ
il
z
, 12
ˆ
il
z
, 03
ˆ
z
l . (2)
ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɢɜ (2) ɧɚ ɤɚɠɞɵɣ ɢɡ ɜɟɤɬɨɪɨɜ ɫɨɫɬɨɹɧɢɹ
1, 2, 3,ɩɨɥɭɱɚɟɦ
ɦɚɬɪɢɱɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɞɥɹ ɨɩɟɪɚɬɨɪɚ ɜ ɛɚɡɢɫɟ, ɨɛɪɚɡɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɹɦɢ
z
l
ˆ
1, 2 ɢ 3:
71
¸
¸
¸
¹
·
¨
¨
¨
©
§
000
00
00
ˆ
i
i
l
z
. (3)
Ⱥɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚɡɨɦ ɜɵɱɢɫɥɹɸɬɫɹ ɦɚɬɪɢɰɵ ɢ :
y
l
ˆ
x
l
ˆ
, .
¸
¸
¸
¹
·
¨
¨
¨
©
§
00
00
000
ˆ
i
il
x
¸
¸
¸
¹
·
¨
¨
¨
©
§
00
000
00
ˆ
i
i
l
y
Ⱦɢɚɝɨɧɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɩɟɪɟɱɢɫɥɟɧɧɵɯ ɦɚɬɪɢɰ ɪɚɜɧɵ ɧɭɥɸ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
0 l (4)
ɞɥɹ ɥɸɛɨɝɨ ɢɡ ɬɪɟɯ ɫɨɫɬɨɹɧɢɣ (1).
ɛ) ɉɨɫɤɨɥɶɤɭ ɜ ɪɟɡɭɥɶɬɚɬɟ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɨɜ ɩɪɨɟɤɰɢɣ ɦɨɦɟɧɬɚ ɧɚ
ɫɨɫɬɨɹɧɢɹ (1) ɧɟ ɨɛɪɚɡɭɸɬɫɹ ɧɨɜɵɟ ɮɭɧɤɰɢɢ, ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵɟ ɩɨ
ɨɬɧɨɲɟɧɢɸ ɤ (1), ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɷɬɢ ɬɪɢ ɫɨɫɬɨɹɧɢɹ ɫɨɫɬɚɜɥɹɸɬ
ɦɭɥɶɬɢɩɥɟɬ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɡɧɚɱɟɧɢɸ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ 1
l
(ɫɨɝɥɚɫɧɨ
ɪɚɜɟɧɫɬɜɭ 312
l
). Ɍɚɤɨɣ ɜɵɜɨɞ ɩɨɞɬɜɟɪɠɞɚɟɬɫɹ ɜɵɱɢɫɥɟɧɢɟɦ ɦɚɬɪɢɰɵ
ɤɜɚɞɪɚɬɚ ɦɨɦɟɧɬɚ:
2222
ˆˆˆ
ˆ
zyx
lll l
¸
¸
¸
¹
·
¨
¨
¨
©
§
200
020
002
.
ȼ ɛɚɡɢɫɟ, ɨɛɪɚɡɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɹɦɢ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɜɟɥɢɱɢɧɨɣ ɦɨɦɟɧɬɚ
l
,
ɦɚɬɪɢɰɚ ɤɜɚɞɪɚɬɚ ɦɨɦɟɧɬɚ ɞɨɥɠɧɚ ɢɦɟɬɶ ɜɢɞ 1, ɬɚɤ ɱɬɨ ɢɡ ɪɚɜɟɧɫɬɜɚ
ˆ
)1( ll
2)1(
l
l
ɦɵ ɜɧɨɜɶ ɩɨɥɭɱɚɟɦ ɡɧɚɱɟɧɢɟ 1
l
.
Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɫɨɜɩɚɞɟɧɢɟ ɦɚɬɪɢɰɵ (3) ɫ ɦɚɬɪɢɰɟɣ ɩɪɨɟɤɰɢɢ
ɦɨɦɟɧɬɚ (9) ɜ ɡɚɞɚɱɟ 3.9. Ɉɧɨ ɝɨɜɨɪɢɬ ɧɚɦ ɨ ɬɨɦ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɡɞɟɫɶ
ɫɨɫɬɨɹɧɢɹ ɨɫɰɢɥɥɹɬɨɪɚ (1) ɦɨɠɧɨ ɨɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɤɚɤ «ɥɢɧɟɣɧɨ
