∂
i Ψ(r, t)= ˆ
HΨ(r, t) (1)
∂t
Ãàìèëüòîíèàí âûðàæàåòñÿ ñóììîé îïåðàòîðîâ êèíåòè÷åñêîé è ïîòåíöèàëüíîé ýíåðãèé ýëåêòðîíà â ïîòåíöèàëüíîì ïîëå [3]:
H =T +ˆ
ˆˆU
22
mvm (mv)2 p
E = ==
2 · m 2m 2m ⇒ p)2 2 2 2
Tˆ=(ˆ={pˆ==Δ
2m −i} =− 2m− 2m ˆ
U =U(r)
2
ˆ
H =Δ+U(r)
− 2m
Ðàçäåëèì ïåðåìåííûå, âîñïîëüçîâàâøèñü ìåòîäîì Ôóðüå:
Ψ(r, t)=f(r)(t)
2
∂
1
t)= ˆ
Hf(
if(r)
(
r)(t)=(t)Δf(r)+U(r)f(r)
− 2m
|·
∂t
f(r)(t)
i ∂ 1
2
(t)= Δf(r)+U(r)f(r) (2)
(t)∂t f(r) − 2m
Ïîñêîëüêó ëåâàÿ ÷àñòü óðàâíåíèÿ (2) çàâèñèò òîëüêî îò t, à ïðàâàÿ òîëüêî îò r, îáå îíè äîëæíû ðàâíÿòüñÿ îäíîé è òîé æå êîíñòàíòå ðàçäåëåíèÿ:
i ∂
E = (t)
(t)
∂t
2
1
E =
Δf(r)+U(r)f(r)
f(r) − 2m
 ïîñòàâëåííîé çàäà÷å òðåáóåòñÿ îïðåäåëèòü õàðàêòåðèñòèêè ñèñòåìû â ñòàöèîíàðíûõ ñîñòîÿíèÿõ, ïîýòîìó ïðàêòè÷åñêèé èíòåðåñ ïðåäñòàâëÿåò ëèøü ðåøåíèå âòîðîãî óðàâíåíèÿ èç äàííîé ñèñòåìû, ñîäåðæàùåãî ðàñïðåäåëåíèå àìïëèòóäû âîëíîâîé ôóíêöèè.
2
Δf(r)+U(r)f(r)=Ef(r)
− 2m
2 ∂2
Ñ ó÷¼òîì òîãî, ÷òî ïîòåíöèàëüíîå ïîëå îäíîìåðíî (r =x è Δ== ∂x2 ):
2 ∂2 f(x)+U(x)f(x)=Ef(x) (3)
−
2m ∂x2
ˆ
Hf(x)
3
Òàêèì îáðàçîì, ðåøåíèå ñâåëîñü ê èçâåñòíîé â ìàòôèçèêå [1] çàäà÷å íà ñîáñòâåí íûå çíà÷åíèÿ (â îäíîìåðíîì ñëó÷àå: çàäà÷à Øòóðìà Ëèóâèëëÿ).
2 ∂2 f (x) = E − U (x) f (x)
− 2m ∂x2 2 2m ∂x2 f (x) = − 2 E − U (x) f (x)
2 2m
∂x2 f (x) + 2 E − U (x) f (x) = 0
(4)
 êâàíòîâîé ìåõàíèêå íà âîëíîâóþ ôóíêöèþ èç ôèçè÷åñêèõ ñîîáðàæåíèé íà êëàäûâàþò äîïîëíèòåëüíûå óñëîâèÿ [4]:
Óñëîâèå íîðìèðîâêè
•
+∞
|Ψ (x, t)2 dx = 1 (5)
|
−∞
Íåïîñðåäñòâåííûì èíòåãðèðîâàíèåì îïðåäåëèì âèä ôóíêöèè (t):
i ∂ ∂t (t) ∂t(t) = E |· i
∂(t) E
= ∂t
(t) iE
ln((t)) = t + C1 i
E iE
(t) = exp t exp C1 = C2 exp t
i−
Ïîñòîÿííóþ èíòåãðèðîâàíèÿ ìîæíî âûáðàòü òàêèì îáðàçîì, ÷òîáû ôóíêöèÿ Ψ (x, t) áûëà íîðìèðîâàííàÿ.
