Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
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ɡɧɚɱɟɧɢɹɦɢ ɦɧɨɠɢɬɟɥɹ Ʌɚɧɞɟ
3
/
2
g
ɞɥɹ -ɫɨɫɬɨɹɧɢɣ ɢ
2/1
p
3
/
4
g
ɞɥɹ -ɫɨɫɬɨɹɧɢɣ. ȼ
ɫɢɥɶɧɨɦ ɩɨɥɟ (P )
ɮɨɪɦɭɥɵ (10) ɨɩɢɫɵɜɚɸɬ
ɷɮɮɟɤɬ ɉɚɲɟɧɚ–Ȼɚɤɚ, ɜ
ɫɨɝɥɚɫɢɢ ɫ ɪɟɡɭɥɶɬɚɬɨɦ (26)
ɡɚɞɚɱɢ 9.6.
2/3
p
A!!B
B
9.9. ȼ ɡɚɞɚɱɟ ɨ ɥɢɧɟɣɧɨɦ ɨɫɰɢɥɥɹɬɨɪɟ, ɞɜɢɠɭɳɟɦɫɹ ɜ ɩɪɢɫɭɬɫɬɜɢɢ
ɜɧɟɲɧɟɣ ɫɢɥɵ )()(
t
E
q
t
F , ɨɩɟɪɚɬɨɪ ɜɨɡɦɭɳɟɧɢɹ ɢɦɟɟɬ ɜɢɞ
xtFtV
ˆ
)()(
ˆ
. (1)
Ɋɢɫ. 9.2
p
1/2
o –A
p
3/2
o A/2
)1(
H
)1(
H
H
ɷɮɮɟɤɬ
ɉɚɲɟɧɚ-Ȼɚɤɚ
ɷɮɮɟɤɬ
Ɂɟɟɦɚɧɚ
2
)1(
4
)1(
6
H
)1(
5
H
)1(
1
H
)1(
3
H
B
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɨɩɟɪɚɬɨɪɚ (1) ɩɨ ɧɟɜɨɡɦɭɳɟɧɧɵɦ
ɫɬɚɰɢɨɧɚɪɧɵɦ ɫɨɫɬɨɹɧɢɹɦ ɨɫɰɢɥɥɹɬɨɪɚ
n ɭɞɨɛɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɢɡɜɟɫɬɧɵɦ
ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɨɩɟɪɚɬɨɪɚ ɤɨɨɪɞɢɧɚɬɵ
x
ˆ
ɥɢɧɟɣɧɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ (ɫ ɦɚɫɫɨɣ
ɢ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɨɣ ) ɱɟɪɟɡ ɨɩɟɪɚɬɨɪɵ ɭɧɢɱɬɨɠɟɧɢɹ ɢ ɪɨɠɞɟɧɢɹ a ,
m
0
Z
ˆ
a :
ˆ
)
ˆˆ
()(
2
)(
ˆ
0
Z
aatF
m
tV
=
. (2)
ɉɨɫɤɨɥɶɤɭ
1
ˆ
nnna , 11
ˆ
nnna , (3)
ɬɨ ɫɪɟɞɢ ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɨɩɟɪɚɬɨɪɚ (2) ɦɨɝɭɬ ɛɵɬɶ ɨɬɥɢɱɧɵɦɢ ɨɬ ɧɭɥɹ
ɥɢɲɶ
ntF
m
V
nn
)(
2
0
,1
Z
=
, 1)(
2
0
,1
Z
ntF
m
V
nn
=
. (4)
ɉɨɷɬɨɦɭ ɜ ɩɟɪɜɨɦ ɩɨɪɹɞɤɟ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɜɨɡɦɨɠɧɵ ɥɢɲɶ ɫɥɟɞɭɸɳɢɟ
ɩɟɪɟɯɨɞɵ: (ɩɪɢ n ),1o nn 1t 1o nn .
ɋ ɭɱɟɬɨɦ ɢɡɜɟɫɬɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɫɩɟɤɬɪɚ ɥɢɧɟɣɧɨɝɨ
ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ ()2/1(
0
Z H n
n
= , ...,2,1,0 n ) ɦɚɬɪɢɱɧɵɟ
ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ ɜɨɡɦɭɳɟɧɢɹ (1) ɜ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
ɩɪɢɧɢɦɚɸɬ ɜɢɞ:
nn
ti
nn
VeV
nn
c
HH
c
c
=/)(
~
nn
tnni
Ve
c
Z
c
0
)(
. (5)
130
ɋɨɝɥɚɫɧɨ ɮɨɪɦɭɥɟ (XV) ɜɜɟɞɟɧɢɹ ɤ ɞɚɧɧɨɣ ɝɥɚɜɟ ɞɥɹ ɚɦɩɥɢɬɭɞ ɜɟɪɨɹɬɧɨɫɬɢ
ɩɟɪɟɯɨɞɨɜ ɢ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɢɦɟɟɦ:
1o nn 1o nn
iC
nn
)1(
,1
ti
etFdt
m
n
0
)(
2
0
Z
f
f
³
Z=
, (6)
iC
nn
)1(
,1
ti
etFdt
m
n
0
)(
2
1
0
Z
f
f
³
Z
=
. (7)
ɉɪɟɞɟɥɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɡɞɟɫɶ ɜɵɛɪɚɧɵ ɜ ɜɢɞɟ
f
r
, ɩɨɫɤɨɥɶɤɭ ɮɭɧɤɰɢɹ )(
t
F
ɦɨɠɟɬ ɛɵɬɶ ɧɟ ɪɚɜɧɚ ɧɭɥɸ ɬɨɥɶɤɨ ɜ ɩɪɨɦɟɠɭɬɤɟ ɜɪɟɦɟɧɢ ɨɬ
ɞɨ , ɢ ɷɬɨɬ
ɩɪɨɦɟɠɭɬɨɤ ɜ ɭɫɥɨɜɢɹɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɢ ɩɨɥɧɨɫɬɶɸ ɫɨɞɟɪɠɢɬɫɹ ɜɧɭɬɪɢ
ɢɧɬɟɪɜɚɥɚ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɩɪɨɫɬɢɪɚɸɳɟɝɨɫɹ ɨɬ
0
t
1
t
0
tt
c
ɞɨ .