ɩɨɥɹɪɢɡɨɜɚɧɧɵɟ ɤɨɥɟɛɚɧɢɹ» ɱɚɫɬɢɰɵ, ɩɪɟɨɛɪɚɡɭɸɳɢɟɫɹ ɩɪɢ ɩɨɜɨɪɨɬɚɯ ɜɨɤɪɭɝ
ɨɫɢ ɩɨɞɨɛɧɨ ɨɪɬɚɦ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɋɨɫɬɨɹɧɢɟ
z 3
ɢɧɜɚɪɢɚɧɬɧɨ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɬɚɤɢɦ ɩɨɜɨɪɨɬɚɦ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢɧɚɞɥɟɠɢɬ
ɪɚɜɧɨɦɭ ɧɭɥɸ ɡɧɚɱɟɧɢɸ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ml
z
{ :
0,13 ml . (5)
ɋɨɫɬɨɹɧɢɹ
1 ɢ 2 ɧɟ ɢɦɟɸɬ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ; ɨɧɢ ɦɨɝɭɬ ɛɵɬɶ
ɡɚɩɢɫɚɧɵ ɜ ɮɨɪɦɟ ɥɢɧɟɣɧɵɯ ɤɨɦɛɢɧɚɰɢɣ «ɰɢɪɤɭɥɹɪɧɨ ɩɨɥɹɪɢɡɨɜɚɧɧɵɯ
m
72
ɤɨɥɟɛɚɧɢɣ» ɫ 1, ɬɨ ɟɫɬɶ – ɜ ɜɢɞɟ ɫɭɩɟɪɩɨɡɢɰɢɢ ɫɨɫɬɨɹɧɢɣr m 1,1 r ml . ȼ
ɫɜɨɸ ɨɱɟɪɟɞɶ, ɫɨɫɬɨɹɧɢɹ
1,1 r ml ɦɨɠɧɨ ɪɚɡɥɨɠɢɬɶ ɩɨ ɛɚɡɢɫɧɵɦ ɜɟɤɬɨɪɚɦ
1 ɢ 2 ɫ ɩɨɦɨɳɶɸ ɮɨɪɦɭɥ, ɚɧɚɥɨɝɢɱɧɵɯ (7) ɜ ɡɚɞɚɱɟ 7.4:
21
2
1
1,1 iml
,
21
2
1
1,1 iml
. (6)
ɂɫɤɨɦɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɡɧɚɱɟɧɢɣ
ml
W
l
ɢ ɜ ɫɨɫɬɨɹɧɢɹɯm k (ɝɞɟ
3,2,1 k ) ɪɚɜɧɚ ɤɜɚɞɪɚɬɭ ɦɨɞɭɥɹ ɚɦɩɥɢɬɭɞɵ ɜɟɪɨɹɬɧɨɫɬɢ
kml,.Ɍɚɤɢɦ
ɨɛɪɚɡɨɦ,
ɜ ɫɨɫɬɨɹɧɢɢ1
10
W 3,
2/1
1,111
WW ɜ ɫɨɫɬɨɹɧɢɹɯ 1 ɢ 2 . (7)
ȼɟɪɨɹɬɧɨɫɬɢ ɞɥɹ ɞɪɭɝɢɯ ɡɧɚɱɟɧɢɣ
ml
W
l
ɢ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɫɨɫɬɨɹɧɢɹɯ
ɪɚɜɧɵ ɧɭɥɸ.
m
Ɍɟ ɠɟ ɪɟɡɭɥɶɬɚɬɵ ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ ɢ ɧɟ ɨɛɪɚɳɚɹɫɶ ɤ ɪɟɲɟɧɢɹɦ
ɩɪɟɞɲɟɫɬɜɭɸɳɢɯ ɡɚɞɚɱ. ɇɚɩɪɢɦɟɪ, ɩɨɥɶɡɭɹɫɶ ɞɚɧɧɵɦɢ ɨ ɞɟɣɫɬɜɢɢ ɨɩɟɪɚɬɨɪɨɜ
ɢ ɧɚ ɛɚɡɢɫɧɨɟ ɫɨɫɬɨɹɧɢɟ
x
l
ˆ
y
l
ˆ
3,ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɤ (5) ɨɩɟɪɚɬɨɪɵ
yx
lill
ˆˆˆ
, .
yx
lill
ˆˆˆ
Ɍɚɤɨɟ ɜɵɱɢɫɥɟɧɢɟ ɫ ɭɱɟɬɨɦ ɫɨɨɬɧɨɲɟɧɢɣ (XIII) ɢɡ ɜɜɟɞɟɧɢɹ ɤ ɞɚɧɧɨɣ ɝɥɚɜɟ ɩɪɢ
l
j ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜɟɞɟɬ ɤ ɮɨɪɦɭɥɚɦ (6). ȿɳɟ ɨɞɢɧ ɫɩɨɫɨɛ ɩɨɥɭɱɟɧɢɹ (6)
ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɫɪɚɜɧɟɧɢɢ (1) ɫ ɹɜɧɵɦɢ ɜɵɪɚɠɟɧɢɹɦɢ ɞɥɹ )ɩɨɫɥɟ ɬɨɝɨ
ɤɚɤ ɮɭɧɤɰɢɢ (1) ɛɭɞɭɬ ɡɚɩɢɫɚɧɵ ɜ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ.
