Лыков С.Н., Гасумянц В.Э., Рыков С.А. Квантовая механика. Учебное пособие
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ɜɵɪɨɠɞɟɧɢɟ. ȼ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɜɨɡɧɢɤɲɟɦɭ
ɭɤɚɡɚɧɧɵɦ ɨɛɪɚɡɨɦ ɫɟɦɟɣɫɬɜɭ ɭɪɨɜɧɟɣ ɷɧɟɪɝɢɢ ɨɬɜɟɱɚɸɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ
ɩɪɚɜɢɥɶɧɵɟ ɫɨɫɬɨɹɧɢɹ – ɥɢɧɟɣɧɵɟ ɤɨɦɛɢɧɚɰɢɢ
g
ɫɨɫɬɨɹɧɢɣ
)0(
k
ɫ
ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ , ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɦɢ ɫɢɫɬɟɦɟ
)0(
k
C
g
ɨɞɧɨɪɨɞɧɵɯ ɥɢɧɟɣɧɵɯ
ɭɪɚɜɧɟɧɢɣ:
0)(
)0(
)1(
GH
¦
c
c
cc
k
k
kkkk
CV , (IX)
ɝɞɟ ;
g
nnkk ,...,,
1
c
)1(
H – ɩɚɪɚɦɟɬɪ, ɡɧɚɱɟɧɢɹɦɢ ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɞɜɢɝɢ
ɭɪɨɜɧɟɣ ɷɧɟɪɝɢɢ. ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (IX) ɢɦɟɟɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ ɩɪɢ
ɭɫɥɨɜɢɢ
0)(det
)1(
GH
cc
kkkk
V , (X)
ɧɚɡɵɜɚɟɦɨɦ
ɫɟɤɭɥɹɪɧɵɦ (ɢɥɢ ɜɟɤɨɜɵɦ) ɭɪɚɜɧɟɧɢɟɦ. ɗɬɨ ɟɫɬɶ ɭɪɚɜɧɟɧɢɟ
ɫɬɟɩɟɧɢ
g
ɞɥɹ . ȿɝɨ ɤɨɪɧɢ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɢɫɤɨɦɵɟ ɩɨɩɪɚɜɤɢ ɩɟɪɜɨɝɨ
ɩɪɢɛɥɢɠɟɧɢɹ ɤ ɜɵɪɨɠɞɟɧɧɨɦɭ ɭɪɨɜɧɸ ɷɧɟɪɝɢɢ.
)1(
H
ȼ ɧɟɫɬɚɰɢɨɧɚɪɧɨɦ ɫɥɭɱɚɟ ɨɩɟɪɚɬɨɪ ɜɨɡɦɭɳɟɧɢɹ )ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ.
Ɋɟɲɟɧɢɟ ɜɨɥɧɨɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɒɪɟɞɢɧɝɟɪɚ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (I)
ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɮɨɪɦɟ ɪɚɡɥɨɠɟɧɢɹ ɩɨ ɧɟɜɨɡɦɭɳɟɧɧɵɦ ɫɬɚɰɢɨɧɚɪɧɵɦ
ɫɨɫɬɨɹɧɢɹɦ ɫ ɡɚɜɢɫɹɳɢɦɢ ɨɬ ɜɪɟɦɟɧɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ :
(
ˆ
tV
)(
tC
k
¦
H
\
k
ti
k
ketCt
k
)0(
/
)0(
)()(
=
. (XI)
ɇɟɢɡɜɟɫɬɧɵɟ ɮɭɧɤɰɢɢ )ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɢɫɬɟɦɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ
ɭɪɚɜɧɟɧɢɣ:
(tC
k
¦
c
cc
k
kkk
k
tCtV
dt
tdC
i )()(
~
)(
= , (XII)
ɝɞɟ )(
~
tV
kk
c
– ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ ɜɨɡɦɭɳɟɧɢɹ ɜ ɬɚɤ ɧɚɡɵɜɚɟɦɨɦ
ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ. Ɍɚɤɢɟ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɜɵɱɢɫɥɹɸɬɫɹ ɩɨ
ɧɟɜɨɡɦɭɳɟɧɧɵɦ ɫɬɚɰɢɨɧɚɪɧɵɦ ɫɨɫɬɨɹɧɢɹɦ
)0(
k
ɢ ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ:
c
c
ketVektV
tHitHi
kk
== /
ˆ
/
ˆ
00
)(
ˆ
)(
~
)(
/)(
)0()0(
tVe
kk
ti
k
k
c
HH
c
=
. (XIII)
ɉɨɞɫɬɚɧɨɜɤɚ ɜ (XII) ɜɟɥɢɱɢɧ )ɜ ɮɨɪɦɟ ɪɹɞɨɜ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ,(tC
k
100
)(tC
k
)0(
k
C )(
)1(
tC
k
...)(
)2(
tC
k
, (XIV)
ɩɨɡɜɨɥɹɟɬ ɪɟɲɚɬɶ ɭɪɚɜɧɟɧɢɹ (XII) ɦɟɬɨɞɨɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɢɬɟɪɚɰɢɣ. ɉɪɢ
ɷɬɨɦ ɞɨɥɠɧɵ ɛɵɬɶ ɡɚɞɚɧɵ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ.