1
tt !
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɟɥɢɱɢɧɚ
ti
etFdtf
0
)()(
0
Z
f
f
³
Z (8)
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɤɨɦɩɨɧɟɧɬɭ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ )(
t
F
ɧɚ ɱɚɫɬɨɬɟ
0
Z Z , ɢ
ɜɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ
0
0
2
)(
Z
Z
D
=m
fi
. (9)
Ɍɨɝɞɚ ɚɦɩɥɢɬɭɞɵ ɜɟɪɨɹɬɧɨɫɬɢ (6) ɢ (7) ɩɪɢɦɭɬ ɜɢɞ
)(
)1(
,1
D nC
nn
, D
1
)1(
,1
nC
nn
. (10)
ȼɨɡɜɨɞɹ ɜ ɤɜɚɞɪɚɬ ɦɨɞɭɥɢ ɚɦɩɥɢɬɭɞ (10), ɧɚɯɨɞɢɦ ɢɫɤɨɦɵɟ ɜɟɪɨɹɬɧɨɫɬɢ
ɩɟɪɟɯɨɞɨɜ:
2
)1(
,1
D
nW
nn
, (11)
22
)1(
,1
DD
nW
nn
. (12)
ɗɬɢ ɮɨɪɦɭɥɵ ɩɪɢɦɟɧɢɦɵ ɜ ɫɥɭɱɚɟ «ɫɥɚɛɨɝɨ» ɢɫɬɨɱɧɢɤɚ ɜɧɟɲɧɟɣ ɫɢɥɵ, ɤɨɝɞɚ
ɧɟɪɚɜɟɧɫɬɜɨ
1
2
D
ɜɵɩɨɥɧɹɟɬɫɹ ɫ ɧɟɨɛɯɨɞɢɦɵɦ ɡɚɩɚɫɨɦ.
ȿɫɥɢ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɱɢɫɥɨ ɤɚɤ ɱɢɫɥɨ ɱɚɫɬɢɰ-ɤɜɚɧɬɨɜ ɫ ɷɧɟɪɝɢɟɣ
, ɬɨ ɮɨɪɦɭɥɟ (11) ɦɨɠɧɨ ɞɚɬɶ ɩɪɨɫɬɨɟ ɨɛɴɹɫɧɟɧɢɟ: ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ
ɢɫɬɨɱɧɢɤɨɦ ɜɨɡɦɭɳɟɧɢɹ ɛɭɞɟɬ ɩɨɝɥɨɳɟɧɚ ɷɧɟɪɝɢɹ
n
0
Z=
0
Z= , ɪɚɜɧɚ ɜɟɪɨɹɬɧɨɫɬɢ
ɩɨɝɥɨɳɟɧɢɹ
2
D
ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɤɜɚɧɬɚ, ɩɨɦɧɨɠɟɧɧɨɣ ɧɚ ɱɢɫɥɨ ɤɜɚɧɬɨɜ ,n
131
ɢɦɟɸɳɢɯɫɹ ɜ ɧɚɱɚɥɶɧɨɦ ɫɨɫɬɨɹɧɢɢ. ɉɪɢ ɷɬɨɦ ɜ ɮɨɪɦɭɥɟ (12) ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ,
ɧɟ ɢɫɱɟɡɚɸɳɟɟ ɩɪɢ 0, ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɟɪɨɹɬɧɨɫɬɶ «ɫɩɨɧɬɚɧɧɨɝɨ»
ɪɨɠɞɟɧɢɹ ɤɜɚɧɬɚ ɷɧɟɪɝɢɢ ɢɫɬɨɱɧɢɤɨɦ ɜɨɡɦɭɳɟɧɢɹ, ɚ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ,
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɟ ɱɢɫɥɭ ɭɠɟ ɢɦɟɸɳɢɯɫɹ ɤɜɚɧɬɨɜ, ɫɥɟɞɭɟɬ ɢɫɬɨɥɤɨɜɚɬɶ ɤɚɤ
ɜɟɪɨɹɬɧɨɫɬɶ «ɫɬɢɦɭɥɢɪɨɜɚɧɧɨɝɨ» ɪɨɠɞɟɧɢɹ ɤɜɚɧɬɚ: ɢɦɟɸɳɢɟɫɹ ɜ ɧɚɱɚɥɶɧɨɦ
ɫɨɫɬɨɹɧɢɢ ɱɚɫɬɢɰɵ-ɤɜɚɧɬɵ ɫɬɢɦɭɥɢɪɭɸɬ ɪɨɠɞɟɧɢɟ ɟɳɟ ɨɞɧɨɣ ɬɚɤɨɣ ɠɟ
ɱɚɫɬɢɰɵ.
n
0
Z=
ɋɨɝɥɚɫɧɨ (4) ɜɫɟ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ ɜɨɡɦɭɳɟɧɢɹ
ɫ
ɪɚɜɧɵ ɧɭɥɸ. ɉɟɪɟɯɨɞɵ, ɞɥɹ ɤɨɬɨɪɵɯ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ
ɜɨɡɦɭɳɟɧɢɹ ɪɚɜɧɵ ɧɭɥɸ, ɜ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ
ɡɚɩɪɟɳɟɧɧɵɦɢ. ɗɬɨɬ ɬɟɪɦɢɧ ɨɞɧɚɤɨ ɧɟ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɡɚɩɪɟɳɟɧɧɵɣ ɩɟɪɟɯɨɞ
ɜɨɨɛɳɟ ɧɟ ɜɨɡɦɨɠɟɧ. ȼɟɪɨɹɬɧɨɫɬɶ ɬɚɤɨɝɨ ɩɟɪɟɯɨɞɚ ɢɫɱɟɡɚɟɬ ɥɢɲɶ ɜ ɩɟɪɜɨɦ
ɩɨɪɹɞɤɟ, ɚ ɜ ɜɵɫɲɢɯ ɩɨɪɹɞɤɚɯ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɜɟɪɨɹɬɧɨɫɬɶ ɡɚɩɪɟɳɟɧɧɨɝɨ
ɩɟɪɟɯɨɞɚ ɭɠɟ ɦɨɠɟɬ ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɧɭɥɹ. ȼ ɷɬɨɦ ɥɟɝɤɨ ɭɛɟɞɢɬɶɫɹ ɧɚ
ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɡɞɟɫɶ ɩɪɢɦɟɪɟ ɨɫɰɢɥɥɹɬɨɪɚ, ɜɨɡɦɭɳɚɟɦɨɝɨ ɡɚɞɚɧɧɨɣ ɜɧɟɲɧɟɣ
ɫɢɥɨɣ (ɭɩɪɚɠɧɟɧɢɟ ɞɥɹ ɱɢɬɚɬɟɥɹ).
