Расчёт волновой функции в квантовой яме сложной формы

[3]. Ïðè ýòîì ñîñòîÿíèå ôèçè÷åñêîé ñèñòåìû â êâàíòîâîé ìåõàíèêå îïèñûâàåò âîëíîâàÿ ôóíêöèÿ Ψ(r, t) [3], êîòîðàÿ â ñâîþ î÷åðåäü îïðåäåëÿåòñÿ âîëíîâûì óðàâíåíèåì Øð¼äèíãå-ðà [4]:

∂

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)^²p

E = ==

2 · m 2m 2m ⇒ _{_p₎}2 2 2 2

T^ˆ=^(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

Â 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 ∂x^²

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 ∂x^²^²2m _∂x_₂f (x) = _₋_₂E _₋U (x) f (x)

^²2m

_∂x_₂f (x) + _₂E _₋U (x) f (x) = 0

(4)

Â êâàíòîâîé ìåõàíèêå íà âîëíîâóþ ôóíêöèþ èç ôèçè÷åñêèõ ñîîáðàæåíèé íà êëàäûâàþò äîïîëíèòåëüíûå óñëîâèÿ [4]:

Óñëîâèå íîðìèðîâêè

•

⁺_{_∞}

_|Ψ (x, t)^²dx = 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)^²dx = _|f (x)(t)^²dx = 1

||

−∞ −∞ ⁺_{_∞}⁺_{_∞}

iE iE

_|f (x)(t)^²dx = f (x)_|^²exp t exp t dx = 1

||−

−∞ −∞ ⁺_{_∞}

_|f (x)^²dx = 1 (6)

|

−∞

_{_•}Óñëîâèå ðåãóëÿðíîñòè
1. Âîëíîâàÿ ôóíêöèÿ íå ìîæåò ïðèíèìàòü áåñêîíå÷íûõ çíà÷åíèé, òàêèõ, ÷òî èíòåãðàë (6) ñòàíåò ðàñõîäÿùèìñÿ.
2. Âîëíîâàÿ ôóíêöèÿ äîëæíà áûòü îäíîçíà÷íîé ôóíêöèåé êîîðäèíàò è âðåìåíè, òàê êàê ïëîòíîñòü âåðîÿòíîñòè îáíàðóæåíèÿ ÷àñòèöû äîëæíà îïðåäåëÿòüñÿ îäíîçíà÷íî.
3. Â ëþáîé ìîìåíò âðåìåíè âîëíîâàÿ ôóíêöèÿ è å¼ ÷àñòíûå ïðîèçâîäíûå äîëæíû áûòü íåïðåðûâíûìè ôóíêöèÿìè ïðîñòðàíñòâåííûõ êîîðäèíàò.
Â 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
1. Ðåøåíèå çàäà÷è ñâîäèòñÿ ê ðåøåíèþ çàäà÷è Øòóðìà Ëèóâèëëÿ.
  1. Ðåøåíèåì ÿâëÿþòñÿ ñîáñòâåííûå ôóíêöèè f (x) è ñîáñòâåííûå çíà÷åíèÿ E ˆ
  2. (ýíåðãèè) îïðåðàòîðà Ãàìèëüòîíà H.
2. Îäíîðîäíûå ãðàíè÷íûå óñëîâèÿ çàäà÷è Øòóðìà Ëèóâèëëÿ îïðåäåëÿþòñÿ èç ôîðìû ïîòåíöèàëà.
3. Ñîáñòâåííûå ôóíêöèè f (x)îáëàäàþò îïðåäåë¼ííîé ÷¼òíîñòüþ â òîì èëè èíîì ñòàöèîíàðíîì ñîñòîÿíèå.
4. Ðåøàòü çàäà÷þ ìîæíî íà ïîëîâèíå èíòåðâàëà, ïåðåíîñÿ ðåøåíèÿ íà îñòàâøóþñü ÷àñòü ñ ïîìîùüþ ñèììåòðèè.
5. Íà ôóíêöèè f (x)íàêëàäûâàþòñÿ óñëîâèÿ ðåãóëÿðíîñòè è íåïðåðûâíîñòè â ëþáîì ñòàöèîíàðíîì ñîñòîÿíèå.
3 Ðàñ÷¼ò

3a

_{_•}Ïðîàíàëèçèðóåì ïîâåäåíèå ôóíêöèè f (x) â òî÷êàõ ñèíãóëÿðíîñòè x =_±:

2

^²2m

_∂x_₂f (x)+ _₂E ₋U (x) f (x)=0

^²2m

_∂x_₂f (x)_₋_₂U (x)₋Ef (x)=0

def 2 def 2m

y = f (x);U (x)_→∞;æ = _₂E ₋U (x)