ɋɮɨɪɦɭɥɢɪɭɟɦ ɪɟɡɭɥɶɬɚɬ, ɩɨɥɭɱɚɸɳɢɣɫɹ ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɩɭɬɟɦ. ȿɫɥɢ ɜ
ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ
t
c
ɤɜɚɧɬɨɜɨɟ ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ ɫɨɜɩɚɞɚɥɨ ɫ ɨɞɧɢɦ
ɢɡ ɧɟɜɨɡɦɭɳɟɧɧɵɯ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɧɚɩɪɢɦɟɪ,
)0(
/
)0(
)( nett
ti
n
c
c
\
c
H
c
=
,
ɬɨ
nk
k
C
c
G
)0(
,
³
c
c
t
t
nk
k
tVdt
i
tC )(
~
)(
11
)1(
=
, (XV)
¦
³³
cc
c
¸
¹
·
¨
©
§
1
2
11
)(
~
)(
~
)(
1212
2
)2(
k
t
t
t
t
nkkk
k
tVtVdtdt
i
tC
=
. (XVI)
Ɉɛɳɚɹ ɮɨɪɦɭɥɚ ɞɥɹ )ɩɪɢ ɷɬɨɦ ɢɦɟɟɬ ɜɢɞ(
)(
tC
m
k
¦
³³ ³
¦
cc
c
c
¸
¹
·
¨
©
§
1
11
2
1
)(
~
)...(
~
......)(
111
)(
m
m
m
k
t
t
t
t
nkmkk
t
t
mm
k
m
m
k
tVtVdtdtdt
i
tC
=
. (XVII)
ȼɟɥɢɱɢɧɚ
2
)(tC
k
ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ ɫɢɫɬɟɦɵ ɡɚ
ɜɪɟɦɹ
t
t
c
ɢɡ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ n
c
ɜ ɤɨɧɟɱɧɨɟ ɧɟɜɨɡɦɭɳɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ
k
ɩɨɞ ɜɥɢɹɧɢɟɦ ɜɨɡɦɭɳɟɧɢɹ ). ȼ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɞɥɹ ɜɟɪɨɹɬɧɨɫɬɢ
ɬɚɤɨɝɨ ɩɟɪɟɯɨɞɚ ɩɪɢ
(
ˆ
tV
n
k
c
z
ɢɦɟɟɦ:
2
)1(
)1(
)(tCW
k
nk
c
, nk
c
z (XVIII)
ȿɫɥɢ ɞɥɹ ɞɚɧɧɨɝɨ ɩɟɪɟɯɨɞɚ ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɨɩɟɪɚɬɨɪɚ ɜɨɡɦɭɳɟɧɢɹ ɪɚɜɟɧ
ɧɭɥɸ, ɬɨ ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ ɫɥɟɞɭɟɬ ɜɵɱɢɫɥɹɬɶ ɜɨ ɜɬɨɪɨɦ ɩɨɪɹɞɤɟ ɬɟɨɪɢɢ
ɜɨɡɦɭɳɟɧɢɣ – ɤɚɤ
2
)2(
)(tC
k
. Ʉɪɢɬɟɪɢɟɦ ɩɪɢɦɟɧɢɦɨɫɬɢ ɩɨɥɭɱɚɟɦɵɯ ɬɚɤɢɦ
ɨɛɪɚɡɨɦ ɪɟɡɭɥɶɬɚɬɨɜ ɹɜɥɹɟɬɫɹ ɦɚɥɨɫɬɶ ɡɧɚɱɟɧɢɣ ɜɟɪɨɹɬɧɨɫɬɢ ɩɟɪɟɯɨɞɚ ɩɨ
ɫɪɚɜɧɟɧɢɸ ɫ ɟɞɢɧɢɰɟɣ. ȼ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɷɬɨ ɭɫɥɨɜɢɟ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɞɥɹ
ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɨ ɜɟɪɨɹɬɧɨɫɬɢ ɩɟɪɟɯɨɞɚ kn o
c
ɫɥɟɞɭɟɬ ɜɟɪɧɭɬɶɫɹ ɤ ɬɨɱɧɵɦ
ɭɪɚɜɧɟɧɢɹɦ (XII) ɢ ɩɨɩɵɬɚɬɶɫɹ ɧɚɣɬɢ ɢɧɨɣ ɩɪɢɛɥɢɠɟɧɧɵɣ ɦɟɬɨɞ ɨɬɵɫɤɚɧɢɹ
ɮɭɧɤɰɢɣ .)(tC
k
ɉɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɩɟɪɟɯɨɞɨɜ ɫɥɨɠɧɨɣ ɤɜɚɧɬɨɜɨɣ ɫɢɫɬɟɦɵ ɜ ɫɨɫɬɨɹɧɢɹ,
101
ɩɪɢɧɚɞɥɟɠɚɳɢɟ ɧɟɩɪɟɪɵɜɧɨɦɭ ɫɩɟɤɬɪɭ ɷɧɟɪɝɢɢ, ɩɪɢɦɟɧɹɸɬ ɡɨɥɨɬɨɟ ɩɪɚɜɢɥɨ
Ɏɟɪɦɢ ɞɥɹ ɜɟɪɨɹɬɧɨɫɬɢ ɩɟɪɟɯɨɞɨɜ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ:w
w )(
2
2
ifif
M HU
S
=
, (XIX)
ɝɞɟ ɢi
f
– ɫɨɜɨɤɭɩɧɨɫɬɢ ɤɜɚɧɬɨɜɵɯ ɱɢɫɟɥ ɫɢɫɬɟɦɵ ɜ ɰɟɥɨɦ ɜ ɧɚɱɚɥɶɧɨɦ
(initial) ɢ ɤɨɧɟɱɧɨɦ (final) ɫɨɫɬɨɹɧɢɹɯ;)(
if
HU – ɩɥɨɬɧɨɫɬɶ ɤɨɧɟɱɧɵɯ ɫɨɫɬɨɹɧɢɣ
ɫ ɷɧɟɪɝɢɟɣ, ɪɚɜɧɨɣ ɷɧɟɪɝɢɢ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ; – ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ
ɩɟɪɟɯɨɞɚ, ɨɩɪɟɞɟɥɟɧɧɵɦ ɨɛɪɚɡɨɦ ɫɨɩɨɫɬɚɜɥɹɟɦɵɣ ɨɩɟɪɚɬɨɪɭ ɜɨɡɦɭɳɟɧɢɹ .