nn
V
c
1rz
c
nn
ɇɟ
ɨɛɪɚɳɚɹɫɶ ɤ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ [8], ɱɬɨ ɟɫɥɢ
ɧɚɱɚɥɶɧɵɦ ɫɨɫɬɨɹɧɢɟɦ ɨɫɰɢɥɥɹɬɨɪɚ ɹɜɥɹɟɬɫɹ ɨɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɬɨ ɩɨɫɥɟ
ɨɤɨɧɱɚɧɢɹ ɞɟɣɫɬɜɢɹ ɜɧɟɲɧɟɣ ɫɢɥɵ ɨɫɰɢɥɥɹɬɨɪ ɨɤɚɠɟɬɫɹ ɜ ɤɨɝɟɪɟɧɬɧɨɦ
ɫɨɫɬɨɹɧɢɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɦɫɹ ɤɨɦɩɥɟɤɫɧɵɦ ɩɚɪɚɦɟɬɪɨɦ (9). ɋɜɨɣɫɬɜɚ ɬɚɤɨɝɨ
ɫɨɫɬɨɹɧɢɹ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɥɢ ɜ ɡɚɞɚɱɟ 6.9.
9.10. Ɋɟɲɟɧɢɟ )(t\ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ
ɜ ɫɥɭɱɚɟ ɞɜɭɯɭɪɨɜɧɟɜɨɣ ɫɢɫɬɟɦɵ ɢɳɟɦ ɜ ɜɢɞɟ)(
ˆˆ
tVH
\
H
1)()(
/
1
1
=ti
etCt 2)(
/
2
2
=ti
etC
H
, (1)
ɝɞɟ ɮɭɧɤɰɢɢ
ɢ )ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ:)(
1
tC (
2
tC
°
¯
°
®
.
~~
,
~~
222121
2
212111
1
CVCV
dt
dC
i
CVCV
dt
dC
i
=
=
(2)
Ɂɞɟɫɶ
nn
ti
nn
VeV
nn
c
HH
c
c
=/)(
~
– ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ ɜɨɡɦɭɳɟɧɢɹ ɜ
ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ. Ɇɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɭɤɚɡɚɧɧɨɝɨ ɜ ɭɫɥɨɜɢɢ
ɡɚɞɚɱɢ ɨɩɟɪɚɬɨɪɚ
titi
eMeMtV
ZZ
ˆˆ
)( , (3)
ɩɪɢɫɭɬɫɬɜɭɸɳɢɟ ɜ ɭɪɚɜɧɟɧɢɹɯ (2), ɢɦɟɸɬ ɜɢɞ:
132
titi
eMeMV
ZZ
111111
~
,
titi
eMeMV
ZZ
222222
~
, (4)
ti
ti
eMeMVV
'
'HH
21
)/)(2(
122112
21
~~
=
, (5)
ɝɞɟ ɩɨɫɪɟɞɫɬɜɨɦ
' ɨɛɨɡɧɚɱɟɧɚ «ɪɚɫɫɬɪɨɣɤɚ» ɱɚɫɬɨɬɵ ɜɨɡɦɭɳɟɧɢɹ ɩɨ
ɨɬɧɨɲɟɧɢɸ ɤ «ɱɚɫɬɨɬɟ ɩɟɪɟɯɨɞɚ»
=/)(
12
HH :
ZHH ' =/)(
12
. (6)
ɋɨɝɥɚɫɧɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ, ɜɟɥɢɱɢɧɚ
' ɦɚɥɚ ɜ ɫɪɚɜɧɟɧɢɢ ɫ ɱɚɫɬɨɬɚɦɢ
,.
=/)(
12
HHZ
ȼɫɟ ɜɟɥɢɱɢɧɵ (4), ɚ ɬɚɤɠɟ ɬɨ ɫɥɚɝɚɟɦɨɟ ɜ (5), ɤɨɬɨɪɨɟ ɫɨɞɟɪɠɢɬ ɜ
ɩɨɤɚɡɚɬɟɥɟ ɷɤɫɩɨɧɟɧɬɵ ɭɞɜɨɟɧɧɭɸ ɱɚɫɬɨɬɭ ɩɟɪɟɯɨɞɚ, ɛɵɫɬɪɨ ɨɫɰɢɥɥɢɪɭɸɬ ɩɨ
ɫɪɚɜɧɟɧɢɸ ɫ ɱɥɟɧɚɦɢ . ɉɪɟɞɩɨɥɚɝɚɹ, ɱɬɨ ɩɪɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɭɪɚɜɧɟɧɢɣ
(2) ɩɨ ɢɧɬɟɪɜɚɥɭ ɜɪɟɦɟɧɢ
ti
e
'r
Z!!
/
1
t
ɛɵɫɬɪɨ ɨɫɰɢɥɥɢɪɭɸɳɢɟ ɫɥɚɝɚɟɦɵɟ ɞɚɞɭɬ
ɥɢɲɶ ɦɚɥɵɣ ɜɤɥɚɞ ɜ ɢɫɤɨɦɵɟ ɮɭɧɤɰɢɢ ɢ ), ɨɬɛɪɨɫɢɦ ɜɫɟ ɬɚɤɢɟ
ɫɥɚɝɚɟɦɵɟ. ɉɟɪɟɣɞɹ ɭɤɚɡɚɧɧɵɦ ɨɛɪɚɡɨɦ ɨɬ ɫɢɫɬɟɦɵ ɬɨɱɧɵɯ ɭɪɚɜɧɟɧɢɣ (2) ɤ
ɩɪɢɛɥɢɠɟɧɧɵɦ ɭɪɚɜɧɟɧɢɹɦ, ɢɦɟɟɦ:
)(
1
tC (
2
tC
°
°
¯
°
°
®
'
'
.
