• 1 Общий вид спектра
  • 1.1 Периодическая задача
  • 1.2 Влияние границ кристалла
• 2 Модель Кронига — Пенни
• 3 Программный код
  • 3.1 Код для Maple
  • 3.2 Код для Scilab
  • 3.3 Код для Matlab
• 4 Ссылки
• 5 Примечания
• 6 Литература

${\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}\psi (x)}{\partial x^{2}}}+V_{a}(x)\psi (x)=E\psi (x)}$

${\displaystyle \left({\begin{matrix}\psi _{1}(x+a)\\\psi _{2}(x+a)\end{matrix}}\right)=\mathrm {T} \left({\begin{matrix}\psi _{1}(x)\\\psi _{2}(x)\end{matrix}}\right),}$

где ${\displaystyle \mathrm {T} }$ — некоторая матрица (матрица монодромии).

${\displaystyle \mathrm {T} =\left({\begin{matrix}e^{\mathrm {i} ka}&0\\0&e^{-\mathrm {i} ka}\end{matrix}}\right).}$

${\displaystyle \psi _{1,2}(x)=e^{\mathrm {i} kx}\phi _{1,2}(x),}$

где ${\displaystyle \phi _{1,2}(x)}$ — периодические функции.

${\displaystyle \mathrm {Tr} \ \mathrm {T} =2\cos(kx)\in [-2;2].}$

${\displaystyle \psi (0)=\psi (L).}$

${\displaystyle kL=2\pi n,\;n\in \mathbb {Z} .}$

${\displaystyle \Delta k={\frac {2\pi n}{L}}.}$

${\displaystyle \Delta k_{x,y,z}={\frac {2\pi n}{L_{x,y,z}}}.}$

${\displaystyle 0<x<a-b:{-\hbar ^{2} \over 2m}\psi _{xx}=E\psi \Rightarrow \psi =Ae^{i\alpha x}+A'e^{-i\alpha x}\quad \left(\alpha ^{2}={2mE \over \hbar ^{2}}\right)}$

${\displaystyle -b<x<0:{-\hbar ^{2} \over 2m}\psi _{xx}=(E+V_{0})\psi \Rightarrow \psi =Be^{i\beta x}+B'e^{-i\beta x}\quad \left(\beta ^{2}={2m(E+V_{0}) \over \hbar ^{2}}\right).}$

${\displaystyle \psi (0<x<a-b)=Ae^{i\alpha x}+A'e^{-i\alpha x}=e^{ikx}\cdot \left(Ae^{i(\alpha -k)x}+A'e^{-i(\alpha +k)x}\right)}$

${\displaystyle \Rightarrow u(0<x<a-b)=Ae^{i(\alpha -k)x}+A'e^{-i(\alpha +k)x}.}$

${\displaystyle u(-b<x<0)=Be^{i(\beta -k)x}+B'e^{-i(\beta +k)x}.}$

${\displaystyle \psi (0^{-})=\psi (0^{+})\quad \psi '(0^{-})=\psi '(0^{+})}$

${\displaystyle u(-b)=u(a-b)\quad u'(-b)=u'(a-b).}$

${\displaystyle {\begin{pmatrix}1&1&-1&-1\\\alpha &-\alpha &-\beta &\beta \\e^{i(\alpha -k)(a-b)}&e^{-i(\alpha +k)(a-b)}&-e^{-i(\beta -k)b}&-e^{i(\beta +k)b}\\(\alpha -k)e^{i(\alpha -k)(a-b)}&-(\alpha +k)e^{-i(\alpha +k)(a-b)}&-(\beta -k)e^{-i(\beta -k)b}&(\beta +k)e^{i(\beta +k)b}\end{pmatrix}}{\begin{pmatrix}A\\A'\\B\\B'\end{pmatrix}}={\begin{pmatrix}0\\0\\0\\0\end{pmatrix}}.}$

${\displaystyle \cos(ka)=\cos(\beta b)\cos[\alpha (a-b)]-{\alpha ^{2}+\beta ^{2} \over 2\alpha \beta }\sin(\beta b)\sin[\alpha (a-b)].\qquad (*)}$

${\displaystyle b\rightarrow 0\ ;\ V_{0}\rightarrow \infty \ ;\ V_{0}b=\mathrm {constant} .}$

${\displaystyle \Rightarrow \beta b\rightarrow 0\ ;\ \beta ^{2}b=\mathrm {constant} \ ;\ \alpha ^{2}b\rightarrow 0\ ;\ \sin(\beta b)\rightarrow \beta b\ ;\ \cos(\beta b)\rightarrow 1.}$

${\displaystyle \cos(ka)=\cos(\alpha a)-P{\sin(\alpha a) \over \alpha a}\qquad \left(P={\beta ^{2}ab \over 2}\right).}$

  restart;
  with(plots):
  with(stats[statplots]):
  eq:=cos(k*a)=cos(beta*b)*cos(alpha*(a-b)) - (alpha^2+beta^2)/(2*alpha*beta)*sin(beta*b)*sin(alpha*(a-b));
  alpha:=sqrt(8*Pi^2*m*(E)*e/h^2):
  beta:=sqrt(8*Pi^2*m*(E+V)*e/h^2):
  e:=1.6*1e-19:
  a:=0.54310*1e-9:
  m:=0.19*9.1*1e-31:
  b:=1/5*a:
  h:=6.6*1e-34:
  k(E,V):=arccos(rhs(evalf(eq)))/a;

  #График
  p:=plot({subs(V=10,k(E,V)),subs(V=10,-k(E,V))},E=-5..50,labels=[ka, E],color=blue):
  xyexchange(p);

  #Анимация, зависимость от глубины ямы
  p:=animate( plot, [{k(E,V),-k(E,V)},E=-10..50, color=blue,labels=[ka, E]], V=0..30 ):
  xyexchange(p);