if
M
)(
ˆ
tV
Ɂɨɥɨɬɨɟ ɩɪɚɜɢɥɨ ɩɪɢɦɟɧɢɦɨ, ɜ ɱɚɫɬɧɨɫɬɢ, ɤ ɨɩɢɫɚɧɢɸ ɪɚɫɫɟɹɧɢɹ ɱɚɫɬɢɰɵ
ɜ ɧɟ ɡɚɜɢɫɹɳɟɦ ɨɬ ɜɪɟɦɟɧɢ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɩɨɥɟ
)(r
U
. ȿɫɥɢ ɧɟɜɨɡɦɭɳɟɧɧɵɟ
ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ɱɚɫɬɢɰɵ ɜɵɛɪɚɬɶ ɜ ɜɢɞɟ ɩɥɨɫɤɢɯ ɜɨɥɧ (ɫɨɫɬɨɹɧɢɹ ɫ
ɨɩɪɟɞɟɥɟɧɧɵɦ ɢɦɩɭɥɶɫɨɦ, ɨɬɜɟɱɚɸɳɢɟ ɫɜɨɛɨɞɧɨɦɭ ɞɜɢɠɟɧɢɸ), ɚ ɜ ɤɚɱɟɫɬɜɟ
ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɩɟɪɟɯɨɞɚ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨ
)(r
U
ɢɫɩɨɥɶɡɨɜɚɬɶ
ɜɟɥɢɱɢɧɵ
pp
c
U
ˆ
, ɬɨ ɪɟɡɭɥɶɬɚɬ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ ɮɨɪɦɭɥɵ Ȼɨɪɧɚ
(ɩɟɪɜɨɟ ɛɨɪɧɨɜɫɤɨɟ ɩɪɢɛɥɢɠɟɧɢɟ) ɞɥɹ ɫɟɱɟɧɢɹ ɪɚɫɫɟɹɧɢɹ ɱɚɫɬɢɰɵ ɜ
ɬɟɥɟɫɧɵɣ ɭɝɨɥ :
Vd
p
:d
2
2
2
)(
2
q
p
u
m
d
d
¸
¸
¹
·
¨
¨
©
§
S
:
V
=
. (XX)
Ɂɞɟɫɶ m – ɦɚɫɫɚ ɱɚɫɬɢɰɵ,
=/)( ppq
c
,
)()(
3
rrq
rq
Uedu
i
³
. (XXI)
ɍɫɥɨɜɢɟɦ ɩɪɢɦɟɧɢɦɨɫɬɢ ɩɪɢɛɥɢɠɟɧɢɹ Ȼɨɪɧɚ ɹɜɥɹɟɬɫɹ ɜɵɩɨɥɧɟɧɢɟ ɯɨɬɹ ɛɵ
ɨɞɧɨɝɨ ɢɡ ɧɟɪɚɜɟɧɫɬɜ
2
2
0
ma
U
=
, ka
ma
U
2
2
0
=
, (XXII)
ɝɞɟ – ɯɚɪɚɤɬɟɪɧɚɹ ɜɟɥɢɱɢɧɚ ɩɨɬɟɧɰɢɚɥɚ, – ɟɝɨ ɪɚɞɢɭɫ,
0
Ua== //2 pmk H .
Ɂɚɞɚɱɢ
9.1. ȼɵɩɢɫɚɬɶ ɩɨɩɪɚɜɤɭ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɤ ɷɧɟɪɝɟɬɢɱɟɫɤɢɦ ɭɪɨɜɧɹɦ
ɱɚɫɬɢɰɵ ɜ ɛɟɫɤɨɧɟɱɧɨɣ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɨɞɧɨɦɟɪɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɹɦɟ
ɲɢɪɢɧɨɣ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɜɨɡɦɭɳɟɧɢɹ
)1(
n
H
a )(
x
V
.
102
9.2. Ⱦɥɹ ɱɚɫɬɢɰɵ ɦɚɫɫɵ ɫ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɡɚɪɹɞɨɦ , ɧɚɯɨɞɹɳɟɣɫɹ ɜ
ɨɫɧɨɜɧɨɦ ɫɨɫɬɨɹɧɢɢ ɜ ɛɟɫɤɨɧɟɱɧɨɣ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɹɦɟ
ɲɢɪɢɧɨɣ , ɧɚɣɬɢ ɜ ɩɟɪɜɵɯ ɞɜɭɯ ɩɨɪɹɞɤɚɯ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɫɞɜɢɝ
ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɭɪɨɜɧɹ, ɜɵɡɜɚɧɧɵɣ ɨɞɧɨɪɨɞɧɵɦ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɩɨɥɟɦ
mq
a
E
.
9.3. ɇɚɣɬɢ ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ ɚɧɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɥɢɧɟɣɧɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ ɫ
ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ
4322
0
2
2
1
2
ˆ
ˆ
xxxm
m
p
H
EDZ ,
ɪɚɫɫɦɚɬɪɢɜɚɹ ɱɥɟɧɵ ɤɚɤ ɜɨɡɦɭɳɟɧɢɟ.
43
xx ED
9.4. ɋɬɚɰɢɨɧɚɪɧɵɦ ɫɨɫɬɨɹɧɢɹɦ ɷɥɟɤɬɪɨɧɚ ɜ ɫɮɟɪɢɱɟɫɤɢ ɫɢɦɦɟɬɪɢɱɧɨɣ
ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɹɦɟ )(
r
U
ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ . ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ
ɩɨɥɟ )
)0(
ln
r
H
(
r
U
ɧɟ ɹɜɥɹɟɬɫɹ ɤɭɥɨɧɨɜɫɤɢɦ, ɬɚɤ ɱɬɨ ɭɪɨɜɧɢ ɷɧɟɪɝɢɢ ɧɟ ɜɵɪɨɠɞɟɧɵ ɩɨ
ɚɡɢɦɭɬɚɥɶɧɨɦɭ ɤɜɚɧɬɨɜɨɦɭ ɱɢɫɥɭ
l
. Ɋɚɫɫɦɨɬɪɟɬɶ ɢɡɦɟɧɟɧɢɟ ɷɬɢɯ
ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ, ɜɨɡɧɢɤɚɸɳɟɟ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɟɝɨ ɞɨɫɬɚɬɨɱɧɨ
ɫɥɚɛɨɝɨ ɨɞɧɨɪɨɞɧɨɝɨ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ (ɷɮɮɟɤɬ ɒɬɚɪɤɚ).
E
9.5.
Ɋɚɫɫɦɨɬɪɟɬɶ ɢɡɦɟɧɟɧɢɟ ɩɟɪɜɨɝɨ ɢɡ ɭɪɨɜɧɟɣ ɷɧɟɪɝɢɢ ɜɨɡɛɭɠɞɟɧɧɵɯ
ɫɨɫɬɨɹɧɢɣ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ, ɜɨɡɧɢɤɚɸɳɟɟ ɜ ɥɢɧɟɣɧɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨ
ɜɧɟɲɧɟɦɭ ɷɥɟɤɬɪɢɱɟɫɤɨɦɭ ɩɨɥɸ (ɥɢɧɟɣɧɵɣ ɷɮɮɟɤɬ ɒɬɚɪɤɚ ɧɚ ɩɪɢɦɟɪɟ
p
s
2,2-
ɫɨɫɬɨɹɧɢɣ ɚɬɨɦɚ ɜɨɞɨɪɨɞɚ).
