Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
Подождите немного. Документ загружается.
J \ ()
x
e
x
J
ˆ
y
e
y
J
ˆ
z
e
z
J
ˆ
\
x
e
x
J
y
e
y
J
z
e
z
J . (1)
Ɂɞɟɫɶ ɧɚɩɪɚɜɥɟɧɢɟ ɨɪɬɨɜ ,, ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɧɟ
ɤɨɧɤɪɟɬɢɡɢɪɨɜɚɧɨ ɢ ɦɨɠɟɬ ɛɵɬɶ ɜɵɛɪɚɧɨ ɩɪɨɢɡɜɨɥɶɧɵɦ ɨɛɪɚɡɨɦ. ȼɵɛɟɪɟɦ
ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɬɚɤ, ɱɬɨɛɵ ɨɫɶ ɛɵɥɚ ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɡɚɪɚɧɟɟ ɡɚɞɚɧɧɨɝɨ
ɟɞɢɧɢɱɧɨɝɨ ɜɟɤɬɨɪɚ , ɬɨ ɟɫɬɶ ɩɨɥɨɠɢɦ
x
e
y
e
z
e
z
n
z
e n . (2)
ɉɨ ɨɬɧɨɲɟɧɢɸ ɤ ɬɚɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɭɤɚɡɚɧɧɨɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ
ɫɨɫɬɨɹɧɢɟ
\ mj
~
, ɹɜɥɹɟɬɫɹ ɨɛɵɱɧɵɦ ɫɨɫɬɨɹɧɢɟɦ mj, ɢ ɩɨɷɬɨɦɭ
ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɪɚɜɟɧɫɬɜɚɦɢ
x
J 0
y
J , mJ
z
. (3)
ɍɱɢɬɵɜɚɹ (2) ɢ (3) ɜ (1), ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɵɣ ɭɫɪɟɞɧɟɧɧɵɣ ɜɟɤɬɨɪ ɦɨɦɟɧɬɚ
ɢɦɩɭɥɶɫɚ
nJ m . (4)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɫɨɫɬɨɹɧɢɢ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɧɚ
ɡɚɞɚɧɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɭɫɪɟɞɧɟɧɧɵɣ ɜɟɤɬɨɪ ɦɨɦɟɧɬɚ
m
n J ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɡɧɚɤɚ ɩɚɪɚɥɥɟɥɟɧ ɢɥɢ ɚɧɬɢɩɚɪɚɥɥɟɥɟɧ (ɥɢɛɨ ɪɚɜɟɧ ɧɭɥɸ, ɟɫɥɢ ).m
n 0 m
7.4. Ɉɛɨɡɧɚɱɢɦ ɩɨɫɪɟɞɫɬɜɨɦ
mj
c
~
,1 ɫɨɫɬɨɹɧɢɹ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦɢ
ɡɧɚɱɟɧɢɹɦɢ mJ
z
c
~
ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ɧɚ ɨɫɶ z
~
, ɫɨɫɬɚɜɥɹɸɳɭɸ ɫ ɢɫɯɨɞɧɨɣ
ɨɫɶɸ ɡɚɞɚɧɧɵɣ ɭɝɨɥ . Ⱥɦɩɥɢɬɭɞɚ ɜɟɪɨɹɬɧɨɫɬɢ ɨɛɧɚɪɭɠɢɬɶ ɡɧɚɱɟɧɢɟ
ɭ ɱɚɫɬɢɰɵ ɜ ɫɨɫɬɨɹɧɢɢ
z T
mJ
z
c
~
mj ,1 ɪɚɜɧɚ ɫɤɚɥɹɪɧɨɦɭ ɩɪɨɢɡɜɟɞɟɧɢɸ
mjmj ,1
~
,1
c
, ɚ ɫɚɦɚ ɜɟɪɨɹɬɧɨɫɬɶ ɪɚɜɧɚ ɤɜɚɞɪɚɬɭ ɦɨɞɭɥɹ ɷɬɨɣ ɚɦɩɥɢɬɭɞɵ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɥɸɛɨɟ ɢɡ ɬɪɟɯ (1,0,1
c
m ) ɫɨɫɬɨɹɧɢɣ
mj
c
~
,1 ɦɨɠɧɨ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɪɟɡɭɥɶɬɚɬ ɞɟɣɫɬɜɢɹ ɨɩɟɪɚɬɨɪɚ ɩɨɜɨɪɨɬɚ )ɧɚ ɫɨɫɬɨɹɧɢɟ(
ˆ
y
R eT
mj
c
,1 ɫ ɬɟɦ ɠɟ ɫɚɦɵɦ ɡɧɚɱɟɧɢɟɦ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ, mJJ
zz
c
~
:
c
mj
~
,)(
ˆ
y
R eT mj
c
,. (1)
ɉɨɷɬɨɦɭ ɢɧɬɟɪɟɫɭɸɳɢɟ ɧɚɫ ɚɦɩɥɢɬɭɞɵ ɜɟɪɨɹɬɧɨɫɬɢ ɛɭɞɭɬ ɧɚɣɞɟɧɵ, ɟɫɥɢ
ɭɞɚɫɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ ɩɨɜɨɪɨɬɚ )ɜ ɛɚɡɢɫɟ(
ˆ
y
R eT
mj ,1 :
60
mjmjmjRmjmj
y
c
c
T
c
,,,)(
ˆ
,
~
,
e )(
ˆ
y
R eT
mj,
mj
c
, )(
ˆ
1
y
R eT
mj, mj
c
,)(
ˆ
y
R eT mj, . (2)
Ɍɪɟɛɭɟɦɭɸ ɦɚɬɪɢɰɭ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɧɟɫɤɨɥɶɤɢɦɢ ɫɩɨɫɨɛɚɦɢ (ɜ ɨɛɳɟɦ
ɫɥɭɱɚɟ ɪɟɡɭɥɶɬɚɬɵ ɨɤɚɠɭɬɫɹ ɪɚɡɥɢɱɧɵɦɢ ɢɡ-ɡɚ ɩɪɨɢɡɜɨɥɚ ɜ ɜɵɛɨɪɟ ɮɚɡɨɜɵɯ
ɦɧɨɠɢɬɟɥɟɣ ɭ ɛɚɡɢɫɧɵɯ ɜɟɤɬɨɪɨɜ ɫɨɫɬɨɹɧɢɹ, ɧɨ ɬɚɤɨɟ ɪɚɡɥɢɱɢɟ ɧɟ ɨɬɪɚɡɢɬɫɹ ɧɚ
ɡɧɚɱɟɧɢɹɯ ɤɜɚɞɪɚɬɨɜ ɦɨɞɭɥɟɣ ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ). Ɋɚɫɫɦɨɬɪɢɦ ɩɨɞɪɨɛɧɨ ɞɜɚ
ɫɩɨɫɨɛɚ, ɬɟɦ ɫɚɦɵɦ ɢɥɥɸɫɬɪɢɪɭɹ ɧɚ ɞɚɧɧɨɦ ɩɪɢɦɟɪɟ ɪɹɞ ɨɛɳɢɯ ɫɨɨɬɧɨɲɟɧɢɣ
ɤɜɚɧɬɨɜɨ-ɦɟɯɚɧɢɱɟɫɤɨɣ
ɬɟɨɪɢɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ.