,( MT
ml
Y
7.8. ɍɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ ɞɥɹ ɱɚɫɬɢɰɵ ɦɚɫɫɵ P
0)(
2
2
\H
P
\' rU
=
(1)
ɫ ɭɤɚɡɚɧɧɵɦ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɩɨɬɟɧɰɢɚɥɨɦ ɡɚɩɢɲɟɦ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ (ɫɦ. ɡɚɞɚɱɭ 4.10):
\
¸
¸
¹
·
¨
¨
©
§
w
w
Mw
w
w
w
w
w
2
2
2
2
2
11
zr
r
r
rr
0
2
2
\
¸
¹
·
¨
©
§
D
H
P
r
=
. (2)
73
ɉɪɢ ɞɜɭɦɟɪɧɨɦ ɞɜɢɠɟɧɢɢ ɱɚɫɬɢɰɵ ɜ ɧɚɩɪɚɜɥɟɧɢɹɯ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɤ ɨɫɢ ,
ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ )
z
,( M\
r
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɤɨɨɪɞɢɧɚɬɵ . ȿɫɥɢ ɬɚɤɭɸ ɮɭɧɤɰɢɸ
ɢɫɤɚɬɶ ɜ ɜɢɞɟ
z
M
K
M\
mi
e
r
r
r
)(
),(
, (3)
ɬɨ ɞɥɹ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɮɭɧɤɰɢɢ )(
r
K ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɟɟ ɭɪɚɜɧɟɧɢɟ:
0
24/12
22
2
22
2
K
¸
¸
¹
·
¨
¨
©
§
DP
HP
K
r
r
m
rd
d
==
. (4)
ɂɡɜɟɫɬɧɨ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ ɒɪɟɞɢɧɝɟɪɚ (1) ɫ ɬɪɟɯɦɟɪɧɵɦ ɤɭɥɨɧɨɜɫɤɢɦ
ɩɨɬɟɧɰɢɚɥɨɦ
r
U
/
D ɩɨɞɫɬɚɧɨɜɤɨɣ ),()(
1
MTF \
lm
Yrr ɩɪɢɜɨɞɢɬɫɹ ɤ ɜɢɞɭ
0
2)1(
2
2222
2
F
¸
¹
·
¨
©
§
DP
HP
F
r
r
ll
rd
d
==
. (5)
ɉɪɢ ɭɫɥɨɜɢɢ ɨɝɪɚɧɢɱɟɧɧɨɫɬɢ ɢɫɤɨɦɨɣ ɮɭɧɤɰɢɢ F ɭɪɚɜɧɟɧɢɟ (5) ɞɚɟɬ
«ɜɨɞɨɪɨɞɨɩɨɞɨɛɧɵɣ» ɫɩɟɤɬɪ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ:
22
2
1
2 n
n
=
PD
H , (6)
ɝɞɟ
...,3,2,11 { lnn
r
. (7)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɡɚɦɟɧɟ
l
ɧɚ 2/1m ɭɪɚɜɧɟɧɢɟ (5) ɩɟɪɟɯɨɞɢɬ ɜ ɭɪɚɜɧɟɧɢɟ (4),
ɢɧɬɟɪɟɫɭɸɳɟɟ ɧɚɫ ɜ ɫɜɹɡɢ ɫ ɚɧɚɥɨɝɢɱɧɨɣ ɞɜɭɦɟɪɧɨɣ ɡɚɞɚɱɟɣ ɨ ɫɜɹɡɚɧɧɵɯ
ɫɨɫɬɨɹɧɢɹɯ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɫɤɨɦɵɟ ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ ɦɨɠɧɨ ɧɚɣɬɢ
ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɨ ɮɨɪɦɭɥɚɦ (6) ɢ (7), ɡɚɦɟɧɢɜ ɜ (7) ɜɟɥɢɱɢɧɭ
H
l
ɧɚ 2/1m :
22
2
,
)2/1(
1
2
PD
H
mn
r
mn
r
=
. (8)
ɋɨɝɥɚɫɧɨ ɨɫɰɢɥɥɹɰɢɨɧɧɨɣ ɬɟɨɪɟɦɟ
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɱɢɫɥɨ ɭɡɥɨɜ
ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ (3) ɩɨ ɩɟɪɟɦɟɧɧɨɣ
r
n
r
ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɹ
0, 1, 2, … . ɋɩɟɤɬɪ ɡɧɚɱɟɧɢɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɫɥɨɜɢɟɦ ɨɞɧɨɡɧɚɱɧɨɫɬɢ ɜɨɥɧɨɜɨɣ
ɮɭɧɤɰɢɢ (3), ɜ ɫɢɥɭ ɤɨɬɨɪɨɝɨ )
m
,( M\
r
ɩɨ ɩɟɪɟɦɟɧɧɨɣ ɞɨɥɠɧɚ ɛɵɬɶ
ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ ɫ ɩɟɪɢɨɞɨɦ
M
S2.ɂɡ ɷɬɨɝɨ ɫɥɟɞɭɟɬ, ɱɬɨ
...,2,1,0 rr m . (9)
74
ɗɧɟɪɝɟɬɢɱɟɫɤɢɣ ɫɩɟɤɬɪ (8) ɜɵɪɨɠɞɟɧ: ɫɨɫɬɨɹɧɢɹ ɫ ɪɚɡɥɢɱɧɵɦɢ
ɡɧɚɱɟɧɢɹɦɢ ɢ , ɧɨ ɫ ɨɞɧɢɦ ɢ ɬɟɦ ɠɟ ɡɧɚɱɟɧɢɟɦ ɫɭɦɦɵ +
r
nm
r
n m ɢɦɟɸɬ
ɨɞɢɧɚɤɨɜɭɸ ɷɧɟɪɝɢɸ. ɐɟɥɟɫɨɨɛɪɚɡɧɨ, ɩɨ ɚɧɚɥɨɝɢɢ ɫ (7), ɜɜɟɫɬɢ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ
«ɝɥɚɜɧɨɟ» ɤɜɚɧɬɨɜɨɟ ɱɢɫɥɨ
N
, ɧɭɦɟɪɭɸɳɟɟ ɪɚɡɥɢɱɧɵɟ ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ (8) ɢ
ɩɪɢɧɢɦɚɸɳɟɟ ɰɟɥɨɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ:
1 mnN
r
. (10)
Ɍɨɝɞɚ (8) ɨɤɨɧɱɚɬɟɥɶɧɨ ɡɚɩɢɲɟɬɫɹ ɜ ɜɢɞɟ
22
2
)2/1(
1
2
PD
H
N
N
=
, ...,3,2,1
N
. (11)
Ʉɪɚɬɧɨɫɬɶ ɜɵɪɨɠɞɟɧɢɹ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɭɪɨɜɧɹ
N
H ɪɚɜɧɚ 21
N
.
Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɤɜɚɧɬɨɜɨɝɨ ɱɢɫɥɚ ɜɢɞɟɧ ɢɡ ɬɨɝɨ, ɱɬɨ ɩɪɢ ɡɚɦɟɧɟ
, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɞɟɣɫɬɜɢɸ ɨɩɟɪɚɬɨɪɚ ɩɨɜɨɪɨɬɚ
m
DMoM
z
li
z
eR
ˆ
)(
ˆ
D
De
ɧɚ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ (3), ɩɨɫɥɟɞɧɹɹ ɥɢɲɶ ɩɪɢɨɛɪɟɬɚɟɬ ɮɚɡɨɜɵɣ ɦɧɨɠɢɬɟɥɶ
. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɜɢɞɚ (3) ɹɜɥɹɟɬɫɹ ɫɨɛɫɬɜɟɧɧɨɣ
ɮɭɧɤɰɢɟɣ ɨɩɟɪɚɬɨɪɚ ɩɪɨɟɤɰɢɢ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ
ɨɩɪɟɞɟɥɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ .
mi
e
D
z
l
ˆ
ml
z
75
Ƚɥɚɜɚ 8. ɋɉɂɇ
ɋɩɢɧ ɱɚɫɬɢɰɵ – ɷɬɨ ɟɟ ɫɨɛɫɬɜɟɧɧɵɣ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ, ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ
ɧɟ ɫɜɹɡɚɧɧɵɣ ɫ ɩɟɪɟɦɟɳɟɧɢɟɦ ɱɚɫɬɢɰɵ ɜ -ɩɪɨɫɬɪɚɧɫɬɜɟ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟ
ɜɵɪɚɠɚɸɳɢɣɫɹ ɱɟɪɟɡ ɤɨɨɪɞɢɧɚɬɵ ɢ ɢɦɩɭɥɶɫ. ɑɚɫɬɢɰɚ ɫɨ ɫɩɢɧɨɦ
r
s
ɦɨɠɟɬ
ɧɚɯɨɞɢɬɶɫɹ ɜ
12
s
ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɹɯ V ɫ ɨɩɪɟɞɟɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ
ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ
,V{
z
s
s
s
s
V ...,,1, . (I)
Ɍɚɤɢɟ ɜɟɤɬɨɪɵ ɫɨɫɬɨɹɧɢɹ ɨɛɪɚɡɭɸɬ ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɧɵɣ ɛɚɡɢɫ ɜ (2 1
s
)-
ɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɣ ɱɚɫɬɢɰɵ.
ɂɡ ɨɩɵɬɚ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɞɥɹ ɱɚɫɬɢɰ ɤɚɠɞɨɝɨ ɜɢɞɚ ɜɟɥɢɱɢɧɚ ɫɩɢɧɚ
s
ɹɜɥɹɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɧɚɪɹɞɭ ɫ ɦɚɫɫɨɣ ɢ ɷɥɟɤɬɪɢɱɟɫɤɢɦ
ɡɚɪɹɞɨɦ. ɋɩɢɧ ɷɥɟɤɬɪɨɧɚ, ɩɪɨɬɨɧɚ, ɧɟɣɬɪɨɧɚ ɢ ɪɹɞɚ ɞɪɭɝɢɯ ɦɢɤɪɨɱɚɫɬɢɰ ɪɚɜɟɧ
1/2. ɉɪɢ 2
/
1
s
(ɧɢɠɟ ɪɟɱɶ ɢɞɟɬ, ɜ ɨɫɧɨɜɧɨɦ, ɨɛ ɷɬɨɦ ɫɥɭɱɚɟ) ɩɪɨɟɤɰɢɹ ɫɩɢɧɚ
ɩɪɢɧɢɦɚɟɬ ɞɜɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɯ ɡɧɚɱɟɧɢɹ, ɪɚɜɧɵɯ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ
ɫɚɦɨɦɭ ɫɩɢɧɭ. ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɝɨɜɨɪɹɬ, ɱɬɨ ɷɥɟɤɬɪɨɧ ɦɨɠɟɬ ɧɚɯɨɞɢɬɶɫɹ ɜ
ɫɨɫɬɨɹɧɢɢ ɫɨ ɫɩɢɧɨɦ ɜɞɨɥɶ ɨɫɢ ɢɥɢ ɜ ɫɨɫɬɨɹɧɢɢ ɫɨ ɫɩɢɧɨɦ ɩɪɨɬɢɜ ɨɫɢ , ɢ
ɩɪɢɦɟɧɹɸɬ ɨɛɨɡɧɚɱɟɧɢɹ
V
zz
n , p ɞɥɹ ɫɨɫɬɨɹɧɢɣ 2/1 V , 2/1 V . Ɉɞɧɚɤɨ
ɫɥɟɞɭɟɬ ɢɦɟɬɶ ɜɜɢɞɭ, ɱɬɨ ɨɬɞɟɥɶɧɨ ɜɡɹɬɨɣ ɱɚɫɬɢɰɟ ɧɟɜɨɡɦɨɠɧɨ ɩɪɢɩɢɫɚɬɶ
ɤɥɚɫɫɢɱɟɫɤɢ ɩɨɧɢɦɚɟɦɵɣ ɜɟɤɬɨɪ ɫɨɛɫɬɜɟɧɧɨɝɨ ɦɟɯɚɧɢɱɟɫɤɨɝɨ ɦɨɦɟɧɬɚ , ɬɚɤ
ɱɬɨ, ɜ ɱɚɫɬɧɨɫɬɢ, ɢɧɨɝɞɚ ɭɩɨɦɢɧɚɟɦɨɣ ɨɛɪɚɡɧɨɣ «ɦɨɞɟɥɢ» ɷɥɟɤɬɪɨɧɚ ɜ ɜɢɞɟ
ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɥɱɤɚ ɧɟɥɶɡɹ ɩɪɢɞɚɬɶ ɬɨɱɧɨɝɨ ɫɦɵɫɥɚ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ,
ɜɫɥɟɞɫɬɜɢɟ ɧɟɤɨɦɦɭɬɚɬɢɜɧɨɫɬɢ ɨɩɟɪɚɬɨɪɨɜ ɩɪɨɟɤɰɢɣ ɫɩɢɧɚ,
s
zyx
siss
ˆ
]
ˆ
,
ˆ
[
,
xzy
siss
ˆ
]
ˆ
,
ˆ
[
,
yxz
siss
ˆ
]
ˆ
,
ˆ
[
, (II)
ɷɥɟɤɬɪɨɧ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ
V
z
s ɧɟ ɢɦɟɟɬ ɨɩɪɟɞɟɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
, ɢ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɟ ɨɛɥɚɞɚɟɬ ɨɩɪɟɞɟɥɟɧɧɵɦ ɜɟɤɬɨɪɨɦ ɫɩɢɧɚ . ɇɨ
ɭɫɪɟɞɧɟɧɧɵɣ ɩɨ ɤɜɚɧɬɨɜɨɦɭ ɚɧɫɚɦɛɥɸ ɜɟɤɬɨɪ ɫɩɢɧɚ
y
s
z
s s
s ɢɦɟɟɬ ɱɟɬɤɢɣ ɫɦɵɫɥ –
ɞɥɹ ɷɥɟɤɬɪɨɧɚ ɜ ɥɸɛɨɦ ɧɨɪɦɢɪɨɜɚɧɧɨɦ ɫɩɢɧɨɜɨɦ ɫɨɫɬɨɹɧɢɢ
F ,
pFnF F
pn
, (III)
ɭɫɪɟɞɧɟɧɧɵɣ ɜɟɤɬɨɪ ɫɩɢɧɚ ɜɵɱɢɫɥɹɟɬɫɹ ɤɚɤ ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɜɟɤɬɨɪɧɨɝɨ
ɨɩɟɪɚɬɨɪɚ ɫɩɢɧɚ:
FF ss
ˆ
, (IV)
ɝɞɟ
zzyyxx
sss
ˆˆˆ
ˆ
eees
, (V)
76
xx
s V
ˆ
2
1
ˆ
,
yy
s V
ˆ
2
1
ˆ
,
zz
s V
ˆ
2
1
ˆ
, (VI)
x
V
ˆ
,, – ɦɚɬɪɢɰɵ ɉɚɭɥɢ:
y
V
ˆ
z
V
ˆ
¸
¸
¹
·
¨
¨
©
§
V
01
10
ˆ
x
,
¸
¸
¹
·
¨
¨
©
§
V
0
0
ˆ
i
i
y
,
¸
¸
¹
·
¨
¨
©
§
V
10
01
ˆ
z
. (VII)
ɍɤɚɡɚɧɧɨɟ ɦɚɬɪɢɱɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨɩɟɪɚɬɨɪɨɜ ɩɪɨɟɤɰɢɣ ɫɩɢɧɚ ɫɥɟɞɭɟɬ ɢɡ
ɮɨɪɦɭɥ (XII) ɜɜɟɞɟɧɢɹ ɤ ɝɥɚɜɟ 7 ɩɪɢ
2
/
1
s
j . ȼ ɪɚɡɜɟɪɧɭɬɨɣ ɮɨɪɦɟ
ɤɨɦɩɨɧɟɧɬɵ ɭɫɪɟɞɧɟɧɧɨɝɨ ɜɟɤɬɨɪɚ ɫɩɢɧɚ (IV) ɞɚɸɬɫɹ ɜɵɪɚɠɟɧɢɹɦɢ:
)()2/1(
**
nppn
FFFF
x
s ,
)()2/(
**
nppn
FFFF is
y
,
)()2/1(
22
pn
FF
z
s . (VIII)
ȼɟɥɢɱɢɧɚ ɜɟɤɬɨɪɚ ɫɩɢɧɚ ɫ ɤɨɦɩɨɧɟɧɬɚɦɢ (VIII) ɩɪɢ ɭɱɟɬɟ ɭɫɥɨɜɢɹ
ɧɨɪɦɢɪɨɜɤɢ,
1
22
FF
pn
, (IX)
ɪɚɜɧɚ 1/2, ɩɨɷɬɨɦɭ ɭɫɪɟɞɧɟɧɧɵɣ ɜɟɤɬɨɪ ɫɩɢɧɚ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɮɨɪɦɟ
ns
2
1
,
ɝɞɟ
F n ı
ˆ
F – ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ:1 nn . ɋɩɢɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ F
ɹɜɥɹɟɬɫɹ ɫɨɫɬɨɹɧɢɟɦ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɧɚ
ɧɚɩɪɚɜɥɟɧɢɟ , ɪɚɜɧɵɦ
n 2/1
n
s ; ɩɪɢ ɷɬɨɦ ɩɪɨɟɤɰɢɹ ɫɩɢɧɚ ɧɚ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɪɚɜɧɚ 2
/
1 . ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɞɥɹ ɥɸɛɨɝɨ
ɫɩɢɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ
F ɬɚɤɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ, ɧɚ ɤɨɬɨɪɨɟ ɩɪɨɟɤɰɢɹ ɫɩɢɧɚ ɜ
ɞɚɧɧɨɦ ɫɨɫɬɨɹɧɢɢ ɢɦɟɟɬ ɨɩɪɟɞɟɥɟɧɧɨɟ ɡɧɚɱɟɧɢɟ (ɪɚɜɧɨɟ 2
/
1),ɩɪɟɞɫɬɚɜɥɹɟɬ
ɫɨɛɨɣ ɯɚɪɚɤɬɟɪɧɨɟ ɫɜɨɣɫɬɜɨ ɫɩɢɧɚ 2
/
1
s
.