9.6. Ⱦɥɹ ɷɥɟɤɬɪɨɧɚ ɜ ɫɮɟɪɢɱɟɫɤɢ ɫɢɦɦɟɬɪɢɱɧɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɹɦɟ )(
r
U
ɨɛɫɭɞɢɬɶ ɪɚɫɱɟɬ ɢɡɦɟɧɟɧɢɹ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɧɟɲɧɟɝɨ
ɨɞɧɨɪɨɞɧɨɝɨ ɦɚɝɧɢɬɧɨɝɨ ɩɨɥɹ (ɷɮɮɟɤɬ Ɂɟɟɦɚɧɚ).
B
ɍɤɚɡɚɧɢɹ. ȼɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ ɉɚɭɥɢ ɫ ɭɱɟɬɨɦ ɷɧɟɪɝɢɢ
ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜɢɞɚ . Ɉɩɪɟɞɟɥɢɬɶ ɥɢɧɟɣɧɵɣ ɢ
ɤɜɚɞɪɚɬɢɱɧɵɣ ɩɨ ɦɚɝɧɢɬɧɨɦɭ ɩɨɥɸ ɨɩɟɪɚɬɨɪɵ ɜɨɡɦɭɳɟɧɢɹ.
sl
ˆ
ˆ
)(
D r
9.7. Ɋɚɫɫɦɨɬɪɟɬɶ ɷɮɮɟɤɬ Ɂɟɟɦɚɧɚ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ
s
1-ɭɪɨɜɧɸ ɚɬɨɦɚ
ɜɨɞɨɪɨɞɚ ɫ ɭɱɟɬɨɦ ɫɜɟɪɯɬɨɧɤɨɣ ɫɬɪɭɤɬɭɪɵ ɷɬɨɝɨ ɭɪɨɜɧɹ.
9.8. ɇɚ ɩɪɢɦɟɪɟ ɫɨɫɬɨɹɧɢɣ ɷɥɟɤɬɪɨɧɚ ɢ ɪɚɫɫɦɨɬɪɟɬɶ ɫ
ɩɨɦɨɳɶɸ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɷɮɮɟɤɬ Ɂɟɟɦɚɧɚ ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɨɦ ɨɬɧɨɲɟɧɢɢ
ɦɚɝɧɢɬɧɨɣ ɷɧɟɪɝɢɢ ɤ ɷɧɟɪɝɢɢ ɫɩɢɧ-ɨɪɛɢɬɚɥɶɧɨɝɨ ɪɚɫɳɟɩɥɟɧɢɹ.
2/1
p
2/3
p
9.9. ɇɚ ɥɢɧɟɣɧɵɣ ɨɫɰɢɥɥɹɬɨɪ ɫ ɡɚɪɹɞɨɦ ɧɚɱɢɧɚɹ ɫ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ
ɢ ɞɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ ɞɟɣɫɬɜɨɜɚɥɨ ɨɞɧɨɪɨɞɧɨɟ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ
q
0
0
t
1
t
103
)(
t
E
, ɢɡɦɟɧɹɜɲɟɟɫɹ ɜɨ ɜɪɟɦɟɧɢ ɩɨ ɩɪɨɢɡɜɨɥɶɧɨɦɭ ɡɚɤɨɧɭ. ɉɪɢ
0
tt
ɨɫɰɢɥɥɹɬɨɪ ɧɚɯɨɞɢɥɫɹ ɜ -ɦ ɫɬɚɰɢɨɧɚɪɧɨɦ ɫɨɫɬɨɹɧɢɢ. ɇɚɣɬɢ ɜ ɩɟɪɜɨɦ ɩɨɪɹɞɤɟ
ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɜɟɪɨɹɬɧɨɫɬɢ ɩɟɪɟɯɨɞɨɜ ɜ ɪɚɡɥɢɱɧɵɟ ɫɨɫɬɨɹɧɢɹ ɞɥɹ .
n
1
tt !
9.10. Ʉɜɚɧɬɨɜɚɹ ɫɢɫɬɟɦɚ, ɢɦɟɸɳɚɹ ɞɜɚ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɹ 1, 2 ɫ
ɧɟɜɵɪɨɠɞɟɧɧɵɦɢ ɭɪɨɜɧɹɦɢ ɷɧɟɪɝɢɢ
1
H ,
2
H (ɞɜɭɯɭɪɨɜɧɟɜɚɹ ɫɢɫɬɟɦɚ),
ɩɨɞɜɟɪɝɚɟɬɫɹ ɩɟɪɢɨɞɢɱɟɫɤɨɦɭ ɜɨɡɦɭɳɟɧɢɸ ɜɢɞɚ
titi
eMeMtV
ZZ
ˆˆ
)(
ˆ
ɫ ɱɚɫɬɨɬɨɣ
Z, ɛɥɢɡɤɨɣ ɤ :=/)(
12
HH
Z'HH
=/)(
12
,
ɝɞɟ – ɦɚɥɚɹ ɜɟɥɢɱɢɧɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ
' =/)(
12
HH . ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ ɫɢɫɬɟɦɚ ɧɚɯɨɞɢɥɚɫɶ ɜ ɨɞɧɨɦ ɢɡ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ:
1)0( \ t . ɇɚɣɬɢ ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ )(t
\
ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
ɜɪɟɦɟɧɢ.
ɍɤɚɡɚɧɢɹ. ȼɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɭɪɚɜɧɟɧɢɹɦɢ (XII) ɜɜɟɞɟɧɢɹ ɤ ɞɚɧɧɨɣ ɝɥɚɜɟ,
ɨɩɭɫɬɢɜ ɜ ɧɢɯ ɛɵɫɬɪɨ ɨɫɰɢɥɥɢɪɭɸɳɢɟ ɫɥɚɝɚɟɦɵɟ. ȼ ɛɨɥɟɟ ɨɛɳɟɣ ɩɨɫɬɚɧɨɜɤɟ
ɩɨɞɨɛɧɭɸ ɡɚɞɚɱɭ ɫɦ. ɜ [1,8].