2

y ₋æ y =0

kx

Ðåøåíèå â âèäå y =e ^{k^x}:k^²e ₋æ ²2e ^{k^x}=0_⇒k^²₋æ ^²=0k =æ

⇒±

_{_y₌_C}₁_{_e}æx

+C₂e−^{æ^x}

Èç óñëîâèÿ ðåãóëÿðíîñòè y :C1 _→0_⇒y =C2e−^{æ^x}ò.ê. U (x)_→∞⇒æ_→∞⇒y _→0

3a

f =0

(7)

±2

_{_•}Ðàññìîòðèì ðåøåíèå óðàâíåíèÿ âèäà:

y +α^²y =0_|·2y (8)

2yy +2α^²yy =0

22

(y) +α²(y ) =0

22

(y)+α^²y =0

2 22

(y)+α^²y =c_₁

y =c_₁₋α²y^²dx

± ^²|·

dy =c_₁₋α²y²dx

± ^²

dy

=dx

2

_₁₋α²y^²^{^±}

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 _₋α^²y = 0 (9)

kx

Ðåøåíèå â âèäå y = e ^{k^x}: k^²e _₋α²2e ^{k^x}= 0 _{_⇒}k^²_₋α^²= 0 _{_⇒}k = _±α αx _₊C₂_e₋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

^²2m

U (x) _₋Ef (x) = 0

∂x^²^{^f⁽^x⁾}− ^²

def

=

1 = f (x); U (x) = 4U0; κ^²^{de^f}^{2^m}4U0 _₋E

2

_₁_₋κ^²y1 = 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

^²2m

_∂x_₂f(x) + _₂E _₋U(x) f(x) = 0

def

2 = f(x); U(x) = 0; γ^²^{de^f}^{²^m}E

=

2

₂+ γ^²y2 = 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 γ^{^a}− ^{3^a}+ πn , n _{_∈}Z

22

aa

y= y

1

2 ^²2

a

Bκ sh κ = Dγ cos γ^{^a}− ^{3^a}+ πn , n _{_∈}Z

22

a

κ th κ = γ ctg γ^{^a}− ^{3^a}+ 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 γ^{^a}− ^{3^a}+ πn , n _{_∈}Z

22

aa

y= y

1

2 ^²2

a

Aκ ch κ = Dγ cos γ^{^a}− ^{3^a}+ πn , n _{_∈}Z

22

a

κ cth κ = γ ctg γ^{^a}− ^{3^a}+ 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_{₍_γ}2

^²+ ξ^²= (γa)^²+ (κa)^²= a + κ²) 2m 2m 8m

^²+ κ^²= E + _₂4U0 _₋E = U0

2 2 ₂

8m 8mπ²^²4π^²2π

0 = ==

2

^²^²2ma^²aa

₂

2π

2

^²+ ξ^²= a = (2π)^²

a

ξ = (2π)^²_₋η^²(15)

±

_{_•} Ïîäñòàâèì (15) â (13)è (14):

(2π)^²_₋η^²th ^{⁽²^π⁾}^²− ^η^²= ₋η ctg(η) (÷¼òíûå ðåøåíèÿ)

2

(2π)^²_₋η^²cth ^{⁽²^π⁾}^²− ^η^²= ₋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)

2ma^²

ξ = κa = (2π)^²_₋η^²

(2π)^²_₋η^²

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_₋η^²2

1

2

ch( (2π)^²_₋η²) _₋= 1

+

2

2 (2π)^²_₋η^²

Èñïîëüçóÿ óðàâíåíèÿ íåïðåðûâíîñòè, ïîëó÷àåì ñèñòåìó äëÿ ÷¼òíûõ ðåøåíèé:

2

1

2

+ sh( (2π)^²_₋η²) + = 1

2

_₋η^²Bch ^{⁽²^π⁾}^²− ^η^²= ₋Dsin(η)2

(2π)²2

È äëÿ íå÷¼òíûõ ðåøåíèé:

2

1

2

+ ch( (2π)^²_₋η²) _₋= 1

2

_₋η^²Ash ^{⁽²^π⁾}^²− ^η^²= ₋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

2ma^²·

2 _η22

4

4 == U0 = 2.855 _{_·}U0

2

2ma^²π·

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)^²dz= _|f(z)^²dz= 0.450

||

1

2

3

2

−

1

2

⁰

_|f(z)^²dz= _|f(z)^²dz= 0.101

||

0

−

1

2

4 :
3 1
−

_|f (z)^²dz = _|f (z)^²dz = 0.449

||
1
2
3 2 − 1
2

⁰

_|f (z)^²dz = _|f (z)^²dz = 0.069

||

0

−
1 2 Ãðàôèêè âîëíîâûõ ôóíêöèé: 3: 3 ñòàöèîíàðíîå ñîñòîÿíèå 4: 4 ñòàöèîíàðíîå ñîñòîÿíèå
13

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.