ɉɟɪɜɵɣ ɫɩɨɫɨɛ ɨɫɧɨɜɚɧ ɧɚ ɨɩɪɟɞɟɥɟɧɢɢ ɫ ɩɨɦɨɳɶɸ ɷɥɟɦɟɧɬɚɪɧɵɯ
ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɫɨɨɛɪɚɠɟɧɢɣ ɦɚɬɪɢɰɵ
R
c
ˆ
ɩɨɜɨɪɨɬɚ ɧɚ ɭɝɨɥ ɜɨɤɪɭɝ ɨɫɢ ɜ
ɛɚɡɢɫɟ, ɨɛɪɚɡɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɹɦɢ ɫ «ɥɢɧɟɣɧɨɣ ɩɨɥɹɪɢɡɚɰɢɟɣ»
T y
x , y , z .
Ɍɚɤɢɟ ɫɨɫɬɨɹɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɥɢɫɶ ɜ ɡɚɞɚɱɟ 3.9 ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɦɚɫɫɢɜɧɨɣ
ɱɚɫɬɢɰɟ ɫ ɫɨɛɫɬɜɟɧɧɵɦ ɦɨɦɟɧɬɨɦ ɢɦɩɭɥɶɫɚ 1 j . Ɇɚɬɪɢɰɚ ɭɤɚɡɚɧɧɨɝɨ
ɩɨɜɨɪɨɬɚ
R
ˆ
ɜ ɛɚɡɢɫɟ
mj, ɛɭɞɟɬ ɫɜɹɡɚɧɚ ɫ ɦɚɬɪɢɰɟɣ
R
c
ˆ
ɮɨɪɦɭɥɨɣ
¦¦
c
ccc
c
kk
mkkRkkmmRm
ˆˆ
, (3)
ɝɞɟ , ɫɨɫɬɨɹɧɢɹzyxkk ,,,
c
mj ,1 ɫɨɤɪɚɳɟɧɧɨ ɨɛɨɡɧɚɱɟɧɵ ɩɨɫɪɟɞɫɬɜɨɦ
m . ɗɥɟɦɟɧɬɚɦɢ ɦɚɬɪɢɰɵ
R
c
ˆ
ɹɜɥɹɸɬɫɹ ɜɟɥɢɱɢɧɵ
kRkR
kk
ˆ
)
ˆ
(
c
c
c
,
ɩɪɢɫɭɬɫɬɜɭɸɳɢɟ ɜ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ ɪɚɜɟɧɫɬɜɚ (3). Ɋɚɫɫɦɚɬɪɢɜɚɹ ɜɟɥɢɱɢɧɵ
mk
ɤɚɤ ɷɥɟɦɟɧɬɵ ɧɟɤɨɬɨɪɨɣ ɭɧɢɬɚɪɧɨɣ ɦɚɬɪɢɰɵ (ɩɪɢɱɟɦ ɦɵ ɭɫɥɨɜɢɦɫɹ
ɧɭɦɟɪɨɜɚɬɶ ɫɨɫɬɨɹɧɢɹ
U
ˆ
m ɜ ɩɨɪɹɞɤɟ ɭɛɵɜɚɧɢɹ ɡɧɚɱɟɧɢɣ 1, ɬɚɤ ɱɬɨ,
ɧɚɩɪɢɦɟɪ,
,0,1 m
1
13
mxU ), ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɮɨɪɦɭɥɭ (3) ɜ ɜɢɞɟ
ɦɚɬɪɢɱɧɨɝɨ ɪɚɜɟɧɫɬɜɚ:
URUR
ˆˆˆˆ
c
. (4)
x
z
ș
ș
zR
ˆ
xR
ˆ
Ɋɢɫ. 7.1
Ȼɭɞɟɦ ɨɩɪɟɞɟɥɹɬɶ ɦɚɬɪɢɰɭ ), ɢɫɯɨɞɹ
ɢɡ ɩɪɟɞɩɨɥɨɠɟɧɢɹ, ɱɬɨ ɫɨɫɬɨɹɧɢɹ
(
ˆ
y
R eT
c
x , y , z
z
ɩɨɞ
ɞɟɣɫɬɜɢɟɦ ɩɨɜɨɪɨɬɨɜ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɤɚɤ ɨɛɵɱɧɵɟ
ɨɪɬɵ . Ɍɚɤɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɩɪɢ
ɩɨɜɨɪɨɬɟ ɧɚ ɭɝɨɥ
yx
eee ,,
T ɜɨɤɪɭɝ ɨɫɢ ɫɯɟɦɚɬɢɱɟɫɤɢ
ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 7.1 (ɨɫɶ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ
ɩɥɨɫɤɨɫɬɢ ɪɢɫɭɧɤɚ ɢ ɧɚɩɪɚɜɥɟɧɚ «ɨɬ ɧɚɫ»).