9.11. Ⱥɬɨɦ ɜɨɞɨɪɨɞɚ, ɧɚɯɨɞɹɳɢɣɫɹ ɜ ɨɫɧɨɜɧɨɦ ɫɨɫɬɨɹɧɢɢ, ɩɨɞɜɟɪɝɚɟɬɫɹ
ɞɟɣɫɬɜɢɸ ɥɢɧɟɣɧɨ ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ
ɜɨɥɧɵ ɜɵɫɨɤɨɣ ɱɚɫɬɨɬɵ:
R
y!!Z= . ȿɫɥɢ ɩɪɢ ɷɬɨɦ ɞɥɢɧɚ ɜɨɥɧɵ ɜɟɥɢɤɚ ɩɨ
ɫɪɚɜɧɟɧɢɸ ɫ ɪɚɡɦɟɪɨɦ ɚɬɨɦɚ (ɱɬɨ ɢɦɟɟɬ ɦɟɫɬɨ ɩɪɢ DSZ
/
4
R
y= , ɝɞɟ
ce =
/
2
D ), ɬɨ ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɜɨɥɧɵ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɨɞɧɨɪɨɞɧɵɦ. ɇɚɣɬɢ ɜ
ɩɟɪɜɨɦ ɩɨɪɹɞɤɟ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɜɟɪɨɹɬɧɨɫɬɶ ɢɨɧɢɡɚɰɢɢ ɚɬɨɦɚ
ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɩɨɥɟɦ ɬɚɤɨɣ ɜɨɥɧɵ.
ɍɤɚɡɚɧɢɹ. ɍɫɥɨɜɢɟ
R
y!!Z= ɩɨɡɜɨɥɹɟɬ ɩɪɢɧɹɬɶ ɭɩɪɨɳɚɸɳɟɟ
ɩɪɟɞɩɨɥɨɠɟɧɢɟ, ɱɬɨ ɤɨɧɟɱɧɨɟ ɫɨɫɬɨɹɧɢɟ ɷɥɟɤɬɪɨɧɚ, ɜɵɥɟɬɟɜɲɟɝɨ ɢɡ ɚɬɨɦɚ,
ɩɪɢɛɥɢɠɟɧɧɨ ɨɩɢɫɵɜɚɟɬɫɹ ɩɥɨɫɤɨɣ ɜɨɥɧɨɣ. ɋɱɢɬɚɬɶ, ɱɬɨ ɨɞɧɨɪɨɞɧɨɟ
ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɩɨɥɟ ɫ ɱɚɫɬɨɬɨɣ
Z ɢ ɜɟɤɬɨɪɧɨɣ ɚɦɩɥɢɬɭɞɨɣ ɡɚɞɚɧɨ
ɩɨɫɪɟɞɫɬɜɨɦ ɩɨɬɟɧɰɢɚɥɨɜ
E
0 M ,
t
c
t
ZZ sin)
/
()( EA .
9.12. ȼ ɭɫɥɨɜɢɹɯ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ, ɧɨ ɞɥɹ ɫɥɭɱɚɹ ɫ ɰɢɪɤɭɥɹɪɧɨ
ɩɨɥɹɪɢɡɨɜɚɧɧɨɣ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɨɣ, ɪɚɫɩɪɨɫɬɪɚɧɹɸɳɟɣɫɹ ɜ
ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ , ɧɚɣɬɢ ɭɝɥɨɜɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɜɟɪɨɹɬɧɨɫɬɢ ɢɨɧɢɡɚɰɢɢ ɚɬɨɦɚ
ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɩɨɥɟɦ ɜɨɥɧɵ.
z
104
9.13. Ɋɚɫɫɦɨɬɪɟɬɶ ɜ ɛɨɪɧɨɜɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɫɟɱɟɧɢɟ
ɪɚɫɫɟɹɧɢɹ ɷɥɟɤɬɪɨɧɚ ɧɚ ɞɜɭɯ ɨɞɢɧɚɤɨɜɵɯ ɫɢɥɨɜɵɯ ɰɟɧɬɪɚɯ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɧɚ
ɪɚɫɫɬɨɹɧɢɢ ɞɪɭɝ ɨɬ ɞɪɭɝɚ.
d
9.14.
ɉɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɨɣ Ȼɨɪɧɚ, ɧɚɣɬɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɫɟɱɟɧɢɟ
ɪɚɫɫɟɹɧɢɹ ɱɚɫɬɢɰɵ ɫ ɡɚɪɹɞɨɦ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ ɧɟɩɨɞɜɢɠɧɨɝɨ ɰɟɧɬɪɚ ɫ
ɡɚɪɹɞɨɦ .
eZ
1
eZ
2
9.15. Ɉɛɫɭɞɢɬɶ ɪɚɫɱɟɬ ɜ ɛɨɪɧɨɜɫɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɫɟɱɟɧɢɹ ɭɩɪɭɝɨɝɨ ɪɚɫɫɟɹɧɢɹ ɛɵɫɬɪɵɯ ɷɥɟɤɬɪɨɧɨɜ ɚɬɨɦɨɦ [1,8,10]. Ɋɟɡɭɥɶɬɚɬ
ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɚɬɨɦɧɵɣ ɮɨɪɦɮɚɤɬɨɪ ,)(
qF
rq
rrq
³
i
endF )()(
3
,
ɝɞɟ ɮɭɧɤɰɢɹ )ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɥɨɬɧɨɫɬɶɸ )ɷɥɟɤɬɪɨɧɧɨɝɨ ɡɚɪɹɞɚ ɜ ɚɬɨɦɟ.(
rn (rne
Ɋɟɲɟɧɢɹ
9.1. ȼɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ ɧɟɜɨɡɦɭɳɟɧɧɵɯ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ
ɱɚɫɬɢɰɵ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɢɦɟɸɬ ɜɢɞ:
¯
®
!
