Задачи по теоретической механике. Манаков Н.Л - 38 стр.

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ϕ(r) =
r
Z
r
min
ρ
dr
r
2
r
1 −
ρ
2
r
2
−
U(r)
E
.
r = r
min
r = ∞
ϕ
0
= ϕ(∞) = ϕ
1
+ ϕ
2
,
r
min
@
@
@
@
@
¡
¡
¡
¡
¡
a
O
ϕ
1
ϕ
2
*
©
©
©
©
©
©
©
©
©
©
A
A
A
A
A
A
ρ
©
©
©
©
©
©
©
©
©
©
-
H
H
H
HH
H
Hj
H
H
H
H
H
H
χ
ϕ
1
=
a
Z
r
min
ρ
r
2
dr
r
1 +
U
0
E
−
ρ
2
r
2
=
π
2
− arcsin
r
min
a
; ϕ
2
=
∞
Z
a
ρ
r
2
dr
r
1 −
ρ
2
r
2
= arcsin
ρ
a
χ = 2ϕ
0
− π = 2ϕ
1
+ 2ϕ
2
− π = 2 arcsin
ρ
a
− 2 arcsin
r
min
a
,
sin
χ
2
=
ρ
a
r
E
E + U
0
Ã
r
E + U
0
E
−
ρ
2
a
2
−
r
1 −
ρ
2
a
2
!
.
χ(ρ = 0) = χ(ρ > a) = 0; χ
max
= χ(ρ = a) = 2 arcsin
r
U
0
E + U
0
.
U(r) =
½
U
0
, 0 6 r 6 a
0, r > a.
−U
0
U
0
ρ < a E > U
0
+
L
2
2ma
2
L
2
= 2mEρ
2
, E > E
cr
E
cr
=
U
0
(1 − ρ
2
/a
2
)
.
r
min
= ρ
r
E
E − U
0
.
Óðàâíåíèå òðàåêòîðèè ìîæíî çàïèñàòü â âèäå:                                     ©χ
                                                                                    ©
                                                                             ©H
                       Zr            dr                                ©
                                                                          ©      H
                                 ρ                                @           -     H
                                                                                        H
                                                                                        H
             ϕ(r) =          r       r2       .            *        @      rmin ¡¡ HH  jH
                                                            A
                                ρ2 U (r)                      A
                                                                         @ ϕ
                                                                          @ 1 ¡
                                                                                  ¡ a
                      rmin    1− 2 −                            A ρ      ϕ2 ©
                                r    E                            A
                                                                            @¡
                                                                         © O
                                                                    AA ©
Óãîë ïîâîðîòà ðàäèóñ-âåêòîðà ÷àñòèöû ïðè å¼                       ©
                                                           ©               ðèñ. 4.4
äâèæåíèè îò r = rmin äî r = ∞:
    ϕ0 = ϕ(∞) = ϕ1 + ϕ2 , ãäå
        Za        ρ                                  Z∞ ρ dr
                   2
                     dr       π         rmin              2                ρ
  ϕ1 =       r r            = − arcsin       ; ϕ2 =    rr         = arcsin
                   U0 ρ 2     2          a                    ρ2           a
       rmin    1+       − 2                          a  1− 2
                    E    r                                    r
 óãëû ïîâîðîòà ðàäèóñ-âåêòîðà ïðè äâèæåíèè ÷àñòèöû âíóòðè è âíå ÿìû
ñîîòâåòñòâåííî.
4)Óãîë îòêëîíåíèÿ òðàåêòîðèè ÷àñòèöû, ïðîõîäÿùåé ÷åðåç ÿìó, îò ïåðâîíà-
÷àëüíîãî íàïðàâëåíèÿ äâèæåíèÿ (ñì. ðèñ. 4.4):
                                                   ρ           rmin
            χ = 2ϕ0 − π = 2ϕ1 + 2ϕ2 − π = 2 arcsin − 2 arcsin       ,
                                                   a             a
òàê ÷òî                  r       Ãr                  r        !
                 χ ρ         E       E + U0 ρ    2        ρ 2
              sin =                           − 2 − 1− 2 .
                 2      a E + U0       E       a          a
Î÷åâèäíî, ÷òî
                                                                       r
                                                                             U0
      χ(ρ = 0) = χ(ρ > a) = 0;            χmax = χ(ρ = a) = 2 arcsin              .
                                                                           E + U0

Çàäà÷à 4.3. Èññëåäîâàòü äâèæåíèå ÷àñòèöû â ïîëå (ñôåðè÷åñêèé áàðüåð):
              ½
                  U0 , 0 6 r 6 a
    U (r) =
                   0, r > a.
Ðåøåíèå . Äîñòàòî÷íî â ôîðìóëàõ ïðåäûäóùåé çàäà÷è çàìåíèòü −U0 íà U0 .
Ïðè ýòîì äâèæåíèå âñåãäà èíôèíèòíî. ×òîáû ÷àñòèöà ïðîíèêëà âíóòðü ñôå-
                                                                    L2
ðû (ïðè ρ < a), äîñòàòî÷íî, ÷òîáû âûïîëíÿëîñü óñëîâèå E > U0 +          , èëè,
                                                                   2ma2
                                              U0
ñ ó÷åòîì L2 = 2mEρ2 , E > Ecr , ãäå Ecr =       2 /a2 )
                                                        .  ýòîì ñëó÷àå
        r                                 (1 − ρ
            E
rmin = ρ         .
          E − U0
                                             37

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