y
y
61
ɋɩɪɨɟɰɢɪɨɜɚɜ ɜɟɤɬɨɪɵ kR
ˆ
ɧɚ ɢɫɯɨɞɧɵɟ ɧɚɩɪɚɜɥɟɧɢɹ k , ɢɦɟɟɦ:
zxxR TT sincos
ˆ
,
yyR
ˆ
,
zxzR TT cossin
ˆ
. (5)
ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɚɹ ɪɚɜɟɧɫɬɜɚ (5) ɧɚ ɛɚɡɢɫɧɵɟ ɜɟɤɬɨɪɵ
x , y , z ɢ ɭɱɢɬɵɜɚɹ
ɫɜɨɣɫɬɜɨ ɨɪɬɨɧɨɪɦɢɪɨɜɚɧɧɨɫɬɢ ɷɬɢɯ ɛɚɡɢɫɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɧɚɯɨɞɢɦ ɦɚɬɪɢɰɭ
:)(
ˆ
y
R eT
c
¸
¸
¸
¹
·
¨
¨
¨
©
§
TT
TT
c
cos0sin
010
sin0cos
ˆ
R . (6)
ɋɨɝɥɚɫɧɨ ɩɨɹɫɧɟɧɢɹɦ ɜ ɡɚɞɚɱɟ 3.9 ɫɨɫɬɨɹɧɢɟ
0,1 mj ɫɨɜɩɚɞɚɟɬ ɫ
z , ɚ ɞɜɚ ɞɪɭɝɢɯ ɫɨɫɬɨɹɧɢɹ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɪɨɟɤɰɢɟɣ ɦɨɦɟɧɬɚ ɩɪɟɞɫɬɚɜɥɹɸɬ
ɫɨɛɨɣ ɥɢɧɟɣɧɵɟ ɤɨɦɛɢɧɚɰɢɢ ɜɟɤɬɨɪɨɜ ɫɨɫɬɨɹɧɢɹ
x ɢ y , ɬɚɤ ɱɬɨ:
yixmj
2
1
1,1
,
zmj 0,1 ,
yixmj
2
1
1,1
. (7)
ȼ ɩɟɪɜɨɣ ɢɡ ɷɬɢɯ ɮɨɪɦɭɥ ɜɜɟɞɟɧ ɞɨɩɨɥɧɢɬɟɥɶɧɵɣ (ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ (6) ɜ ɡɚɞɚɱɟ
3.9) ɮɚɡɨɜɵɣ ɦɧɨɠɢɬɟɥɶ 1, ɤɨɬɨɪɵɣ ɩɨɡɜɨɥɢɬ ɧɚɦ ɢɡɛɟɠɚɬɶ ɭɩɨɦɢɧɚɜɲɟɝɨɫɹ
ɜɵɲɟ ɪɚɡɥɢɱɢɹ ɜɵɪɚɠɟɧɢɣ ɞɥɹ ɦɚɬɪɢɰɵ
R
ˆ
, ɩɨɥɭɱɚɟɦɵɯ ɪɚɡɧɵɦɢ ɫɩɨɫɨɛɚɦɢ.
ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɚɹ ɪɚɜɟɧɫɬɜɚ (7) ɧɚ ɛɚɡɢɫɧɵɟ ɜɟɤɬɨɪɵ
x , y , z , ɧɚɯɨɞɢɦ
ɜɟɥɢɱɢɧɵ
mk , ɫɨɫɬɚɜɥɹɸɳɢɟ ɦɚɬɪɢɰɭ U :
ˆ
¸
¸
¸
¹
·
¨
¨
¨
©
§
010
2/02/
2/102/1
ˆ
iiU . (8)
ɗɬɚ ɦɚɬɪɢɰɚ, ɤɚɤ ɥɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɭɧɢɬɚɪɧɚ. ɗɪɦɢɬɨɜɨ ɫɨɩɪɹɠɟɧɧɚɹ ɦɚɬɪɢɰɚ
ɢɦɟɟɬ ɜɢɞ
62
¸
¸
¸
¹
·
¨
¨
¨
©
§
02/2/1
100
02/2/1
ˆ
i
i
U . (9)
ɉɨɞɫɬɚɜɢɜ (6), (8) ɢ (9) ɜ (4) ɢ ɩɟɪɟɦɧɨɠɢɜ ɭɤɚɡɚɧɧɵɟ ɦɚɬɪɢɰɵ, ɩɨɥɭɱɚɟɦ
ɢɫɤɨɦɭɸ ɦɚɬɪɢɰɭ
R
ˆ
, ɷɥɟɦɟɧɬɚɦɢ ɤɨɬɨɪɨɣ ɹɜɥɹɸɬɫɹ ɜɟɥɢɱɢɧɵ
mj
c
,)(
ˆ
y
R eT mj, ɩɪɢ :1 j
¸
¸
¸
¸
¸
¸
¸
¹
·
¨
¨
¨
¨
¨
¨
¨
©
§
T
T
T
T
TT
T
T
T
T
2
cos1
sin
2
1
2
cos1
sin
2
1
cossin
2
1
2
cos1
sin
2
1
2
cos1
)(
ˆ
y
R e . (10)
ȼɵɜɟɞɟɦ ɮɨɪɦɭɥɭ (10) ɞɪɭɝɢɦ ɫɩɨɫɨɛɨɦ, ɧɟ ɨɩɢɪɚɹɫɶ ɧɚ ɪɟɡɭɥɶɬɚɬɵ
ɩɪɟɞɲɟɫɬɜɭɸɳɢɯ ɡɚɞɚɱ. ɋɨɝɥɚɫɧɨ ɪɚɜɟɧɫɬɜɚɦ (XII) ɢɡ ɜɜɟɞɟɧɢɹ ɤ ɞɚɧɧɨɣ ɝɥɚɜɟ
ɦɚɬɪɢɰɚ ɨɩɟɪɚɬɨɪɚ ɜ ɛɚɡɢɫɟ, ɨɛɪɚɡɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɹɦɢ
y
J
ˆ
mj,,ɢɦɟɟɬ ɜɢɞ:
¸
¸
¸
¹
·
¨
¨
¨
©
§
02/0
2/02/
02/0
ˆ
i
ii
i
J
y
. (11)
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɨɩɪɟɞɟɥɟɧɢɟɦ (I) ɞɥɹ ɨɩɟɪɚɬɨɪɚ ɩɨɜɨɪɨɬɚ ɜɨɤɪɭɝ ɨɫɢ ɢ
ɪɚɡɥɨɠɢɦ ɩɪɢɫɭɬɫɬɜɭɸɳɭɸ ɜ ɮɨɪɦɭɥɟ (I) ɷɤɫɩɨɧɟɧɬɭ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɩɨ
ɫɬɟɩɟɧɹɦ ɭɝɥɚ ɩɨɜɨɪɨɬɚ:
y
)(
ˆ
y
R eT
T
y
Ji
e
ˆ
n
y
n
n
Ji
n
ˆ
)(
!