ddS
\
,,0;0
,0;)/(sin/2
)(
axx
axaxna
x
n
(1)
ɝɞɟ . ɉɨɞɫɬɚɜɢɜ (1) ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɨɩɪɚɜɤɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɤ
ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ
...,3,2,1
n
³
f
f
\\ H )()()(
ˆ
)1(
xxVxxdnVn
nnn
, (2)
ɩɨɥɭɱɚɟɦ:
³
S
H
a
n
a
xn
xVxd
a
0
2)1(
sin)(
2
. (3)
Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɯ ɤɜɚɞɪɚɬ ɫɢɧɭɫɚ ɛɵɫɬɪɨ
ɨɫɰɢɥɥɢɪɭɟɬ, ɢ ɟɝɨ ɦɨɠɧɨ ɡɚɦɟɧɢɬɶ ɫɪɟɞɧɢɦ ɡɧɚɱɟɧɢɟɦ, ɪɚɜɧɵɦ 2
n
/
1;ɬɨɝɞɚ:
VxVxd
a
a
n
|H
³
0
)1(
)(
1
, . (4) 1!!n
105
ȼɢɞɧɨ, ɱɬɨ ɩɪɢ 1ɩɨɩɪɚɜɤɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɨɦɟɪɚ ɭɪɨɜɧɹ ɢ ɪɚɜɧɚ
ɭɫɪɟɞɧɟɧɧɨɣ ɜɞɨɥɶ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɹɦɵ ɜɟɥɢɱɢɧɟ ɜɨɡɦɭɳɚɸɳɟɝɨ ɩɨɬɟɧɰɢɚɥɚ.
!!n
)1(
n
H
9.2. ȼɨɡɦɭɳɟɧɢɟɦ ɹɜɥɹɟɬɫɹ ɷɧɟɪɝɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɱɚɫɬɢɰɵ ɫ ɜɧɟɲɧɢɦ
ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɩɨɥɟɦ
E
:
x
E
q
x
V
)( . (1)
ɉɨɩɪɚɜɤɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɤ -ɦɭ ɷɧɟɪɝɟɬɢɱɟɫɤɨɦɭ ɭɪɨɜɧɸ ɪɚɜɧɚn
H nVn
n
ˆ
)1(
aEq
a
xn
xxd
a
Eq
a
2
1
sin
2
0
2
S
³
. (2)
ɗɬɚ ɜɟɥɢɱɢɧɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɨɦɟɪɚ ɭɪɨɜɧɹ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɯɚɪɚɤɬɟɪɢɡɭɟɬ
ɫɞɜɢɝ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɫɩɟɤɬɪɚ ɤɚɤ ɰɟɥɨɝɨ.
ɉɨɩɪɚɜɤɚ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɤ ɷɧɟɪɝɢɢ ɨɫɧɨɜɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ
)0(
1
H
¦
f
HH
H
2
)0(
)0(
1
2
)2(
1
1
ˆ
n
n
Vn
¦
f
HH
2
)0(
)0(
1
2
22
1
ˆ
n
n
xn
Eq , (3)
ɝɞɟ
HH
)0(
)0(
1
n
S
2
22
2ma
=
S
2
2
22
2
n
ma
=
)1(
2
2
2
22
n
ma
S
=
, (4)
1
ˆ
xn
a
x
a
xn
xxd
a
a
SS
³
sinsin
2
0
. (5)
ɇɟ ɩɪɢɜɨɞɹ ɡɞɟɫɶ ɞɟɬɚɥɟɣ ɜɵɱɢɫɥɟɧɢɹ ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ (5), ɜɵɩɢɲɟɦ
ɪɟɡɭɥɶɬɚɬ ɞɥɹ :2tn
°
¯
°
®
S
....,7,5,3;0
,...,6,4,2;
)1()1(
8
1
ˆ
222
n
n
nn
an
xn (6)
ɂɫɩɨɥɶɡɭɹ (4) ɢ (6) ɜ (3), ɩɨɥɭɱɚɟɦ:
¦
S
H
,...6,4,2
244
2
2
224
6
)2(
1
)1()1()1(
128
n
nnn
nEqma
=
2
2
1
EE{ . (7)
Ɋɹɞ (7) ɛɵɫɬɪɨ ɫɯɨɞɢɬɫɹ, ɩɨɷɬɨɦɭ ɜɟɥɢɱɢɧɚ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɪɚɜɧɚ ɩɟɪɜɨɦɭ
ɱɥɟɧɭ ɷɬɨɝɨ ɪɹɞɚ; ɜ ɬɚɤɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ
E ɟɫɬɶ
)2(
1
H
106
2243
/104.4 =qma
|E . (8)
9.3. ɉɪɟɠɞɟ ɜɫɟɝɨ ɩɨɹɫɧɢɦ ɩɪɨɢɫɯɨɠɞɟɧɢɟ ɫɥɚɝɚɟɦɵɯ
43
ˆ
xxV ED (1)
ɜ ɝɚɦɢɥɶɬɨɧɢɚɧɟ
4322
0
2
2
1
2
ˆ
ˆ
xxxm
m
p
H EDZ . (2)
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɡɚɞɚɱɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɪɨɜɧɟɣ
ɱɚɫɬɢɰɵ ɜɛɥɢɡɢ ɞɧɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɹɦɵ )(
x
U
ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɜɢɞɚ.