,
1
212
2
211
Ce
Mi
dt
dC
Ce
Mi
dt
dC
ti
ti
=
=
(7)
Ɋɟɲɢɦ ɫɢɫɬɟɦɭ ɩɪɢɛɥɢɠɟɧɧɵɯ ɭɪɚɜɧɟɧɢɣ (7) ɬɨɱɧɨ. ɂɫɤɥɸɱɢɦ ɢɡ ɷɬɢɯ
ɭɪɚɜɧɟɧɢɣ ɨɞɧɭ ɢɡ ɞɜɭɯ ɧɟɢɡɜɟɫɬɧɵɯ ɮɭɧɤɰɢɣ, ɧɚɩɪɢɦɟɪ,.
Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨ
)(
1
tC
t
ɢ ɢɡɛɚɜɥɹɹɫɶ ɨɬ ɢ ɫ
ɩɨɦɨɳɶɸ ɨɛɨɢɯ ɭɪɚɜɧɟɧɢɣ (7), ɩɨɥɭɱɚɟɦ ɨɞɧɨɪɨɞɧɨɟ ɥɢɧɟɣɧɨɟ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɞɥɹ :
1
CdtdC /
1
)(
2
tC
2
2
2
21
2
2
2
2
C
M
dt
dC
i
d
t
Cd
=
'
. (8)
ɑɚɫɬɧɵɟ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (8) ɢɳɟɦ ɜ ɜɢɞɟ
ti
eAtC
Q
)(
2
,0z
A
. (9)
ɉɨɞɫɬɚɜɢɜ (9) ɜ (8), ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɩɚɪɚɦɟɬɪɚ
Q:
0/
2
2
21
2
'QQ =M
. (10)
ɇɚɣɞɹ ɟɝɨ ɤɨɪɧɢ ɢ ,
1
Q
2
Q
133
2
2
21
2
1
/4/2/ =M'' Q ,
2
2
21
2
2
/4/2/ =M'' Q , (11)
ɩɨɥɭɱɚɟɦ ɞɜɚ ɥɢɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵɯ ɪɟɲɟɧɢɹ ɜɢɞɚ (9). Ɉɛɳɟɟ ɪɟɲɟɧɢɟ
ɭɪɚɜɧɟɧɢɹ (8) ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɥɢɧɟɣɧɭɸ ɤɨɦɛɢɧɚɰɢɸ ɧɚɣɞɟɧɧɵɯ ɱɚɫɬɧɵɯ
ɪɟɲɟɧɢɣ ɫ ɩɪɨɢɡɜɨɥɶɧɵɦɢ ɩɨɫɬɨɹɧɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ
A
ɢ
B
:
Q ti
eAtC
1
)(
2
ti
eB
2
Q
. (12)
ɉɪɢ ɷɬɨɦ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ ɫɢɫɬɟɦɵ (7) ɞɚɟɬ ɧɚɦ ɮɭɧɤɰɢɸ :)(
1
tC
titi
ti
eBeAe
M
tC
21
21
21
1
)(
QQ
'
QQ
=
. (13)
ɉɨɫɬɨɹɧɧɵɟ
A
ɢ
B
ɨɩɪɟɞɟɥɹɸɬɫɹ ɧɚɱɚɥɶɧɵɦɢ ɞɚɧɧɵɦɢ, ɤɨɬɨɪɵɟ ɫɨɝɥɚɫɧɨ
ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɢɦɟɸɬ ɜɢɞ:
1)0(
1
C ,0)0(
2
C . (14)
ɋ ɩɨɦɨɳɶɸ (12)–(14) ɧɚɯɨɞɢɦ, ɱɬɨ : =2/
21
MBA , ɝɞɟ ɩɨɫɪɟɞɫɬɜɨɦ :
ɨɛɨɡɧɚɱɟɧɚ ɜɟɥɢɱɢɧɚ :2/)(
12
QQ
2
2
21
2
4
=
M
'
: . (15)
ȼ ɪɟɡɭɥɶɬɚɬɟ, ɧɚɣɞɟɧɧɵɟ ɮɭɧɤɰɢɢ
ɢ )ɩɪɢɧɢɦɚɸɬ ɫɥɟɞɭɸɳɢɣ
ɨɤɨɧɱɚɬɟɥɶɧɵɣ ɜɢɞ:
)(
1
tC (
2
tC
¸
¸
¹
·
¨
¨
©
§
:
:
'
:
'
t
i
tetC
ti
sin
2
cos)(
2/
1
, (16)
te
Mi
tC
ti
:
:
'
sin)(
2/
21
2
=
. (17)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɚɱɚɥɶɧɨɟ ɫɬɚɰɢɨɧɚɪɧɨɟ ɫɨɫɬɨɹɧɢɟ
1 ɞɜɭɯɭɪɨɜɧɟɜɨɣ
ɫɢɫɬɟɦɵ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɜɨɡɦɭɳɟɧɢɹ ɫ ɱɚɫɬɨɬɨɣ , ɛɥɢɡɤɨɣ ɤ
, ɩɟɪɟɯɨɞɢɬ ɜ ɫɭɩɟɪɩɨɡɢɰɢɸ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ
Z
=/)(
12
HH 1 ɢ 2 ɫ
ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ (16) ɢ (17). Ʉɜɚɞɪɚɬɵ ɦɨɞɭɥɟɣ ɷɬɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɜɟɪɨɹɬɧɨɫɬɢ ɢ )ɨɛɧɚɪɭɠɟɧɢɹ ɫɢɫɬɟɦɵ ɤ
ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ
)(
1
tW (
2
tW
t
ɜ ɫɨɫɬɨɹɧɢɹɯ 1 ɢ 2,ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ɋɨɝɥɚɫɧɨ (15)–(17)
ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɷɬɢɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢɦɟɸɬ ɜɢɞ:
134
)(1sin
2
cos)(
2
2
1
tWt
i
ttW :
:
'
:
, (18)
)2cos1(
2
sin)(
22
2
21
2
21
2
t
M
t
M
tW :
:
:
:
=
=
. (19)
ɂɡ (19) ɜɢɞɧɨ, ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ 2 – ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ
ɜɪɟɦɟɧɢ ɫ ɱɚɫɬɨɬɨɣ . ɉɪɢ ɬɨɱɧɨɦ ɪɟɡɨɧɚɧɫɟ (ɬɨ ɟɫɬɶ ɩɪɢ 0) ɜɟɪɨɹɬɧɨɫɬɶ
ɨɛɧɚɪɭɠɟɧɢɹ ɫɢɫɬɟɦɵ ɜ ɫɨɫɬɨɹɧɢɢ
1
o
:
2 '
2 ɩɪɢɧɢɦɚɟɬ ɜɢɞ
¸
¸
¹
·
¨
¨
©
§
=
tM
tW
21
2
2
cos1
2
1
)( . (20)
Ɍɨɬ ɮɚɤɬ, ɱɬɨ ɜɟɪɨɹɬɧɨɫɬɶ (20) ɢɡɦɟɧɹɟɬɫɹ ɨɬ ɧɭɥɹ ɞɨ ɟɞɢɧɢɰɵ, ɨɡɧɚɱɚɟɬ, ɱɬɨ
ɫɢɫɬɟɦɚ ɩɟɪɢɨɞɢɱɟɫɤɢ ɫ ɞɨɫɬɨɜɟɪɧɨɫɬɶɸ ɩɟɪɟɯɨɞɢɬ ɢɡ ɫɨɫɬɨɹɧɢɹ
1 ɜ
ɫɨɫɬɨɹɧɢɟ
2 ɢ ɜɨɡɜɪɚɳɚɟɬɫɹ ɢɡ 2 ɜ 1.
ɉɨ ɯɨɞɭ ɜɵɜɨɞɚ ɮɨɪɦɭɥ (18), (19) ɧɟ ɞɟɥɚɥɨɫɶ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɨ ɦɚɥɨɫɬɢ
ɦɚɬɪɢɱɧɨɝɨ ɷɥɟɦɟɧɬɚ . Ɍɟɩɟɪɶ ɫɞɟɥɚɟɦ ɬɚɤɨɟ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɢ ɭɛɟɞɢɦɫɹ ɜ
ɬɨɦ, ɱɬɨ ɟɫɥɢ ɡɚ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɟ ɜɪɟɦɹ
21
M
t
ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ ,
ɨɩɢɫɵɜɚɟɦɚɹ ɜɵɪɚɠɟɧɢɟɦ (19), ɡɚ ɫɱɟɬ ɦɚɥɨɫɬɢ ɦɚɬɪɢɱɧɨɝɨ ɷɥɟɦɟɧɬɚ
ɨɫɬɚɟɬɫɹ ɦɚɥɨɣ ɜ ɫɪɚɜɧɟɧɢɢ ɫ ɟɞɢɧɢɰɟɣ, ɬɨ ɮɨɪɦɭɥɚ (19) ɩɪɢɜɨɞɢɬ ɤ ɢɡɜɟɫɬɧɨɦɭ
ɪɟɡɭɥɶɬɚɬɭ ɩɟɪɜɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ ɧɟɫɬɚɰɢɨɧɚɪɧɨɣ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ.
21
o
21
M
ɉɪɢ
' =
21
M ɢɦɟɟɦ == 2/)(2/
12
ZHH '|: , ɬɚɤ ɱɬɨ ɜ ɩɟɪɜɨɦ
ɧɟɢɫɱɟɡɚɸɳɟɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨ ɜɵɪɚɠɟɧɢɟ (19) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
21
M
2
12
12
2
2
212
)(
)2/)((sin4
)(
ZHH
ZHH
=
== t
MtW
. (21)
ȿɝɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤɠɟ ɜ ɞɪɭɝɨɦ ɜɢɞɟ:
),(
2
)(
12
2
212
ZHH
S
=
=
tFMttW , (22)
ɝɞɟ ɮɭɧɤɰɢɹ
=
=
2/
)2/(sin
1
),(
2
2
t
t
tF
H
H
S
{H
(23)
ɨɬ ɚɪɝɭɦɟɧɬɚ ɜɟɞɟɬ ɫɟɛɹ ɩɨɞɨɛɧɨ «ɪɚɡɦɵɬɨɣ» ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɢ ɨɛɪɚɳɚɟɬɫɹ ɜ
ɞɟɥɶɬɚ-ɮɭɧɤɰɢɸ ɩɪɢ
H
fo
t
:
135
)(),( HGoH
t
F , fo
t
. (24)
Ⱦɥɹ ɤɚɠɞɨɝɨ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɢɧɬɟɪɜɚɥɚ ɜɪɟɦɟɧɢ
W
t
ɲɢɪɢɧɚ GH
ɪɚɡɦɵɬɨɣ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ (23) ɩɨ ɩɨɪɹɞɤɭ ɜɟɥɢɱɢɧɵ ɫɨɫɬɚɜɥɹɟɬ
WGH
/
~ = . (25)
ȼɨ ɦɧɨɝɢɯ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱɚɯ ɯɚɪɚɤɬɟɪɧɵɣ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ ɦɟɠɞɭ
ɚɤɬɨɦ ɩɪɢɝɨɬɨɜɥɟɧɢɹ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɢ ɪɟɝɢɫɬɪɚɰɢɟɣ ɤɨɧɟɱɧɨɝɨ
ɫɨɫɬɨɹɧɢɹ ɬɚɤɨɜ, ɱɬɨ
W
ZHHGH =
~)(
12
ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɨɠɧɨ ɜ (22)
ɩɪɨɢɡɜɟɫɬɢ ɡɚɦɟɧɭ (24). ȼ ɪɟɡɭɥɶɬɚɬɟ ɦɵ ɩɨɥɭɱɚɟɦ ɢɡɜɟɫɬɧɭɸ ɮɨɪɦɭɥɭ ɞɥɹ
ɜɟɪɨɹɬɧɨɫɬɢ ɩɟɪɟɯɨɞɚ
ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ:ttWw /)(
221
)(
2
12
2
2121
ZHHG
S
=
=
Mw . (26)
ɉɨɹɫɧɢɦ ɪɨɥɶ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɜ (26) ɫɥɟɞɭɸɳɢɦ ɩɪɢɦɟɪɨɦ. Ɋɚɫɫɦɨɬɪɢɦ
ɷɥɟɤɬɪɨɧ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɹɦɟ ɫ ɞɜɭɦɹ ɭɪɨɜɧɹɦɢ ɷɧɟɪɝɢɢ (ɞɜɭɯɭɪɨɜɧɟɜɚɹ
ɦɨɞɟɥɶ ɚɬɨɦɚ). ɉɭɫɬɶ ɩɟɪɟɯɨɞ 2ɨɛɭɫɥɨɜɥɟɧ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ
ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɞɢɩɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɚɬɨɦɚ
1
o
rd
ˆ
ˆ
e (ɝɞɟ – ɨɩɟɪɚɬɨɪ ɤɨɨɪɞɢɧɚɬ
ɷɥɟɤɬɪɨɧɚ) ɫ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɩɨɥɟɦ
r
ˆ
titi
eett
ZZ
Z EEEE
2
1
2
1
cos)((27)
ɥɢɧɟɣɧɨ ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ (
E –
ɚɦɩɥɢɬɭɞɧɵɣ ɜɟɤɬɨɪ ɜɨɥɧɵ). ɉɪɢ ɷɬɨɦ ɨɩɟɪɚɬɨɪ ɜɨɡɦɭɳɟɧɢɹ ɢɦɟɟɬ ɜɢɞ:
titi
eettV
ZZ
EdEdEd
ˆ
2
1
ˆ
2
1
)(
ˆ
)(
ˆ
. (28)
Ɂɚɜɢɫɢɦɨɫɬɶ ɩɨɥɹ ɨɬ ɤɨɨɪɞɢɧɚɬ ɡɞɟɫɶ ɧɟ ɭɱɬɟɧɚ, ɬɚɤ ɤɚɤ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ
ɞɥɢɧɚ ɜɨɥɧɵ ɦɧɨɝɨ ɛɨɥɶɲɟ ɪɚɡɦɟɪɚ ɚɬɨɦɚ.
ɉɪɢ 0
12
!HH|Z ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɤ ɩɪɟɞɫɬɚɜɥɟɧɢɸ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟɞɥɟɧɧɨ ɨɫɰɢɥɥɢɪɭɸɳɢɦ ɫɬɚɧɨɜɢɬɫɹ ɩɟɪɜɵɣ ɱɥɟɧ ɜ ɩɪɚɜɨɣ
ɫɬɨɪɨɧɟ (28); ɨɞɧɢɦ ɷɬɢɦ ɱɥɟɧɨɦ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɩɟɪɟɯɨɞɚ:
2/)(
2
1
1
ˆ
2
2
1
212121
EM edEdEd , (29)
ɝɞɟ
E
/
Ee – ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɧɚɩɪɚɜɥɟɧɢɹ ɩɨɥɹɪɢɡɚɰɢɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ
ɜɨɥɧɵ,
E
– ɚɦɩɥɢɬɭɞɧɨɟ ɡɧɚɱɟɧɢɟ ɧɚɩɪɹɠɟɧɧɨɫɬɢ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ, –
ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɨɩɟɪɚɬɨɪɚ ɞɢɩɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɚɬɨɦɚ ɞɥɹ ɩɟɪɟɯɨɞɚ .
ȼɵɪɚɡɢɦ ɦɧɨɠɢɬɟɥɶ
21
d
21 o
2
E
ɜ
136
4/
2
2
21
2
21
EM ed
ɱɟɪɟɡ ɭɫɪɟɞɧɟɧɧɭɸ ɡɚ ɩɟɪɢɨɞ ɢɧɬɟɧɫɢɜɧɨɫɬɶ
I
ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɵ
(ɭɫɪɟɞɧɟɧɧɚɹ ɩɨ ɜɪɟɦɟɧɢ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɷɧɟɪɝɢɢ),
22
2
8
cos
4
)()(
4
E
c
t
Ec
tt
c
I
S
Z
S
u
S
BE , (30)
ɢ ɩɨɞɫɬɚɜɢɦ ɪɟɡɭɥɶɬɚɬ ɜ (26). Ɉɛɨɡɧɚɱɚɹ
1212
/)( ZZ HH = ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ
)/1()(
12
== ZHHG )(
12
ZZZG , ɩɨɥɭɱɚɟɦ:
)(
4
12
2
21
2
2
21
ZZZG
S
I
c
w ed
=
. (31)
Ɍɟɩɟɪɶ ɨɬɤɚɠɟɦɫɹ ɨɬ ɭɫɥɨɜɢɹ ɫɬɪɨɝɨɣ ɦɨɧɨɯɪɨɦɚɬɢɱɧɨɫɬɢ ɜɨɥɧɵ (27). Ȼɭɞɟɦ
ɫɱɢɬɚɬɶ, ɱɬɨ ɷɧɟɪɝɢɹ ɜɨɥɧɵ ɪɚɫɩɪɟɞɟɥɟɧɚ ɩɨ ɧɟɤɨɬɨɪɨɦɭ ɢɧɬɟɪɜɚɥɭ ɱɚɫɬɨɬ,
ɜɤɥɸɱɚɸɳɟɦɭ ɡɧɚɱɟɧɢɟ
12
ZZ , ɢ ɱɬɨ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɜ (31) ɦɨɠɟɬ ɛɵɬɶ
ɡɚɩɢɫɚɧɚ ɤɚɤ ZZ d
J
I
)(,ɝɞɟ )(Z
J
– ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɢɧɬɟɧɫɢɜɧɨɫɬɢ. ȼ
ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɥɧɚɹ ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɩɪɟɞɫɬɚɜɥɹɸɳɚɹ
ɫɨɛɨɣ ɫɭɦɦɭ ɜɤɥɚɞɨɜ ɨɬ ɫɩɟɤɬɪɚɥɶɧɵɯ ɫɨɫɬɚɜɥɹɸɳɢɯ ɩɨɥɹ ɫ ɛɥɢɡɤɢɦɢ
ɡɧɚɱɟɧɢɹɦɢ ɱɚɫɬɨɬ, ɞɨɥɠɧɚ ɨɩɪɟɞɟɥɹɬɶɫɹ ɩɭɬɟɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɜɵɪɚɠɟɧɢɹ
(31) ɩɨ :
Z
ZZZGZZ
S
³
)()(
4
12
2
21
2
2
Jd
c
w ed
=
)(
4
12
2
21
2
2
ZZ
S
J
c
ed
=
. (32)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɩɪɢɦɟɪɟ ɪɨɥɶ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɜ (26)
ɫɨɫɬɨɢɬ ɜ ɜɵɞɟɥɟɧɢɢ ɢɡ ɫɩɟɤɬɪɚ ɱɚɫɬɨɬ ɜɨɡɦɭɳɚɸɳɟɝɨ ɩɨɥɹ ɬɟɯ ɫɨɫɬɚɜɥɹɸɳɢɯ,
ɱɚɫɬɨɬɚ ɤɨɬɨɪɵɯ ɞɨɫɬɚɬɨɱɧɨ ɛɥɢɡɤɚ ɤ ɱɚɫɬɨɬɟ ɩɟɪɟɯɨɞɚ
ZZ
12
=/)(
12
HH .