1
0
T
¦
f
. (12)
ȼɵɱɢɫɥɹɹ ɫ ɩɨɦɨɳɶɸ (11) ɪɚɡɥɢɱɧɵɟ ɫɬɟɩɟɧɢ ɦɚɬɪɢɰɵ , ɢɦɟɟɦ:
y
J
ˆ
¸
¸
¸
¹
·
¨
¨
¨
©
§
2/102/1
010
2/102/1
ˆ
2
y
J , (13)
yy
JJ
ˆˆ
3
. (14)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
63
°
¯
°
®
...,5,3,1,
ˆ
...,6,4,2,
ˆ
ˆ
2
nJ
nJ
J
y
y
n
y
(15)
ɋ ɭɱɟɬɨɦ (15) ɫɬɟɩɟɧɧɨɣ ɪɹɞ (12) ɩɪɢɧɢɦɚɟɬ ɜɢɞ:
n
y
n
n
Ji
n
ˆ
)(
!
1
0
T
¦
f
¸
¸
¹
·
¨
¨
©
§
T
T
¸
¸
¹
·
¨
¨
©
§
T
T
...
!3
ˆ
...
!4!2
ˆ
1
ˆ
342
2
yy
JiJ
TT sin
ˆ
1cos
ˆ
1
ˆ
2
yy
JiJ . (16)
ɉɨɞɫɬɚɜɢɜ ɜ ɩɪɚɜɭɸ ɫɬɨɪɨɧɭ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɹɜɧɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɦɚɬɪɢɰ (11),
(13) ɢ ɞɥɹ ɟɞɢɧɢɱɧɨɣ ɦɚɬɪɢɰɵ 1, ɩɨɥɭɱɚɟɦ ɪɟɡɭɥɶɬɚɬ (10).
ˆ
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ, ɬɪɟɛɭɟɦɵɯ ɭɫɥɨɜɢɟɦ ɡɚɞɚɱɢ, ɞɨɫɬɚɬɨɱɧɨ ɜ
ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɨɣ (2) ɩɪɨɢɡɜɟɫɬɢ ɜ (10) ɡɚɦɟɧɭ To
T
ɢ ɜɨɡɜɟɫɬɢ
ɷɥɟɦɟɧɬɵ ɩɨɥɭɱɢɜɲɟɣɫɹ ɦɚɬɪɢɰɵ ɜ ɤɜɚɞɪɚɬ. ɍɤɚɡɚɧɧɨɟ ɢɡɦɟɧɟɧɢɟ ɡɧɚɤɚ T , ɤɚɤ
ɜɢɞɧɨ, ɧɟ ɛɭɞɟɬ ɜɥɢɹɬɶ ɧɚ ɨɤɨɧɱɚɬɟɥɶɧɵɣ ɪɟɡɭɥɶɬɚɬ. ȼ ɫɨɫɬɨɹɧɢɢ
1,1 mj
ɜɟɪɨɹɬɧɨɫɬɢ ɡɧɚɱɟɧɢɣ 1,0,1
~
z
J ɪɚɜɧɵ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɤɜɚɞɪɚɬɚɦ ɷɥɟɦɟɧɬɨɜ
ɩɟɪɜɨɝɨ ɫɬɨɥɛɰɚ ɦɚɬɪɢɰɵ (10):
2
)cos1(
4
1
T , T
2
sin
2
1
,
2
)cos1(
4
1
T . (17)
ȼ ɫɨɫɬɨɹɧɢɢ
0,1 mj ɜɟɪɨɹɬɧɨɫɬɢ ɡɧɚɱɟɧɢɣ 1,0,1
~
z
J ɪɚɜɧɵ ɤɜɚɞɪɚɬɚɦ
ɷɥɟɦɟɧɬɨɜ ɜɬɨɪɨɝɨ ɫɬɨɥɛɰɚ ɦɚɬɪɢɰɵ (10):
T
2
sin
2
1
, T
2
cos , T
2
sin
2
1
. (18)
ɇɚɤɨɧɟɰ, ɜ ɫɨɫɬɨɹɧɢɢ
1,1 mj ɜɟɪɨɹɬɧɨɫɬɢ ɡɧɚɱɟɧɢɣ ɪɚɜɧɵ,
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɤɜɚɞɪɚɬɚɦ ɷɥɟɦɟɧɬɨɜ ɬɪɟɬɶɟɝɨ ɫɬɨɥɛɰɚ ɦɚɬɪɢɰɵ (10):
1,0,1
~
z
J
2
)cos1(
4
1
T , T
2
sin
2
1
,
2
)cos1(
4
1
T . (19)
ȼ ɤɚɱɟɫɬɜɟ ɩɪɨɜɟɪɤɢ ɦɨɠɧɨ, ɩɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɚɦɢ (17) – (19), ɩɨɞɫɱɢɬɚɬɶ
ɫɪɟɞɧɢɟ ɡɧɚɱɟɧɢɹ
z
J
~
ɜ ɫɨɫɬɨɹɧɢɹɯ mj ,1 ɢ ɭɛɟɞɢɬɶɫɹ ɜ ɬɨɦ, ɱɬɨ ɞɥɹ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɡɞɟɫɶ ɫɥɭɱɚɹ (1 j ) ɧɚɣɞɟɧɧɨɟ ɜ ɡɚɞɚɱɟ 7.2 ɫɨɨɬɧɨɲɟɧɢɟ
T cos
~
mJ
z
, (20)
ɤɚɤ ɢ ɞɨɥɠɧɨ ɛɵɬɶ, ɜɵɩɨɥɧɹɟɬɫɹ.
