Finite Square Well

#Physics

$$

U(x)=\begin{cases}
-U_{0}, & -a\leq x\leq a \\0, & |x|>a
\end{cases}
$$

\psi(x)=\begin{cases}
Ae^{kx}, & x\leq a \\C\sin(lx)+D\cos(lx), & \lvert x\rvert \leq a\\ Be^{-kx}, & x\geq a
\end{cases}
$$

Even solutions to finite square well
$\displaystyle z=la$
$\displaystyle z_{0}=\frac{\sqrt{ 2mU_{0} }}{\hbar}a$
- As $\displaystyle z_{0}\rightarrow \infty$ ($\displaystyle U_{0}\lor a\rightarrow \infty$, so deep wide well), the number of energy states $\displaystyle \rightarrow\infty$
- As $\displaystyle z_{0}\rightarrow0$, the number of energy states $\displaystyle \rightarrow 1$
Desmos Demo

\psi(x)=\begin{cases}
Ae^{ikx}+Be^{-ikx}, & x<-a \\C\sin(lx)+D\cos(lx), & \lvert x\rvert <a\\Fe^{ikx}+G^{-ikx}, & x>a
\end{cases}
$$

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