Ɉɩɟɪɚɬɨɪɵ ɩɨɜɨɪɨɬɨɜ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɫɩɢɧɨɜɵɟ ɫɨɫɬɨɹɧɢɹ ɫ 2
/
1
s
, ɜ
ɛɚɡɢɫɟ
n , p , ɫɜɹɡɚɧɧɨɦ ɫ ɨɫɶɸ , ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:z
)(
ˆ
x
R eD
¸
¸
¹
·
¨
¨
©
§
DD
DD
)2/(cos)2/(sin
)2/(sin)2/(cos
i
i
,
77
)(
ˆ
y
R eD
¸
¸
¹
·
¨
¨
©
§
DD
DD
)2/(cos)2/(sin
)2/(sin)2/(cos
,
)(
ˆ
z
R eD
¸
¸
¹
·
¨
¨
©
§
D
D
)2/(exp0
0)2/(exp
i
i
. (X)
ɋɸɞɚ ɜɯɨɞɢɬ ɭɝɨɥ ɩɨɜɨɪɨɬɚ , ɞɟɥɟɧɧɵɣ ɧɚ ɞɜɚ, ɬɚɤ ɱɬɨ ɩɪɢDS D 2 ɷɬɢ
ɦɚɬɪɢɰɵ ɪɚɜɧɵ 1, ɚ ɧɟ ɟɞɢɧɢɱɧɨɣ ɦɚɬɪɢɰɟ 1. Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɫɩɢɧɨɜɨɟ
ɫɨɫɬɨɹɧɢɟ
ˆ
ˆ
F ɩɪɢ ɩɨɜɨɪɨɬɟ ɧɚ ɭɝɨɥ S2 ɜɨɤɪɭɝ ɥɸɛɨɣ ɨɫɢ ɩɟɪɟɯɨɞɢɬ ɜ F .
ɉɨɫɤɨɥɶɤɭ ɬɚɤɨɣ ɡɚɤɨɧ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɩɪɚɜɢɥ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɪɢ ɜɪɚɳɟɧɢɹɯ ɨɛɵɱɧɵɯ ɜɟɤɬɨɪɨɜ, ɞɥɹ ɫɩɢɧɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ
F ɩɪɢɧɹɬ ɫɩɟɰɢɚɥɶɧɵɣ ɬɟɪɦɢɧ – ɫɩɢɧɨɪ. ȼ ɭɤɚɡɚɧɧɨɣ ɧɟɨɞɧɨɡɧɚɱɧɨɫɬɢ
ɫɩɢɧɨɪɨɜ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɜɪɚɳɟɧɢɹɦ ɧɚ ɭɝɨɥ S2 ɫɤɚɡɵɜɚɟɬɫɹ ɬɨɬ ɮɚɤɬ, ɱɬɨ
ɫɩɢɧ ɹɜɥɹɟɬɫɹ ɜɧɭɬɪɟɧɧɟɣ ɫɬɟɩɟɧɶɸ ɫɜɨɛɨɞɵ ɱɚɫɬɢɰɵ ɢ ɧɟ ɨɩɢɫɵɜɚɟɬɫɹ
ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɟɣ ɜ ɤɨɨɪɞɢɧɚɬɧɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɢ; ɢɦɟɧɧɨ ɩɨɷɬɨɦɭ ɡɞɟɫɶ ɧɟ
ɜɨɡɧɢɤɚɟɬ ɬɪɟɛɨɜɚɧɢɹ ɨɞɧɨɡɧɚɱɧɨɫɬɢ, ɤɨɬɨɪɨɟ ɜ ɩɪɢɦɟɧɟɧɢɢ ɤ ɜɨɥɧɨɜɨɣ
ɮɭɧɤɰɢɢ )ɜɟɞɟɬ ɤ ɞɨɩɭɫɬɢɦɨɫɬɢ ɥɢɲɶ ɰɟɥɨɱɢɫɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ
ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɢ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɯ
ɫɩɢɧɨɪɚ ɩɨɫɪɟɞɫɬɜɨɦ ɦɚɬɪɢɰ (X) ɤɨɦɩɨɧɟɧɬɵ ɭɫɪɟɞɧɟɧɧɨɝɨ ɜɟɤɬɨɪɚ ɫɩɢɧɚ
(VIII) ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɤɚɤ ɩɪɨɟɤɰɢɢ ɨɛɵɱɧɨɝɨ ɬɪɟɯɦɟɪɧɨɝɨ ɜɟɤɬɨɪɚ.