64
7.5. ɋɨɝɥɚɫɧɨ ɮɨɪɦɭɥɚɦ (XIX)–(XXI) ɜɜɟɞɟɧɢɹ ɤ ɞɚɧɧɨɣ ɝɥɚɜɟ ɪɟɲɟɧɢɟ
ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ
\H \'
m2
2
=
, (1)
ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟɫɹ ɨɩɪɟɞɟɥɟɧɧɨɣ ɜɟɥɢɱɢɧɨɣ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
l
ɢ
ɩɪɨɟɤɰɢɟɣ ɦɨɦɟɧɬɚ (ɧɟ ɩɭɬɚɬɶ ɫ ɦɚɫɫɨɣ ɱɚɫɬɢɰɵ!), ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟm
\ )(
r
r
r)(F
),( MT
lm
Y . (2)
ȼɫɩɨɦɨɝɚɬɟɥɶɧɚɹ ɮɭɧɤɰɢɹ )(
r
F ɞɨɥɠɧɚ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɭɪɚɜɧɟɧɢɸ
0)(
)1()(
2
2
2
2
F
¸
¹
·
¨
©
§
F
r
r
ll
k
rd
rd
(3)
ɢ ɝɪɚɧɢɱɧɨɦɭ ɭɫɥɨɜɢɸ
0)0(
F . (4)
ȼɨɥɧɨɜɨɟ ɱɢɫɥɨ
=/2 H mk , ɤɚɤ ɨɛɵɱɧɨ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɱɟɧɢɟɦ ɷɧɟɪɝɢɢ
ɱɚɫɬɢɰɵ H .
Ɋɚɡɞɟɥɢɦ ɨɛɟ ɫɬɨɪɨɧɵ ɭɪɚɜɧɟɧɢɹ (3) ɧɚ
2
k ɢ ɩɟɪɟɣɞɟɦ ɤ ɛɟɡɪɚɡɦɟɪɧɨɣ
ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɟɪɟɦɟɧɧɨɣ
r
k [ . (5)
ȼɵɩɨɥɧɢɜ ɜ (3) ɩɨɞɫɬɚɧɨɜɤɭ
)()( [[ [F
D
w (6)
ɢ ɜɵɛɪɚɜ ɡɚɬɟɦ 2
/
1 D , ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ (3) ɜ ɮɨɪɦɟ ɭɪɚɜɧɟɧɢɹ Ȼɟɫɫɟɥɹ ɞɥɹ
ɧɨɜɨɣ ɜɫɩɨɦɨɝɚɬɟɥɶɧɨɣ ɮɭɧɤɰɢɢ )([w :
0)(
)2/1(
1
)(1)(
2
2
2
2
[
¸
¸
¹
·
¨
¨
©
§
[
[
[
[
[
[
w
l
d
dw
d
wd
. (7)
ȼ ɤɚɱɟɫɬɜɟ ɞɜɭɯ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵɯ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ (7) ɞɨɫɬɚɬɨɱɧɨ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɮɭɧɤɰɢɸ Ȼɟɫɫɟɥɹ )([
Q
J ɢ ɮɭɧɤɰɢɸ ɇɟɣɦɚɧɚ )ɩɨɪɹɞɤɚ([
Q
N
2
/
1 Q
l
. Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɬɨɝɞɚ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ ɥɢɧɟɣɧɨɣ ɤɨɦɛɢɧɚɰɢɢ ɷɬɢɯ
ɮɭɧɤɰɢɣ:
)()()(
2/122/11
[[ [
ll
NCJCw , (8)
ɝɞɟ ...,2,1,0
l
. ɉɨɫɬɨɹɧɧɵɟ ɢ ɜ (8) ɧɟɨɛɯɨɞɢɦɨ ɜɵɛɪɚɬɶ ɬɚɤ, ɱɬɨɛɵ
1
C
2
C
65
ɜɵɩɨɥɧɹɥɨɫɶ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ (4), ɬɨ ɟɫɬɶ ɫɥɟɞɭɟɬ ɬɪɟɛɨɜɚɬɶ, ɱɬɨɛɵ
0)(
2/1
o[[ w ɩɪɢ 0o[ . (9)
ɂɡɜɟɫɬɧɨ, ɱɬɨ ɩɪɢ ɦɚɥɨɦ ɡɧɚɱɟɧɢɢ ɚɪɝɭɦɟɧɬɚ ɮɭɧɤɰɢɹ )([
Q
J ɜɟɞɟɬ ɫɟɛɹ ɤɚɤ
Q
[ ,
ɚ ɮɭɧɤɰɢɹ )ɩɨɪɹɞɤɚ 0 – ɤɚɤ([
Q
N zQ
Q
[ (ɟɫɥɢ 0 Q , ɬɨ ɮɭɧɤɰɢɹ ɇɟɣɦɚɧɚ
ɜɟɞɟɬ ɫɟɛɹ ɤɚɤ ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɨɫɤɨɥɶɤɭ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɟ[ln
2
/
1tQ , ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɭɫɥɨɜɢɹ (9) ɦɵ ɞɨɥɠɧɵ ɩɨɫɬɨɹɧɧɭɸ ɩɨɥɨɠɢɬɶ
ɪɚɜɧɨɣ ɧɭɥɸ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɢɫɤɨɦɵɟ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ (2) ɡɚɩɢɲɭɬɫɹ ɜ ɜɢɞɟ
2
C
\ )(
r
mlk
)(
1
2/11
rkJ
rk
C
l
),(
M
T
lm
Y , (10)
ɝɞɟ – ɧɨɪɦɢɪɨɜɨɱɧɚɹ ɩɨɫɬɨɹɧɧɚɹ.