Ⱦɪɭɝɨɣ ɩɪɢɦɟɪ ɩɪɢɦɟɧɟɧɢɹ ɮɨɪɦɭɥɵ (26) ɫɜɹɡɚɧ ɫ ɡɚɞɚɱɟɣ ɨ ɩɟɪɟɯɨɞɚɯ
ɜɨɡɦɭɳɚɟɦɨɣ ɫɢɫɬɟɦɵ ɫ ɧɚɱɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣ
1
H
ɜ ɥɸɛɨɟ ɫɨɫɬɨɹɧɢɟ ɫɪɟɞɢ
ɦɧɨɠɟɫɬɜɚ ɤɨɧɟɱɧɵɯ ɫɨɫɬɨɹɧɢɣ ɫ ɧɟɩɪɟɪɵɜɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɷɧɟɪɝɢɢ
2
H .
ɉɭɫɬɶ – ɱɢɫɥɨ ɬɚɤɢɯ ɫɨɫɬɨɹɧɢɣ ɜ ɢɧɬɟɪɜɚɥɟ ɷɧɟɪɝɢɢ ɜɛɥɢɡɢ
ɡɧɚɱɟɧɢɹ
22
)( HHU d
f 2
Hd
ZH H =
12
. Ɉɛɨɡɧɚɱɢɦ:
f
H
H
2
,
i
H ZH =
1
(ɜɟɥɢɱɢɧɭ ɦɨɠɧɨ
ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɤɚɤ ɷɧɟɪɝɢɸ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɫ ɭɱɟɬɨɦ ɢɦɟɸɳɟɝɨɫɹ
«ɤɜɚɧɬɚ» ɜɨɡɦɭɳɚɸɳɟɝɨ ɩɨɥɹ ɫ ɷɧɟɪɝɢɟɣ
i
H
Z= , ɩɪɢ ɩɨɝɥɨɳɟɧɢɢ ɤɨɬɨɪɨɝɨ
ɫɢɫɬɟɦɚ ɩɟɪɟɯɨɞɢɬ ɜ ɤɨɧɟɱɧɨɟ ɫɨɫɬɨɹɧɢɟ ɫ ɷɧɟɪɝɢɟɣ
f
H ). ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɥɧɚɹ
ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɨɜ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɩɪɨɫɭɦɦɢɪɨɜɚɧɧɚɹ ɩɨ ɜɫɟɦ
ɭɤɚɡɚɧɧɵɦ ɤɨɧɟɱɧɵɦ ɫɨɫɬɨɹɧɢɹɦ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ:
137
)(
2
2
ifif
Mw HHG
S
³
=
fff
dHHU )(
2
2
if
M
=
S
)(
if
HU . (33)
Ɏɨɪɦɭɥɚ (33) ɢɡɜɟɫɬɧɚ ɤɚɤ ɡɨɥɨɬɨɟ ɩɪɚɜɢɥɨ ɤɜɚɧɬɨɜɨɣ ɦɟɯɚɧɢɤɢ. ɇɚɥɢɱɢɟɦ
ɞɟɥɶɬɚ-ɮɭɧɤɰɢɢ ɜ ɷɬɨɦ ɪɚɜɟɧɫɬɜɟ, ɨɱɟɜɢɞɧɨ, ɜɵɪɚɠɟɧ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ
ɷɧɟɪɝɢɢ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ.
