Finite Square Well

$$U(x)=\{\begin{array}{ll}-U_0,& -a\le x\le a\\ 0,& |x|>a\end{array}$$

• $U_0>0$

Bounded: $U_0<E<0$

$$\psi (x)=\{\begin{array}{ll}A{e}^{kx},& x\le a\\ C\sin (lx)+D\cos (lx),& |x|\le a\\ B{e}^{-kx},& x\ge a\end{array}$$

• For even solutions, $A=B$ and $C=0$
• For odd solutions, $A=-B$ and $D=0$
• $k=\frac{\sqrt{-2mE}}{\hslash }$
• $l=\frac{\sqrt{2m(E+U_0)}}{\hslash }$

$\tan z=\sqrt{{\left(\frac{z_0}{z}\right)}^2-1}$

• Even solutions to finite square well
• $z=la$
• $z_0=\frac{\sqrt{2m{U}_0}}{\hslash }a$
  • As $z_0\to \infty$ ($U_0\vee a\to \infty$, so deep wide well), the number of energy states $\to \infty$
  • As $z_0\to 0$, the number of energy states $\to 1$
• Desmos Demo

$-\cot z=\sqrt{{\left(\frac{z_0}{z}\right)}^2-1}$

• Odd solutions to finite square well

$E_n=\frac{n^2{\pi }^2{\hslash }^2}{8m{a}^2}-U_0$

Scattering: $E>0$

$$\psi (x)=\{\begin{array}{ll}A{e}^{ikx}+B{e}^{-ikx},& x<-a\\ C\sin (lx)+D\cos (lx),& |x|<a\\ F{e}^{ikx}+G^{-ikx},& x>a\end{array}$$

• $k=\frac{\sqrt{2mE}}{\hslash }$
• $l=\frac{\sqrt{2m(E+U_0)}}{\hslash }$
• For wave traveling rightward:
  • $G=0$
  • $T=\frac{|F{|}^2}{|A{|}^2}={\left[1+\frac{U_0^2}{4E(E+U_0)}{\sin }^2\left(\frac{2a}{\hslash }\sqrt{2m(E+U_0)}\right)\right]}^{-1}$
    • $T$ here is the transmission coefficient
    • When $E=\frac{n^2{\pi }^2{\hslash }^2}{8m{a}^2}-U_0$, $T=1$. I.e., when the energy matches that of an infinite square well, transmission probability is 100%