1
C
ɍɱɢɬɵɜɚɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɨɟ ɩɨɜɟɞɟɧɢɟ ɮɭɧɤɰɢɣ Ȼɟɫɫɟɥɹ
)1(
1
2
)(
Q*
¸
¹
·
¨
©
§
[
|[
Q
Q
J
ɩɪɢ 1[ , (11)
¸
¹
·
¨
©
§
S
SQ
[
¸
¸
¹
·
¨
¨
©
§
[S
|[
Q
42
cos
2
)(
2/1
J ɩɪɢ fo[ , (12)
ɦɵ ɜɢɞɢɦ, ɱɬɨ ɜɛɥɢɡɢ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɡɚɜɢɫɢɦɨɫɬɶ ɧɚɣɞɟɧɧɨɣ ɜɨɥɧɨɜɨɣ
ɮɭɧɤɰɢɢ ɨɬ
mlk
\
r
ɢɦɟɟɬ ɜɢɞ
l
r
, ɚ ɧɚ ɛɨɥɶɲɢɯ ɪɚɫɫɬɨɹɧɢɹɯ ɨɬ ɧɚɱɚɥɚ
ɤɨɨɪɞɢɧɚɬ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ ɜɢɞɭ ɬɚɤ
ɧɚɡɵɜɚɟɦɨɣ ɫɮɟɪɢɱɟɫɤɨɣ ɜɨɥɧɵ )(sin
1
G
rkr :
|\ )(
r
mlk
¸
¹
·
¨
©
§
S
S 2
sin
12
1
l
rk
rk
C ),( MT
lm
Y , fo
r
k . (13)
ɗɬɢ ɪɟɡɭɥɶɬɚɬɵ ɫɨɝɥɚɫɭɸɬɫɹ ɫ ɨɛɳɢɦɢ ɭɬɜɟɪɠɞɟɧɢɹɦɢ ɨ ɩɨɜɟɞɟɧɢɢ ɜɨɥɧɨɜɨɣ
ɮɭɧɤɰɢɢ , ɭɤɚɡɚɧɧɵɦɢ ɜɨ ɜɜɟɞɟɧɢɢ ɤ ɞɚɧɧɨɣ ɝɥɚɜɟ.
mlk
\
Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɢɡ ɜɨɡɦɨɠɧɵɯ ɫɩɨɫɨɛɨɜ ɧɨɪɦɢɪɨɜɤɢ ɜɨɥɧɨɜɨɣ
ɮɭɧɤɰɢɢ (10). Ɇɵ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɨɨɛɪɚɠɟɧɢɹɦɢ, ɚɧɚɥɨɝɢɱɧɵɦɢ ɢɡɥɨɠɟɧɧɵɦ ɜ
ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ 5.3 (ɩɭɧɤɬ ɛ). ɉɨɫɤɨɥɶɤɭ ɧɨɪɦɢɪɨɜɨɱɧɵɣ ɢɧɬɟɝɪɚɥ ɞɥɹ
ɮɭɧɤɰɢɢ (10) ɪɚɫɯɨɞɢɬɫɹ ɜ ɨɛɥɚɫɬɢ fo
r
, ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɜɤɥɚɞ ɷɬɨɣ ɨɛɥɚɫɬɢ
ɨɫɧɨɜɧɵɦ, ɢ ɡɚɦɟɧɢɬɶ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɮɭɧɤɰɢɸ (10) ɟɟ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦ
ɜɵɪɚɠɟɧɢɟɦ (13). ɗɬɨ ɞɚɟɬ:
\:
³³
f
2
0
2
mlk
drdr
2
0
2
2
1
)2/(sin
12
³
f
¸
¸
¹
·
¨
¨
©
§
S
S
lkr
rk
rdrC
66
«Ɉɛɪɟɠɟɦ» ɜɟɪɯɧɢɣ ɩɪɟɞɟɥ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɡɚɦɟɧɢɜ f ɩɪɨɢɡɜɨɥɶɧɨ ɛɨɥɶɲɨɣ
ɜɟɥɢɱɢɧɨɣ . ȼɟɥɢɱɢɧɚ ɢɦɟɟɬ ɫɦɵɫɥ ɪɚɞɢɭɫɚ ɫɮɟɪɢɱɟɫɤɨɣ ɩɨɥɨɫɬɢ, ɜɧɭɬɪɢ
ɤɨɬɨɪɨɣ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɫɜɨɛɨɞɧɨɟ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ. Ɂɚɦɟɧɢɜ ɬɚɤɠɟ ɤɜɚɞɪɚɬ
ɫɢɧɭɫɚ ɟɝɨ ɫɪɟɞɧɢɦ ɡɧɚɱɟɧɢɟɦ 2
aa
/
1,ɩɨɥɭɱɚɟɦ ɧɨɪɦɢɪɨɜɨɱɧɵɣ ɢɧɬɟɝɪɚɥ ɜ ɜɢɞɟ:
2
2
1
2
3
k
a
Cd
mlk
S
\
³
r
, ka
/
1!! . (14)
Ⱦɥɹ ɧɨɪɦɢɪɨɜɤɢ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ ɧɚ ɟɞɢɧɢɰɭ ɜ ɲɚɪɟ ɪɚɞɢɭɫɚ
ɩɪɢɪɚɜɧɹɟɦ ɜɵɪɚɠɟɧɢɟ (14) ɟɞɢɧɢɰɟ:
a
1
2
2
1
Sk
a
C
. (15)
Ɍɨɝɞɚ
akC /
2
1
S , ɢ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ (10) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
\ )(
r
mlk
)(
1
2/1
2
rkJ
rk
a
k
l
S
),( MT
lm
Y . (16)