(
r\
ɋɩɢɧ – ɨɞɧɨ ɢɡ ɧɚɢɛɨɥɟɟ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɩɨɧɹɬɢɣ ɜ ɤɜɚɧɬɨɜɨɣ ɬɟɨɪɢɢ.
əɪɤɢɦ ɩɨɞɬɜɟɪɠɞɟɧɢɟɦ ɬɨɦɭ ɫɥɭɠɢɬ ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɫɜɹɡɶ ɫɩɢɧɚ ɫɨ
ɫɬɚɬɢɫɬɢɤɨɣ: ɱɚɫɬɢɰɵ ɫ ɩɨɥɭɰɟɥɵɦ
ɫɩɢɧɨɦ – ɮɟɪɦɢɨɧɵ, ɚ ɫ ɰɟɥɨɱɢɫɥɟɧɧɵɦ
ɫɩɢɧɨɦ – ɛɨɡɨɧɵ. Ȼɨɡɨɧɚɦɢ ɧɚɡɵɜɚɸɬ ɱɚɫɬɢɰɵ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɜɢɞɚ
(ɬɨɠɞɟɫɬɜɟɧɧɵɟ ɱɚɫɬɢɰɵ), ɩɨɞɱɢɧɹɸɳɢɟɫɹ ɫɬɚɬɢɫɬɢɤɟ Ȼɨɡɟ–ɗɣɧɲɬɟɣɧɚ. Ɉɧɢ
ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɬɟɦ, ɱɬɨ ɢɯ ɤɜɚɧɬɨɜɵɟ ɫɨɫɬɨɹɧɢɹ ɫɢɦɦɟɬɪɢɱɧɵ ɩɨ ɨɬɧɨɲɟɧɢɸ
ɤ ɩɟɪɟɫɬɚɧɨɜɤɟ ɩɨɥɧɵɯ ɧɚɛɨɪɨɜ ɤɜɚɧɬɨɜɵɯ ɱɢɫɟɥ ɱɚɫɬɢɰ (ɢɥɢ ɭɫɥɨɜɧɨ
ɜɜɨɞɢɦɵɯ ɧɨɦɟɪɨɜ ɱɚɫɬɢɰ); ɞɥɹ ɞɜɭɯ ɱɚɫɬɢɰ ɫ ɤɜɚɧɬɨɜɵɦɢ ɱɢɫɥɚɦɢ
ɢ
ɭɤɚɡɚɧɧɨɟ ɬɪɟɛɨɜɚɧɢɟ ɫɢɦɦɟɬɪɢɢ ɢɦɟɟɬ ɜɢɞ:
a
b
baab ,, . (XI)
Ɏɟɪɦɢɨɧɚɦɢ ɧɚɡɵɜɚɸɬ ɬɨɠɞɟɫɬɜɟɧɧɵɟ ɱɚɫɬɢɰɵ, ɩɨɞɱɢɧɹɸɳɢɟɫɹ ɫɬɚɬɢɫɬɢɤɟ
Ɏɟɪɦɢ–Ⱦɢɪɚɤɚ. Ɉɧɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɬɟɦ, ɱɬɨ ɢɯ ɤɜɚɧɬɨɜɵɟ ɫɨɫɬɨɹɧɢɹ
ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɵ:
baab ,, . (XII)
ɋɭɳɟɫɬɜɟɧɧɨɟ ɪɚɡɥɢɱɢɟ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɛɨɡɨɧɨɜ ɢ ɮɟɪɦɢɨɧɨɜ ɫɨɫɬɨɢɬ
ɭɠɟ ɜ ɬɨɦ, ɱɬɨ ɛɨɡɨɧɵ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɫɨɫɬɨɹɧɢɢ
0, zaa ɫ ɨɞɢɧɚɤɨɜɵɦɢ
ɤɜɚɧɬɨɜɵɦɢ ɱɢɫɥɚɦɢ ɱɚɫɬɢɰ, ɬɨɝɞɚ ɤɚɤ ɜ ɫɥɭɱɚɟ ɮɟɪɦɢɨɧɨɜ ɢɡ (XII) ɫɥɟɞɭɟɬ, ɱɬɨ
aaaa ,, , ɬɨ ɟɫɬɶ 0, aa . Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɜ ɥɸɛɨɣ ɫɢɫɬɟɦɟ ɬɨɠɞɟɫɬ-
78
ɜɟɧɧɵɯ ɮɟɪɦɢɨɧɨɜ ɧɟ ɦɨɝɭɬ ɨɞɧɨɜɪɟɦɟɧɧɨ ɧɚɯɨɞɢɬɶɫɹ ɜ ɨɞɢɧɚɤɨɜɵɯ ɫɨɫɬɨɹ-
ɧɢɹɯ ɞɜɟ (ɢɥɢ ɛɨɥɟɟ) ɱɚɫɬɢɰɵ; ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɩɪɢɧɰɢɩɨɦ ɉɚɭɥɢ.