+∞ +∞
|Ψ (x, t)2 dx = |f (x)(t)2 dx = 1
||
−∞ −∞ +∞ +∞
iE iE
|f (x)(t)2 dx = f (x)|2 exp t exp t dx = 1
||−
−∞ −∞ +∞
|f (x)2 dx = 1 (6)
|
−∞
• Óñëîâèå ðåãóëÿðíîñòè
 ïîñòàâëåííîé çàäà÷å ïîòåíöèàëüíîå ïîëå (êâàíòîâàÿ ÿìà) àíàëèòè÷åñêè îïèñûâàåòñÿ ôóíêöèåé U (x), èìåþùåé ñèììåòðèþ îòíîñèòåëüíî çàìåíû −x íà x:
U (−x)=U (x)
Ïðè íàëè÷èè èíâåðñèè ñîáñòâåííûå ôóíêöèè îïåðàòîðà Ãàìèëüòîíà ëèáî àâòî-ìàòè÷åñêè èìåþò îïðåäåë¼ííóþ ÷¼òíîñòü, ëèáî ìîãóò áûòü ïðåîáðàçîâàíû â ôóíê öèè, èìåþùèå îïðåäåë¼ííóþ ÷¼òíîñòü [3].
Ðåçþìå
3 Ðàñ÷¼ò
3a
• Ïðîàíàëèçèðóåì ïîâåäåíèå ôóíêöèè f (x) â òî÷êàõ ñèíãóëÿðíîñòè x =±:
2
2 2m
∂x2 f (x)+ 2 E −U (x) f (x)=0
2 2m
∂x2 f (x)− 2 U (x)−Ef (x)=0
def 2 def 2m
y = f (x);U (x)→∞;æ = 2 E −U (x)
2
y −æ y =0
kx
Ðåøåíèå â âèäå y =e kx :k2 e −æ 22e kx =0⇒k2 −æ 2 =0k =æ
⇒±
y =C1e æx
+C2e−æx
Èç óñëîâèÿ ðåãóëÿðíîñòè y :C1 →0⇒y =C2e−æx ò.ê. U (x)→∞⇒æ→∞⇒y →0
3a
f =0
(7)
±2
• Ðàññìîòðèì ðåøåíèå óðàâíåíèÿ âèäà:
y +α2 y =0|·2y (8)
2yy +2α2 yy =0
22
(y) +α2(y ) =0
22
(y)+α2 y =0
2 22
(y)+α2 y =c1
y =c1 −α2y2 dx
± 2 |·
dy =c1 −α2y2dx
± 2
dy
=dx
2
1 −α2y2 ±
1 dy
=dx 1 1−(α )y)2 ±
c1
1 c1 α
arcsin y =x +c2
1 αc1 ±1
y = sin(±αx +αc2)⇒y =A sin(αx +)
α
• Ðàññìîòðèì ðåøåíèå óðàâíåíèÿ âèäà:
y − α2 y = 0 (9)
kx
Ðåøåíèå â âèäå y = e kx : k2 e − α22e kx = 0 ⇒ k2 − α2 = 0 ⇒ k = ±α αx + C2e−αx
y = C1e
x
e = sh(x) + ch(x) y = C1 sh(αx) + C1 ch(αx) + C2 sh(−αx) + C2 ch(−αx) y = (C1 − C2) sh(αx) + (C1 + C2) ch(αx) = A sh(αx) + B ch(αx)
• Óðàâíåíèå (4) äëÿ ó÷àñòêà 0 < x < a :
2
2 2m
U (x) − Ef (x) = 0
∂x2 f (x) − 2
def
=
1 = f (x); U (x) = 4U0; κ2 def 2m 4U0 − E
2
1 − κ2 y1 = 0
y
Ïîëó÷åííîå óðàâíåíèå òèïà (9), îáùåå ðåøåíèå: y1 = A sh(κx) + B ch(κx) Åñëè f (x) ÷¼òíàÿ: y1(0) = 0; åñëè f (x) íå÷¼òíàÿ: y1(0) = 0
f (x) ÷¼òíàÿ:
y= Aκ ch(κx) + Bκ sh(κx)
1 1(0) = Aκ = 0 ò.ê. κ = 0 A = 0
→
1 = B ch(κx)
(10)
f (x) íå÷¼òíàÿ:
y1(0) = B = 0
1 = A sh(κx)
(11)
a 3a
• Óðàâíåíèå (4) äëÿ ó÷àñòêà 2 < x < :
2
2 2m
∂x2 f(x) + 2 E − U(x) f(x) = 0
def
2 = f(x); U(x) = 0; γ2 def 2mE
=
2
2+ γ2 y2 = 0