ȼ ɫɥɭɱɚɟ ɧɨɪɦɢɪɨɜɤɢ ɧɚ G-ɮɭɧɤɰɢɸ ɨɬ ɜɨɥɧɨɜɨɝɨ ɱɢɫɥɚ ɩɪɚɜɭɸ ɱɚɫɬɶ
ɪɚɜɟɧɫɬɜɚ (14) ɫɥɟɞɭɟɬ ɩɨɧɢɦɚɬɶ ɤɚɤ ɜɟɥɢɱɢɧɭ )(2 kk
c
GS ɩɪɢ
0
c
k
k
. ɗɬɚ
ɜɟɥɢɱɢɧɚ, ɫ ɭɱɟɬɨɦ ɩɪɟɞɫɬɚɜɥɟɧɢɹ
G-ɮɭɧɤɰɢɢ ɜ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɨɪɦɟ
rkki
edrkk
)(
)(2
c
f
f
³
c
GS ,
ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɪɚɜɧɟɧɚ , ɝɞɟa2 foa . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɚɦɟɧɢɜ ɜ ɩɪɚɜɨɣ
ɫɬɨɪɨɧɟ (15) ɟɞɢɧɢɰɭ ɜɟɥɢɱɢɧɨɣ , ɩɨɥɭɱɚɟɦa2
2
1
2 kC S , ɢ
\ )(
r
mlk
)(
2
2/1
rkJ
r
k
l
S
),( MT
lm
Y . (17)
Ɏɭɧɤɰɢɢ (17) ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɥɟɞɭɸɳɢɦ ɭɫɥɨɜɢɹɦ ɧɨɪɦɢɪɨɜɤɢ ɢ ɜɡɚɢɦɧɨɣ
ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ:
mmllmlkmlk
kkd
cc
ccc
GG
c
GS \\
³
)(2
3
r . (18)
ɇɚɤɨɧɟɰ, ɭɛɟɞɢɦɫɹ ɜ ɬɨɦ, ɱɬɨ ɩɪɢ 0
l
ɩɨɥɭɱɟɧɧɵɟ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ
ɩɪɢɧɢɦɚɸɬ ɜɢɞ ɫɮɟɪɢɱɟɫɤɢɯ ɜɨɥɧ, ɤɨɬɨɪɵɟ ɨɛɫɭɠɞɚɥɢɫɶ ɜ ɡɚɞɚɱɟ 4.11.
Ɏɭɧɤɰɢɢ Ȼɟɫɫɟɥɹ ɩɨɥɭɰɟɥɨɝɨ ɩɨɪɹɞɤɚ, ɤɚɤ ɢɡɜɟɫɬɧɨ, ɦɨɝɭɬ ɛɵɬɶ ɜɵɪɚɠɟɧɵ
ɱɟɪɟɡ ɷɥɟɦɟɧɬɚɪɧɵɟ ɮɭɧɤɰɢɢ:
67
[
[
¸
¸
¹
·
¨
¨
©
§
[[
[
S
[
sin
12
)(
2/1
2/1
l
l
l
d
d
J . (19)
ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɜɵɩɢɫɵɜɚɬɶ ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ )(
r
mlk
\ ɜ ɛɨɥɟɟ ɪɚɡɜɟɪɧɭɬɨɣ
ɮɨɪɦɟ. ȼ ɱɚɫɬɧɨɫɬɢ, ɩɪɢ 0
l
ɫ ɭɱɟɬɨɦ ɪɚɜɟɧɫɬɜɚ S 4/1
00
Y ɮɨɪɦɭɥɚ (16)
ɞɚɟɬ:
\
0
)(
l
mlk
r )(
1
2/1
2
rkJ
rk
a
kS
S4
1
S
F
4
1
)(
r
r
k
, (20)
ɝɞɟ
rk
a
r
k
sin
2
)( F . (21)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɞɟɫɶ ɜɨɫɩɪɨɢɡɜɨɞɢɬɫɹ ɧɚɣɞɟɧɧɨɟ ɜ ɡɚɞɚɱɟ 4.11 ɨɩɢɫɚɧɢɟ
ɫɜɨɛɨɞɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɫ ɪɚɜɧɵɦ ɧɭɥɸ ɦɨɦɟɧɬɨɦ ɢɦɩɭɥɶɫɚ.
7.6. ɚ) ɋɨɝɥɚɫɧɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ ɱɚɫɬɢɰɵ )
ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ (ɛɟɡ ɭɱɟɬɚ ɧɨɪɦɢɪɨɜɨɱɧɨɝɨ ɦɧɨɠɢɬɟɥɹ) ɜ ɜɢɞɟ:
(
rI
T
I
cos
)(
rkizki
k
eer . (1)
Ɏɭɧɤɰɢɹ (1) ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɭɝɥɨɜɨɣ ɤɨɨɪɞɢɧɚɬɵ M ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɞ
ɞɟɣɫɬɜɢɟɦ ɨɩɟɪɚɬɨɪɚ ɩɨɜɨɪɨɬɨɜ ɜɨɤɪɭɝ ɨɫɢ ɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɨɣ. ɗɬɨ
ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜ ɫɨɫɬɨɹɧɢɢ (1) ɱɚɫɬɢɰɚ ɨɛɥɚɞɚɟɬ ɪɚɜɧɨɣ ɧɭɥɸ ɩɪɨɟɤɰɢɟɣ
ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɧɚ ɨɫɶ .
