Задачи по квантовой механике. Часть 2. Алмалиев А.Н - 15 стр.

UptoLike

R(r) = R
I
(r) 0 6 r 6 a R(r) = R
II
(r) r > a
U(r) = r > a R
II
(r) = 0
0 6 r 6 a U(r) = 0
R
I
(0) = 0 R
II
(a) = 0
R(r) r = a
R
00
I
l(l + 1)
r
2
R
I
+ k
2
R
I
= 0.
k
2
= 2µE/}
2
R
I
(r) =
rZ(r)
x = kr
x
2
d
2
Z
l
dx
2
+ x
dZ
l
dx
+
"
x
2
µ
l +
1
2
2
#
Z
l
= 0.
Z
l
(r) = C
1
J
l+
1
2
(kr) + C
2
J
l
1
2
(kr).
R
I
(0) = 0 C
2
= 0 J
ν
(x)
x
ν
R
I
(a) = 0
J
l+
1
2
(ka) = J
l+
1
2
Ã
r
2mE
}
2
a
!
= 0.
l = 0 s
J
1
2
(x) =
sin x
x
sin(ka) = 0
ka = πn n = 1, 2, . . . s
E
ns
=
π}
2
2µa
2
n
2
(n = 1, 2, . . . )
n = 0
l
E
nl
=
}
2
X
2
nl
2µa
2
,
R(r) = RI (r) äëÿ 0 6 r 6 a è R(r) = RII (r) äëÿ r > a.
 ñâÿçè ñ òåì, ÷òî U (r) = ∞ ïðè r > a, RII (r) = 0, è óðàâíåíèå (1.12) íåîá-
õîäèìî ðåøàòü â îáëàñòè 0 6 r 6 a ñ U (r) = 0 è ãðàíè÷íûìè óñëîâèÿìè
RI (0) = 0 è RII (a) = 0. Âòîðîå ãðàíè÷íîå óñëîâèå ñëåäóåò èç ñòàíäàðò-
íûõ óñëîâèé, à èìåííî èç óñëîâèÿ íåïðåðûâíîñòè R(r) â òî÷êå r = a. Â
ðåçóëüòàòå èìååì:

                                 l(l + 1)
                        RI00 −        2
                                          RI + k 2 RI = 0.             (1.31)
                                    r
                                                          √
Çäåñü k 2 = 2µE/}2 . Èñïîëüçîâàíèå ïîäñòàíîâêè RI (r) = rZ(r) ñ ââåäå-
íèåì íîâîé ïåðåìåííîé x = kr ïðèâîäèò óðàâíåíèå (1.31) ê óðàâíåíèþ
äëÿ ôóíêöèé Áåññåëÿ ñ ïîëóöåëûì ïàðàìåòðîì:
                                "     µ      ¶2 #
                    2
                  d  Zl    dZl             1
               x2 2 + x        + x2 − l +         Zl = 0.       (1.32)
                   dx      dx              2

Ðåøåíèå (1.32) ìîæíî çàïèñàòü â âèäå

                    Zl (r) = C1 Jl+ 21 (kr) + C2 J−l− 21 (kr).

Èç óñëîâèÿ RI (0) = 0 íåîáõîäèìî ïîëîæèòü C2 = 0 (ò.ê. ôóíêöèÿ Jν (x) â
íóëå âåäåò ñåáÿ êàê xν ). Èñïîëüçóÿ ãðàíè÷íîå óñëîâèå RI (a) = 0, èìååì:
                                         Ãr      !
                                            2mE
                    Jl+ 12 (ka) = Jl+ 12        a = 0.             (1.33)
                                             }2

Òàêèì îáðàçîì, çíà÷åíèÿ óðîâíåé ýíåðãèè ÷àñòèöû, çàêëþ÷åííîé â ñôå-
ðè÷åñêè ñèììåòðè÷íîé ÿìå ñ áåñêîíå÷íî âûñîêèìè ñòåíêàìè, ÿâëÿþòñÿ
ðåøåíèÿìè óðàâíåíèÿ (1.33).  ÷àñòíîì ñëó÷àå l = 0 (s-ñîñòîÿíèå), ó÷è-
                      sin x
òûâàÿ, ÷òî J 21 (x) =       , âìåñòî (1.33) ïîëó÷àåì óðàâíåíèå sin(ka) = 0,
                        x
îòêóäà ka = πn (n = 1, 2, . . . ). Òîãäà ñïåêòð s-ñîñòîÿíèé èìååò âèä:

                              π}2 2
                      Ens =        n          (n = 1, 2, . . . )
                              2µa2

(n = 0 íå óäîâëåòâîðÿåò ôèçè÷åñêîé ïîñòàíîâêå çàäà÷è).
   Äëÿ ïðîèçâîëüíîãî çíà÷åíèÿ l

                                          }2 Xnl
                                              2
                                  Enl   =        ,
                                          2µa2


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