9.11. ɍɤɚɡɚɧɧɵɦ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɩɨɬɟɧɰɢɚɥɚɦ ɜɧɟɲɧɟɝɨ ɩɨɥɹ
0
M ,
t
c
t
ZZ sin)
/
()( EA (1)
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ
t
t
c
t
Z M
w
w
cos
1
)( E
A
E . (2)
ɋ ɭɱɟɬɨɦ ɷɧɟɪɝɢɢ ɤɭɥɨɧɨɜɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɷɥɟɤɬɪɨɧɚ ɢ ɹɞɪɚ ɜ ɚɬɨɦɟ
ɜɨɞɨɪɨɞɚ,
, ɝɚɦɢɥɶɬɨɧɢɚɧ ɚɬɨɦɚ ɜɨ ɜɧɟɲɧɟɦ ɩɨɥɟ (1) ɡɚɩɢɲɟɦ ɜ
ɮɨɪɦɟ
rerU /)(
2
¸
¹
·
¨
©
§
)()(
2
1
ˆ
2
rUt
c
e
i
m
H A= )(
2
ˆ
)()(
2
ˆ
2
2
22
t
mc
e
t
mc
e
rU
m
ApA
p
. (3)
Ɂɞɟɫɶ , ɩɪɢɱɟɦ ɜ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ (3) ɦɵ ɭɱɥɢ, ɱɬɨ =i
p
ˆ
0)(
t
A . ȼ
ɥɢɧɟɣɧɨɦ ɩɨ ɜɧɟɲɧɟɦɭ ɩɨɥɸ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨɫɥɟɞɧɢɦ ɫɥɚɝɚɟɦɵɦ ɜ (3)
ɩɪɟɧɟɛɪɟɝɚɟɦ. Ɍɨɝɞɚ
)(
ˆˆˆ
0
tVHH , (4)
ɝɞɟ ) – ɧɟɜɨɡɦɭɳɟɧɧɵɣ ɝɚɦɢɥɶɬɨɧɢɚɧ ɚɬɨɦɚ, ɚ ɨɩɟɪɚɬɨɪ
ɜɨɡɦɭɳɟɧɢɹ ɢɦɟɟɬ ɜɢɞ
(2/
ˆ
ˆ
2
0
rUmH p
)()(
ˆ
t
mc
ie
tV A
=
Z
Z ti
e
m
e
2
E=
ti
e
m
e
Z
Z
2
E=
. (5)
ɉɪɢ 0ɩɟɪɟɯɨɞɵ ɫ ɩɨɝɥɨɳɟɧɢɟɦ ɚɬɨɦɨɦ ɷɧɟɪɝɢɢ ɜɧɟɲɧɟɝɨ ɩɨɥɹ
ɜɵɡɵɜɚɸɬɫɹ ɜɨɡɦɭɳɟɧɢɟɦ, ɤɨɬɨɪɨɦɭ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɜ ɩɪɚɜɨɣ
ɫɬɨɪɨɧɟ ɪɚɜɟɧɫɬɜɚ (5). Ɇɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɩɟɪɟɯɨɞɚ ɟɫɬɶ
!Z
ifif
m
e
M \\
Z
E
2
=
. (6)
ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɡɚɞɚɱɟ
i
\ – ɷɬɨ
s
1-ɫɨɫɬɨɹɧɢɟ ɷɥɟɤɬɪɨɧɚ ɜ ɚɬɨɦɟ
ɜɨɞɨɪɨɞɚ; ɟɝɨ ɜɨɥɧɨɜɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɜɢɞ:
138
ar
i
e
a
/
3
1
)(
S
\
r , . (7)
22
/ meaa
B
=
Ʉɨɧɟɱɧɨɟ ɫɨɫɬɨɹɧɢɟ
f
\ ɩɪɢɛɥɢɠɟɧɧɨ ɨɩɢɫɵɜɚɟɬɫɹ ɤɚɤ ɫɨɫɬɨɹɧɢɟ ɫɜɨɛɨɞɧɨ
ɞɜɢɠɭɳɟɝɨɫɹ ɷɥɟɤɬɪɨɧɚ ɫ ɬɟɦ ɢɥɢ ɢɧɵɦ ɢɦɩɭɥɶɫɨɦ
k
p = :
rk
r
\
i
f
e
V
1
)(
. (8)
ɇɚɩɪɚɜɢɦ ɨɫɶ ɜɞɨɥɶ , ɬɨ ɟɫɬɶ – ɩɚɪɚɥɥɟɥɶɧɨ ɧɚɩɪɚɜɥɟɧɢɸ ɩɨɥɹɪɢɡɚɰɢɢ
ɜɨɥɧɵ. Ɍɨɝɞɚ
z E
z
E
ww
/
E , ɢ ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ (6) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
ari
if
e
z
ed
aV
m
Ee
M
/3
3
1
2
³
w
w
S
Z
rk
r
=
. (9)
Ɉɛɨɡɧɚɱɢɜ J a
/
1,ɩɪɢɫɬɭɩɢɦ ɤ ɜɵɱɢɫɥɟɧɢɸ ɩɪɢɫɭɬɫɬɜɭɸɳɟɝɨ ɜ (9)
ɢɧɬɟɝɪɚɥɚ:
w
w
J
³
ri
e
z
ed
rk
r
3
J
J
³
ri
e
r
z
ed
rk
r
3
ri
z
e
r
ed
k
i
J
³
w
w
J
1
3 rk
r . (10)
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɢɧɬɟɝɪɚɥɚ ɜ ɩɪɚɜɨɣ ɫɬɨɪɨɧɟ (10) ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɮɟɪɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɨɣ ɤɨɨɪɞɢɧɚɬ ɫ ɩɨɥɹɪɧɨɣ ɨɫɶɸ, ɩɚɪɚɥɥɟɥɶɧɨɣ ɜɟɤɬɨɪɭ
k
. ɍɱɢɬɵɜɚɹ, ɱɬɨ
ɜ ɬɚɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ T
c
cosrkrk ,,ɢ ɨɛɨɡɧɚɱɚɹ
, ɢɦɟɟɦ:
M
c
T
c
T
c
dddrrd sin
23
r
K T
c
cos
KS
³³³
KJ
f
J
1
10
3
2
1
rkirri
ededrre
r
ed
rk
r
22
4
J
S
k
. (11)
ɉɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɡɚɜɢɫɢɬ ɨɬ ɤɜɚɞɪɚɬɚ ɜɨɥɧɨɜɨɝɨ ɜɟɤɬɨɪɚ
()ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɟɟɬ ɨɞɢɧ ɢ ɬɨɬ ɠɟ ɜɢɞ ɜ ɥɸɛɨɣ ɫɢɫɬɟɦɟ
ɤɨɨɪɞɢɧɚɬ. ɉɨɞɫɬɚɧɨɜɤɚ (11) ɜ ɩɪɚɜɭɸ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ (10) ɞɚɟɬ:
2222
zyx
kkkk
w
w
³
ari
e
z
ed
/3 rk
r
z
k
aka
i
222
)(
8
S
. (12)
Ɂɞɟɫɶ ,T coskk
z
T – ɭɝɨɥ ɦɟɠɞɭ ɧɚɩɪɚɜɥɟɧɢɹɦɢ ɢɦɩɭɥɶɫɚ ɷɥɟɤɬɪɨɧɚ
k
= ɢ
ɜɧɟɲɧɟɝɨ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ E .
139