ȿɫɥɢ ɮɭɧɤɰɢɹ )(
x
U
ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɫɜɨɟɝɨ ɦɢɧɢɦɭɦɚ ɢ ɟɟ ɩɪɨɢɡɜɨɞɧɵɟ ɧɟ
ɢɦɟɸɬ ɨɫɨɛɟɧɧɨɫɬɟɣ, ɬɨ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ )(
x
U
ɜ ɮɨɪɦɟ ɪɹɞɚ Ɍɟɣɥɨɪɚ ɩɨ
ɫɬɟɩɟɧɹɦ ɨɬɤɥɨɧɟɧɢɹ ɤɨɨɪɞɢɧɚɬɵ
x
ɨɬ ɬɨɱɤɢ ɦɢɧɢɦɭɦɚ . ɍɞɨɛɧɨ ɜɵɛɪɚɬɶ
ɷɬɭ ɬɨɱɤɭ ɜ ɤɚɱɟɫɬɜɟ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ, ɢ ɫɱɢɬɚɬɶ ɡɧɚɱɟɧɢɟ ɩɨɬɟɧɰɢɚɥɚ
0
x
0
x
U
ɜ
ɬɨɱɤɟ ɦɢɧɢɦɭɦɚ ɪɚɜɧɵɦ ɧɭɥɸ. ɍɱɬɟɦ ɬɚɤɠɟ, ɱɬɨ ɩɟɪɜɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ
)(
x
U
ɜ ɬɨɱɤɟ ɦɢɧɢɦɭɦɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ. Ɍɨɝɞɚ ɭɤɚɡɚɧɧɵɣ ɫɬɟɩɟɧɧɨɣ ɪɹɞ
ɡɚɩɢɲɟɬɫɹ ɜ ɜɢɞɟ
...)0(
!4
1
)0(
!3
1
)0(
2
1
)(
432
ccc
cc
xUxUxUxU
IV
. (3)
Ɂɞɟɫɶ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ, ɨɱɟɜɢɞɧɨ, ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɬɟɧɰɢɚɥɭ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ
ɨɫɰɢɥɥɹɬɨɪɚ ɫ ɤɜɚɞɪɚɬɨɦ ɫɨɛɫɬɜɟɧɧɨɣ ɱɚɫɬɨɬɵ , ɚ ɫɥɟɞɭɸɳɢɟ
ɫɥɚɝɚɟɦɵɟ ɜ ɬɨɣ ɨɛɥɚɫɬɢ ɡɧɚɱɟɧɢɣ
mU /)0(
2
0
cc
Z
x
, ɝɞɟ ɨɧɢ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ
ɩɟɪɜɵɦ ɱɥɟɧɨɦ, ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɚɧɝɚɪɦɨɧɢɱɟɫɤɢɟ ɩɨɩɪɚɜɤɢ ɤ
ɩɨɬɟɧɰɢɚɥɭ ɨɫɰɢɥɥɹɬɨɪɚ. Ɉɛɨɡɧɚɱɢɜ D
ccc
!3/)0(U ,,ɢ ɨɬɛɪɨɫɢɜ
ɱɥɟɧɵ ɪɹɞɚ (3) ɜɵɫɨɤɨɣ ɫɬɟɩɟɧɢ, ɩɪɢɯɨɞɢɦ ɤ ɡɚɞɚɱɟ ɫ ɝɚɦɢɥɶɬɨɧɢɚɧɨɦ (2).
E !4/)0(
IV
U
ȼ ɧɭɥɟɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɜɨɡɦɭɳɟɧɢɸ (1)
ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɭɪɨɜɧɢ ɨɫɰɢɥɥɹɬɨɪɚ ɢɡɜɟɫɬɧɵ:
)2/1(
0
)0(
Z H n
n
= , ...,2,1,0 n . (4)
ɉɨɩɪɚɜɤɚ ɤ ɷɧɟɪɝɢɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɩɨ D ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ,
D H nxn
n
3)1(
ˆ
0)(
23
\D
³
f
f
xxxd
n
, (5)
ɬɚɤ ɤɚɤ ɩɨɞ ɡɧɚɤɨɦ ɢɧɬɟɝɪɚɥɚ ɫ ɫɢɦɦɟɬɪɢɱɧɵɦɢ ɩɪɟɞɟɥɚɦɢ ɜ (5) ɩɪɢɫɭɬɫɬɜɭɟɬ
ɧɟɱɟɬɧɚɹ ɮɭɧɤɰɢɹ ɩɟɪɟɦɟɧɧɨɣ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ. Ɂɞɟɫɶ ɢ ɧɢɠɟ
n –
107
ɧɟɜɨɡɦɭɳɟɧɧɵɟ ɫɬɚɰɢɨɧɚɪɧɵɟ ɫɨɫɬɨɹɧɢɹ ɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ. Ⱦɥɹ
ɜɵɱɢɫɥɟɧɢɹ ɩɨɩɪɚɜɤɢ ɤ ɷɧɟɪɝɢɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɩɨ D ɧɚɣɞɟɦ ɫɧɚɱɚɥɚ
ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ
3
ˆ
x
. ɉɪɨɳɟ ɜɫɟɝɨ ɷɬɨ ɞɟɥɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ
ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɨɩɟɪɚɬɨɪɚ ɤɨɨɪɞɢɧɚɬɵ ɜ ɮɨɪɦɟ
)
ˆˆ
(2/
ˆ
0
Z aamx = , (6)
ɝɞɟ ɢ
– «ɛɨɡɨɧɧɵɟ» ɨɩɟɪɚɬɨɪɵ ɭɧɢɱɬɨɠɟɧɢɢ ɢ ɪɨɠɞɟɧɢɹ (ɫɦ. ɡɚɞɚɱɭ 6.7),
ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟɫɹ ɫɨɨɬɧɨɲɟɧɢɹɦɢ:
a
ˆ
a
ˆ
1
ˆ
nnna , 11
ˆ
nnna . (7)
ɇɟɫɥɨɠɧɨɟ ɜɵɱɢɫɥɟɧɢɟ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɪɚɜɟɧɫɬɜ (7) ɞɚɟɬ:
Z nxm
32/3
0
ˆ
)/2(
= 3)2)(1( nnnn 13
2/3
nn
1)1(3
2/3
nn 3)3)(2)(1( nnnn . (8)
ɋɤɚɥɹɪɧɨ ɭɦɧɨɠɚɹ (8) ɧɚ
n
c
ɢ ɭɱɢɬɵɜɚɹ ɭɫɥɨɜɢɟ
nn
nn
c
G
c
, ɢɦɟɟɦ:
2/3
0
3
)2/(
ˆ
3 Z mnxn = )2)(1( nnn ,
2/3
0
3
)2/(
ˆ
1 Z mnxn =