Ɂɚɞɚɱɢ
8.1. Ⱦɥɹ ɱɚɫɬɢɰɵ ɫɨ ɫɩɢɧɨɦ 1/2 ɧɚɣɬɢ ɫɨɫɬɨɹɧɢɹ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ
ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ ɧɚ ɨɫɶ
x
, ɩɨɥɶɡɭɹɫɶ ɛɚɡɢɫɧɵɦɢ ɫɩɢɧɨɜɵɦɢ ɫɨɫɬɨɹɧɢɹɦɢ ɫ
ɨɩɪɟɞɟɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɩɪɨɟɤɰɢɢ ɫɩɢɧɚ ɧɚ ɨɫɶ .
z
8.2. ȼ ɫɥɭɱɚɟ ɱɚɫɬɢɰɵ ɫɨ ɫɩɢɧɨɦ 1/2 ɧɚɣɬɢ: ɚ) ɫɨɫɬɨɹɧɢɹ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ
ɩɪɨɟɤɰɢɟɣ ɫɩɢɧɚ ; ɛ) ɜɟɪɨɹɬɧɨɫɬɶ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ ɜ ɫɨɫɬɨɹɧɢɹɯ ɫ
ɨɩɪɟɞɟɥɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ .
y
s
x
s
y
s
8.3. Ⱦɥɹ ɡɚɞɚɧɢɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ
ɤɨɨɪɞɢɧɚɬ zyx
~
,
~
,
~
ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɟɪɜɨɧɚɱɚɥɶɧɵɯ ɞɟɤɚɪɬɨɜɵɯ ɨɫɟɣ zy
x
,,
ɧɟɨɛɯɨɞɢɦɨ ɭɤɚɡɚɬɶ ɬɪɢ ɭɝɥɚ. Ɉɩɪɟɞɟɥɢɦ ɬɚɤɢɟ ɭɝɥɵ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ɩɭɫɬɶ ɢ ɹɜɥɹɸɬɫɹ ɭɝɥɨɜɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɨɫɢTM z
~
ɜ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ
ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɨɣ ɫ zy
x
,,,ɚ ɭɝɨɥ J ɡɚɞɚɟɬ ɩɨɥɨɠɟɧɢɟ ɨɫɟɣ yx
~
,
~
ɜ ɩɥɨɫɤɨɫɬɢ
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɨɫɢ z
~
(ɫɦ. ɪɢɫ. 8.1).
z
ɚ) ȼɵɩɢɫɚɬɶ ɜ ɫɩɢɧɨɪɧɨɦ ɛɚɡɢɫɟ
2/1,2/1 r
z
ss ɦɚɬɪɢɰɭ ɜɪɚɳɟɧɢɹ
x
~
y
y
~
z
~
-
M
J
R
ˆ
,
ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɬɪɟɯ
ɩɨɜɨɪɨɬɨɜ ɧɚ ɭɝɥɵ MTJ ,,,ɜ ɪɟɡɭɥɶɬɚɬɟ ɤɨɬɨɪɵɯ
ɨɫɢ zy
x
,, ɩɟɪɟɯɨɞɹɬ ɜ zyx
~
,
~
,
~
.
x
ɛ) ɇɚɣɬɢ ɭɫɪɟɞɧɟɧɧɵɣ ɜɟɤɬɨɪ ɫɩɢɧɚ s ɜ
ɫɨɫɬɨɹɧɢɢ, ɨɛɪɚɡɭɸɳɟɦɫɹ ɢɡ ɫɨɫɬɨɹɧɢɹ ɫɨ
ɫɩɢɧɨɦ ɜɞɨɥɶ ɨɫɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɭɤɚɡɚɧɧɨɝɨ
ɜɵɲɟ ɜɪɚɳɟɧɢɹ
z
R
ˆ
.
Ɋɢɫ. 8.1
8.4. Ƚɚɦɢɥɶɬɨɧɢɚɧ ɱɚɫɬɢɰɵ ɫ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɡɚɪɹɞɨɦ , ɫɩɢɧɨɦq
s
ɢ
ɫɩɢɧɨɜɵɦ ɦɚɝɧɢɬɧɵɦ ɦɨɦɟɧɬɨɦ
0
P ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ ɢɦɟɟɬ
ɜɢɞ (ɜ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɤɜɚɧɬɨɜɨɣ ɬɟɨɪɢɢ):
B
BsrA
P
¸
¹
·
¨
©
§
ˆ
)(
2
1
ˆ
0
2
s
U
c
q
i
m
H = ,
ɝɞɟ , – ɦɚɫɫɚ ɱɚɫɬɢɰɵ. ɈɩɟɪɚɬɨɪAB u m
H
ˆ
ɷɬɨɝɨ ɜɢɞɚ ɧɚɡɵɜɚɸɬ
ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ ɉɚɭɥɢ. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɜ ɨɞɧɨɪɨɞɧɨɦ ɦɚɝɧɢɬɧɨɦ ɩɨɥɟ
ɞɨɩɭɫɤɚɟɬɫɹ ɪɚɡɞɟɥɶɧɨɟ ɨɩɢɫɚɧɢɟ ɨɪɛɢɬɚɥɶɧɵɯ ɢ ɫɩɢɧɨɜɵɯ ɫɨɫɬɨɹɧɢɣ ɱɚɫɬɢɰɵ.
B
79