Ïîëó÷åííîå óðàâíåíèå òèïà (8), îáùåå ðåøåíèå: y2 = Dsin(γx+ D)
3a 3a
Èç êðàåâûõ óñëîâèé: y2 = Dsin γ + D = 0
± 2 ± 2 3a
γ + D+ = πn,n ∈ Z
2 3a
−γ + D= πn,n ∈ Z
2 − 3a
D+ = πn− γ,n ∈ Z
2 3a
D= πn+ γ,n ∈ Z
− 2
3a
2 = Dsin γx+ πn± γ ,n ∈ Z
(12)
2
• Òàêèì îáðàçîì, Ψ(x) ïðèìåò âèä:
3a a 3a
y2+ = Dsin γx− + πn ,n ∈ Z, ïðè < x <
22
2
aa
y1+ = Bch(κx), ïðè −
2 ≤ x ≤ 2 3a 3aa
y2+ = Dsin γx++ πn ,n ∈ Z, ïðè − < x < −
2 22
3a a 3a
y2− = Dsin γx− 2+ πn ,n ∈ Z, ïðè < x <
22
aa
y1− = Ash(κx), ïðè −
2 ≤ x ≤ 2 3a 3aa
y2= Dsin γx++ πn ,n ∈ Z, ïðè − < x < −
− 2 22
• Èç óñëîâèé ðåãóëÿðíîñòè f(x) âûòåêàþò ñëåäóþùèå ñîîòíîøåíèÿ:
+ f(x) ÷¼òíàÿ:
aa
1 = y2
22
a
B ch κ = D sin γa − 3a + πn , n ∈ Z
22
aa
y= y
1
2 2 2
a
Bκ sh κ = Dγ cos γa − 3a + πn , n ∈ Z
22
a
κ th κ = γ ctg γa − 3a + πn , n ∈ Z a
22 |·
a
κa th κ = −γa ctg γa
2
Äëÿ óäîáñòâà ðåøåíèÿ äàííîãî òðàíñöåíäåíòíîãî óðàâíåíèÿ ââåä¼ì ïåðåìåííûå:
ξ = κa; η = γa
ξ
ξ th = −η ctg(η) (13)
2
− f(x) íå÷¼òíàÿ:
aa
1 = y2
22
a
A sh κ = D sin γa − 3a + πn , n ∈ Z
22
aa
y= y
1
2 2 2
a
Aκ ch κ = Dγ cos γa − 3a + πn , n ∈ Z
22
a
κ cth κ = γ ctg γa − 3a + πn , n ∈ Z a
22 |·
a
κa cth κ = −γa ctg γa
2
Äëÿ óäîáñòâà ðåøåíèÿ äàííîãî òðàíñöåíäåíòíîãî óðàâíåíèÿ ââåä¼ì ïåðåìåííûå:
ξ = κa; η = γa
ξ
ξ cth = −η ctg(η) (14)
2
9
• Äëÿ ðåøåíèÿ (13) è (14) ðàññìîòðèì åù¼ îäíî óðàâíåíèå:
2(γ2
2 + ξ2 = (γa)2 + (κa)2 = a + κ2) 2m 2m 8m
2 + κ2 = E + 2 4U0 − E = U0
2 2 2
8m 8mπ22 4π2 2π
0 = ==
2
2 2 2ma2 aa
2
2π
2
2 + ξ2 = a = (2π)2
a
ξ = (2π)2 − η2 (15)
±
• Ïîäñòàâèì (15) â (13)è (14):
(2π)2 − η2 th (2π)2 − η2 = −η ctg(η) (÷¼òíûå ðåøåíèÿ)
2
(2π)2 − η2 cth (2π)2 − η2 = −η ctg(η) (íå÷¼òíûå ðåøåíèÿ)
2
Ðèñ. 2: Ðåøåíèå òðàíñöåíäåíòíûõ óðàâíåíèé
• Ðåøàÿ ãðàôè÷åñêè òðàíñöåíäåíòíîå óðàâíåíèå, íàõîäèì ñîáñòâåííûå çíà÷åíèÿ:
η
η = γa γ =
⇒ a
2
η 2m
= E
a 2
2 η2 E =
(16)
2ma2
ξ = κa = (2π)2 − η2
(2π)2 − η2
κ =
a
x
• Ââåä¼ì íîðìèðîâàííóþ êîîðäèíàòó z = a è îïðåäåëèì âèä ñîáñòâåííûõ ôóíêöèé f(z):
3
13
,n ∈ Z, ïðè < z <
2
= Dsin η + πn
z −
y2+
2
2
11
(2π)2 − η2z), ïðè −
2 ≤ z ≤
= Bch(
y1+
2
3
31
,n ∈ Z, ïðè − < z < −
2