2/3
3n ,
2/3
0
3
)2/(
ˆ
1 Z mnxn =
2/3
)1(3 n ,
2/3
0
3
)2/(
ˆ
3 Z mnxn = )3)(2)(1( nnn . (9)
Ʌɢɲɶ ɩɟɪɟɱɢɫɥɟɧɧɵɟ ɡɞɟɫɶ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ
3
ˆ
x
ɦɨɝɭɬ ɛɵɬɶ
ɨɬɥɢɱɧɵ ɨɬ ɧɭɥɹ; ɜ ɷɬɨɦ ɩɟɪɟɱɧɟ ɨɬɫɭɬɫɬɜɭɸɬ ɞɢɚɝɨɧɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ, ɱɬɨ
ɫɨɝɥɚɫɭɟɬɫɹ ɫ (5). ɂɫɩɨɥɶɡɭɹ (4) ɢ (9) ɜ ɮɨɪɦɭɥɟ ɞɥɹ ɩɨɩɪɚɜɤɢ ɤ ɷɧɟɪɝɢɢ ɜɬɨɪɨɝɨ
ɩɨɪɹɞɤɚ ɩɨ
D, ɩɨɥɭɱɚɟɦ (ɫɭɦɦɚ ɫɨɞɟɪɠɢɬ ɱɟɬɵɪɟ ɫɥɚɝɚɟɦɵɯ):
HH
D
c
¦
z
c
c
nn
n
n
nxn
)0(
)0(
2
3
ˆ
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
ZZ
D
30
11
4
15
2
3
00
2
nn
m
=
=
. (10)
Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɨɩɪɚɜɤɢ ɤ ɷɧɟɪɝɢɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɩɨ ɧɚɞɨ ɧɚɣɬɢ
ɞɢɚɝɨɧɚɥɶɧɵɟ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ . ɉɪɢɦɟɧɹɹ ɤ ɨɛɟɢɦ
ɫɬɨɪɨɧɚɦ ɪɚɜɟɧɫɬɜɚ (8) ɨɩɟɪɚɬɨɪ (6) ɢ ɫɥɟɞɹ ɬɟɩɟɪɶ ɬɨɥɶɤɨ ɡɚ ɬɟɦɢ ɱɥɟɧɚɦɢ,
ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɞɚɬɶ ɜɤɥɚɞ ɜ ɞɢɚɝɨɧɚɥɶɧɵɣ ɦɚɬɪɢɱɧɵɣ ɷɥɟɦɟɧɬ ɨɩɟɪɚɬɨɪɚ
E
4
ˆ
xE
4
ˆ
x
, ɜ
ɢɬɨɝɟ ɧɚɣɞɟɦ:
108
E nxn
4
ˆ
2/1
2
3
2
2
0
¸
¸
¹
·
¨
¨
©
§
Z
E
nn
m
=
. (11)
ɋɭɦɦɢɪɭɹ (4), (10) ɢ (11), ɩɨɥɭɱɚɟɦ ɫɥɟɞɭɸɳɟɟ ɩɪɢɛɥɢɠɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ
ɭɪɨɜɧɟɣ ɷɧɟɪɝɢɢ ɚɧɝɚɪɦɨɧɢɱɟɫɤɨɝɨ ɨɫɰɢɥɥɹɬɨɪɚ:
¸
¹
·
¨
©
§
Z H
2
1
0
n
n
=
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
ZZ
D
30
11
4
15
2
3
00
2
nn
m
=
=
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
Z
E
2
1
2
3
2
2
0
nn
m
=
9.4. ɗɧɟɪɝɢɸ ɱɚɫɬɢɰɵ ɫ ɡɚɪɹɞɨɦ ɜ ɨɞɧɨɪɨɞɧɨɦ ɷɥɟɤɬɪɢɱɟɫɤɨɦ ɩɨɥɟ
ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɨɩɟɪɚɬɨɪ ɜɨɡɦɭɳɟɧɢɹ V :
q
E
ˆ
rE
ˆ
ˆ
qV . (1)
ɇɚɩɪɚɜɢɦ ɨɫɶ ɜɞɨɥɶ
E ; ɬɨɝɞɚz
zEqV
ˆ
ˆ
. (2)
ɋɬɚɰɢɨɧɚɪɧɵɟ ɫɨɫɬɨɹɧɢɹ ɱɚɫɬɢɰɵ ɜ ɧɟɜɨɡɦɭɳɟɧɧɨɦ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɩɨɥɟ )(
r
U
ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɤɜɚɧɬɨɜɵɦɢ ɱɢɫɥɚɦɢ ,
r
n
l
,:m
),()()( MT \
mllnmln
YrR
rr
r . (3)
ɋɧɚɱɚɥɚ ɪɚɫɫɦɨɬɪɢɦ ɞɢɚɝɨɧɚɥɶɧɵɟ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɬɨɪɚ
ɜɨɡɦɭɳɟɧɢɹ (2):
mlnVmln
rr
ˆ
mlnzmlnEq
rr
ˆ
. (4)
ɗɬɢ ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɛɪɚɳɚɸɬɫɹ ɜ ɧɭɥɶ:
0)(
ˆ
2
3
\
³
rr
mlnrr
r
zdmlnzmln . (5)
ȼ ɬɨɦ, ɱɬɨ ɢɧɬɟɝɪɚɥ (5) ɪɚɜɟɧ ɧɭɥɸ, ɩɪɨɳɟ ɜɫɟɝɨ ɭɛɟɞɢɬɶɫɹ ɫ ɩɨɦɨɳɶɸ
ɫɥɟɞɭɸɳɢɯ ɫɨɨɛɪɚɠɟɧɢɣ. Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɢɧɜɟɪɫɢɢ
ɤɨɨɪɞɢɧɚɬɧɵɯ ɩɟɪɟɦɟɧɧɵɯ (
rr o ) ɫɨɫɬɨɹɧɢɹ ɫ ɨɩɪɟɞɟɥɟɧɧɨɣ ɜɟɥɢɱɢɧɨɣ
ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ
l
ɨɛɥɚɞɚɸɬ ɨɩɪɟɞɟɥɟɧɧɨɣ ɱɟɬɧɨɫɬɶɸ,,ɢ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɮɭɧɤɰɢɹ
l
)1(
2
mln
r
\ ɱɟɬɧɚ. Ɍɚɤ ɤɚɤ ɦɟɧɹɟɬ ɫɜɨɣ ɡɧɚɤ ɩɪɢ ɡɚɦɟɧɟ
ɜ (5) ɩɟɪɟɦɟɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
z
rr o , ɬɨ ɢɦɟɸɳɚɹɫɹ ɩɨɞ ɢɧɬɟɝɪɚɥɨɦ
ɮɭɧɤɰɢɹ
2
mln
r
z \ ɧɟɱɟɬɧɚ, ɢ ɢɧɬɟɝɪɚɥ ɞɨɥɠɟɧ ɨɛɪɚɬɢɬɶɫɹ ɜ ɧɭɥɶ.
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