z
z
ɛ) Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɧɚɣɬɢ ɜɟɪɨɹɬɧɨɫɬɶ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ ɜɟɥɢɱɢɧɵ
ɦɨɦɟɧɬɚ
l
, ɪɚɡɥɨɠɢɦ ɜɨɥɧɨɜɭɸ ɮɭɧɤɰɢɸ (1) ɩɨ ɜɨɥɧɨɜɵɦ ɮɭɧɤɰɢɹɦ
mlk
\ ,
ɨɬɜɟɱɚɸɳɢɦ ɫɜɨɛɨɞɧɨɦɭ ɞɜɢɠɟɧɢɸ ɫ ɨɩɪɟɞɟɥɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɜɟɥɢɱɢɧɵ
ɢɦɩɭɥɶɫɚ k
= , ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ
l
ɢ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ . ȼ ɬɚɤɨɟ
ɪɚɡɥɨɠɟɧɢɟ ɜɨɣɞɭɬ ɬɨɥɶɤɨ ɫɨɫɬɨɹɧɢɹ ɫ
ml
z
0
z
l :
)()(
0
0
rr
¦
f
\ I
l
lklk
A . (2)
ɉɪɟɠɞɟ ɱɟɦ ɩɪɢɫɬɭɩɢɬɶ ɤ ɜɵɱɢɫɥɟɧɢɸ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɚɡɥɨɠɟɧɢɹ ,
ɪɚɫɫɦɨɬɪɢɦ ɧɨɪɦɢɪɨɜɨɱɧɵɣ ɢɧɬɟɝɪɚɥ ɞɥɹ ɩɥɨɫɤɨɣ ɜɨɥɧɵ (1), ɡɚɩɢɫɚɜ ɟɝɨ ɜ
ɜɢɞɟ:
l
A
)(2)()(
2)(3
kkLedzdydxd
zkki
kk
c
GS II
c
f
f
f
f
f
f
c
³³³³
rrr . (3)
68
Ɂɞɟɫɶ ɩɨɫɪɟɞɫɬɜɨɦ
2
L
ɨɛɨɡɧɚɱɟɧɚ ɛɟɫɤɨɧɟɱɧɚɹ ɜɟɥɢɱɢɧɚ ɩɥɨɳɚɞɢ ɜ ɩɥɨɫɤɨɫɬɢ
xy
, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɧɚɩɪɚɜɥɟɧɢɸ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ. ɇɚɥɢɱɢɟ ɜ (3) G-
ɮɭɧɤɰɢɢ ɭɤɚɡɵɜɚɟɬ ɧɚ ɰɟɥɟɫɨɨɛɪɚɡɧɨɫɬɶ ɜɵɛɨɪɚ ɜ ɪɨɥɢ
mlk
\ ɬɟɯ ɮɭɧɤɰɢɣ,
ɤɨɬɨɪɵɟ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɸ ɧɨɪɦɢɪɨɜɤɢ ɧɚ G-ɮɭɧɤɰɢɸ ɨɬ ɜɨɥɧɨɜɨɝɨ
ɱɢɫɥɚ (ɫɦ. ɮɨɪɦɭɥɵ (17) ɢ (18) ɡɚɞɚɱɢ 7.5):
\ )(
0
r
lk
)(
1
2
2/1
2
rkJ
rk
k
l
S
)(
0
T
l
Y
ɉɪɢ ɷɬɨɦ ɩɨɞɫɬɚɧɨɜɤɚ ɪɚɡɥɨɠɟɧɢɹ (2) ɜ ɥɟɜɭɸ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ (3) ɞɚɟɬ:
\\ II
cc
c
c
c
¦¦
³³
00
33
)()(
lklk
ll
llkk
dAAd rrrr
¦¦¦
f
c
c
c
c
GS
c
GSG
0
2
)(2)(2
l
l
ll
llll
kkAkkAA . (4)
ɋɪɚɜɧɢɜɚɹ (3) ɢ (4), ɩɨɥɭɱɚɟɦ:
¦
f
0
2
2
l
l
LA , fo
L
. (5)
ɂɡ ɪɚɜɟɧɫɬɜɚ (5) ɜɢɞɧɨ, ɱɬɨ «ɜɟɪɨɹɬɧɨɫɬɶ»
2
l
A ɛɭɞɟɬ ɢɦɟɬɶ ɪɚɡɦɟɪɧɨɫɬɶ
ɩɥɨɳɚɞɢ ɢ ɦɨɠɟɬ ɛɵɬɶ ɧɚɝɥɹɞɧɨ ɢɫɬɨɥɤɨɜɚɧɚ ɤɚɤ «ɩɪɢɰɟɥɶɧɚɹ ɩɥɨɳɚɞɶ» ɜ
ɩɥɨɫɤɨɫɬɢ
xy
, ɧɚ ɤɨɬɨɪɭɸ ɞɨɥɠɧɚ ɩɨɩɚɫɬɶ ɱɚɫɬɢɰɚ, ɞɜɢɠɭɳɚɹɫɹ ɩɚɪɚɥɥɟɥɶɧɨ
ɨɫɢ , ɟɫɥɢ ɟɟ ɦɨɦɟɧɬ ɪɚɜɟɧ
z
l
. ɋɥɟɞɭɹ [1], ɨɛɨɡɧɚɱɢɦ ɷɬɭ ɜɟɥɢɱɢɧɭ
ɩɨɫɪɟɞɫɬɜɨɦ :
l
V
V
l
2
l
A . (6)
ɉɟɪɟɣɞɟɦ ɤ ɜɵɱɢɫɥɟɧɢɸ ɚɦɩɥɢɬɭɞ ɜɟɪɨɹɬɧɨɫɬɢ . Ɂɚɩɢɲɟɦ ɪɚɡɥɨɠɟɧɢɟ
(2) ɜ ɹɜɧɨɣ ɮɨɪɦɟ:
l
A
¦
f
c
c
T
0
cos
l
l
rki
Ae
)(
1
2
2/1
2
rkJ
rk
k
l
c
S
)(
0
T
c
l
Y . (7)
ɍɦɧɨɠɢɦ ɨɛɟ ɫɬɨɪɨɧɵ (7) ɧɚ ɫɮɟɪɢɱɟɫɤɭɸ ɮɭɧɤɰɢɸ
T
)(
0l
Y )(cos
4
12
)(
0
T
S
T
ll
P
l
Y (8)
ɢ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɩɨ ɭɝɥɨɜɵɦ ɤɨɨɪɞɢɧɚɬɚɦ ɫ ɭɱɟɬɨɦ ɨɪɬɨɧɨɪɦɢɪɨɜɤɢ
69