= Dsin ηz ++ πn
2
y2+
2
3
13
,n ∈ Z, ïðè < z <
2
= Dsin η + πn
z −
y2
−
2
2
11
(2π)2 − η2z), ïðè −
2 ≤ z ≤
= Ash(
y1
−
2
3
31
,n ∈ Z, ïðè − < z < −
2
= Dsin ηz ++ πn
2
y2
−
2
• Èç óñëîâèÿ íîðìèðîâêè îïðåäåëèì êîíñòàíòû A, B è D, äëÿ óäîáñòâà âçÿâ ñèñòåìó ñ n = 0:
3
1
1
−
2
22
2
2
2
3
3
(2π)2 − η2
Dsin η
dz +
Bch(
z) dz +
Dsin η
dz = 1
z +
z −
2
2
−
3
2
1
1 2
2
−
3
1
1
−
2
22
2
2
2
3
3
(2π)2 − η2
Dsin η
dz +
Ash(
z) dz +
Dsin η
dz = 1
z +
z −
2
2
−
3
2
1
1
−
2
2
2
1
2
sh( (2π)2 − η2) + = 1
2
+
2 (2π)2 − η2 2
1
2
ch( (2π)2 − η2) − = 1
+
2
2 (2π)2 − η2
Èñïîëüçóÿ óðàâíåíèÿ íåïðåðûâíîñòè, ïîëó÷àåì ñèñòåìó äëÿ ÷¼òíûõ ðåøåíèé:
2
1
2
+ sh( (2π)2 − η2) + = 1
2
− η2 Bch (2π)2 − η2 = −Dsin(η)2
(2π)22
È äëÿ íå÷¼òíûõ ðåøåíèé:
2
1
2
+ ch( (2π)2 − η2) − = 1
2
− η2 Ash (2π)2 − η2 = −Dsin(η)2
11
(2π)22
×èñëåííîå ðåøåíèå
1. ×¼òíîå ðåøåíèå:
3 = 5.261
2. Íå÷¼òíîå ðåøåíèå:
4 = 5.308
3. Ýíåðãèè òðåòüåãî è ÷åòâåðòîãî ñòàöèîíàðíûõ ñîñòîÿíèé ýëåêòðîíà â ïîòåíöèàëüíîé ÿìå:
2 η22
3
3 == U0 = 2.804 · U0
2
2ma2 ·
2 η22
4
4 == U0 = 2.855 · U0
2
2ma2 π·
4. Âîëíîâûå ôóíêöèè:
3 :
3 13
=
−0.911 sin 5.261 z− , ïðè <
22
y
2+
2
11
1+ = 0.270 ch(3.435z), ïðè −
y2 ≤ z≤ 2 3 31
y2+ = 0.911 sin 5.261 z+ , ïðè − <−
2 22
4 :
3 13
y2= 0.909 sin 5.308 z− , ïðè <
− 2 22
11
= 0.290 sh(3.361z), ïðè −
y1− 2 ≤ z≤ 2 3 31
y2= 0.909 sin 5.308 z+ , ïðè − <−
− 2 22
5. Âåðîÿòíîñòü íàõîæäåíèÿ ÷àñòèöû â ñåêòîðàõ ÿìû:
3 :
3
1
−
22
|f(z)2 dz= |f(z)2 dz= 0.450
||
1
2
3
2
−
1
2
0
|f(z)2 dz= |f(z)2 dz= 0.101
||
0
−
1
2
4 :
3 1−
|f (z)2 dz = |f (z)2 dz = 0.449
||
12
3 2 − 12
0
|f (z)2 dz = |f (z)2 dz = 0.069
||
0
−
1 2 Ãðàôèêè âîëíîâûõ ôóíêöèé: 3: 3 ñòàöèîíàðíîå ñîñòîÿíèå 4: 4 ñòàöèîíàðíîå ñîñòîÿíèå13
Ëèòåðàòóðà
[1] Â.Ñ. Âëàäèìèðîâ. Óðàâíåíèÿ ìàòåìàòè÷åñêîé ôèçèêè.Ì.: Íàóêà, 1988.
[2] Ëàíäàó Ë.Ä., Ëèâøèö Å.Ì. Òåîðåòè÷åñêàÿ ôèçèêà: Ìåõàíèêà: â 10 ò.Ì.: Íàóêà, 1988.
[3] Ëàíäàó Ë.Ä., Ëèâøèö Å.Ì. Òåîðåòè÷åñêàÿ ôèçèêà: Êâàíòîâàÿ ìåõàíèêà (íåðåëÿòèâèéñêàÿ òåîðèÿ): â 10 ò.Ì.: Íàóêà, 1989.
[4] Ï.À.Ì. Äèðàê. Ïðèíöèïû êâàíòîâîé ìåõàíèêè.Ì.: Íàóêà, 1979.