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)2 p

    E = ==

    2 · m 2m 2m ⇒ p)2 2 2 2

    Tˆ=={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

    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.  ëþáîé ìîìåíò âðåìåíè âîëíîâàÿ ôóíêöèÿ è å¼ ÷àñòíûå ïðîèçâîäíûå äîëæíû áûòü íåïðåðûâíûìè ôóíêöèÿìè ïðîñòðàíñòâåííûõ êîîðäèíàò.

     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

    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 =0k2 æ 2 =0k =æ

    ⇒±

    y =C1e æx

    +C2e−æx

    Èç óñëîâèÿ ðåãóëÿðíîñòè y :C1 0y =C2e−æx ò.ê. U (x)→∞⇒æ→∞⇒y 0

    3a

    f =0

    (7)

    ±2

    Ðàññìîòðèì ðåøåíèå óðàâíåíèÿ âèäà:

    y +α2 y =02y (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 + C2eαx

    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+ = πnγ,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 γx2+ 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 2

    0 = ==

    2

    2 2 2ma2 aa

    2

    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

    ||

    1

    2

    3 2 − 1

    2

    0

    |f (z)2 dz = |f (z)2 dz = 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.

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