ВУЗ:
Составители:
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.  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 15
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