Day 6 In-Class Activity

The Finite Square Well

The next most complicated potential is where we drop the walls of the infinite square well down to $V_0>0$. In this case, we will observe bound states ($E<V_0$) and unbound states ($E>V_0$). For now, we focus on bound states, which lead to quantized energy eigenstates. The potential we are given is:

$${\displaystyle V(x)=\{\begin{array}{ll}{V}_0& \text{if }x<-a\\ 0& \text{if }-a<x<a\\ V_0& \text{if }x>a\end{array}}$$

We can show that the general solutions for energy eigenstates are:

$${\displaystyle {\varphi }_E(x)=\{\begin{array}{ll}A{e}^{qx}+B{e}^{-qx}& \text{if }x<-a\\ C\sin kx+D\cos kx& \text{if }-a<x<a\\ F{e}^{qx}+G{e}^{-qx}& \text{if }x>a\end{array}}$$

where $k=\sqrt{{\displaystyle \frac{2mE}{{\hslash }^2}}}>0$ and $q=\sqrt{{\displaystyle \frac{2m(V_0-E)}{{\hslash }^2}}}>0$.

1. Question: There are 3 “boundary” conditions on ${\varphi }_E(x)$. What are they?
   • Discussion: What is the physical or practical origin of each of these boundary conditions? Why are there not additional conditions on the derivatives of ${\varphi }_E(x)$?
2. Question: We can apply these boundary conditions (as in McIntyre or my notes), and show that the energy equations we must satisfy are: $ka\tan (ka)=qa$ and $-ka\cot (ka)=qa$. How do we know these equations are transcendental? That is, how do we recognize that we won’t be able to find an analytical solution?
   • Discussion: What are some ways we can find approximate solutions to these equations? What are the benefits and tradeoffs to each method you come up with?
3. Question: Using McIntyre’s definitions for $z=\sqrt{{\displaystyle \frac{2ME{a}^2}{{\hslash }^2}}}$ and $z_0=\sqrt{{\displaystyle \frac{2M{V}_0{a}^2}{{\hslash }^2}}}$, derive the transcendental relationships: $z\tan (z)=\sqrt{z_0^2-z^2}$ and $-z\cot (z)=\sqrt{z_0^2-z^2}$.
   • Discussion: For a strongly bound particle, what would your expect for the values of $z_0$ and $z$? What about for a weakly bound particle?