Undergrad Finite square well bound states

[andrewtz98]

Let's suppose I have a finite potential well: $$
V(x)=
\begin{cases}
\infty,\quad x<0\\
0,\quad 0<x<a\\
V_o,\quad x>a.
\end{cases}
$$

I solved the time-independent Schrödinger equation for each region and after applying the continuity conditions of $\Psi$ and its derivative I ended up with:

$$ \tan(k_1a)=-\frac{k_1}{k_2},$$ where $k_1=\sqrt{\frac{2mE}{\hbar^2}}$ and $k_2=\sqrt{\frac{2m(V_o-E)}{\hbar^2}}$.

I'm aware of the fact that solutions can only be calculated graphically, but what's the relation between the value of $V_o$ and the bound states? What if I want to find the acceptable values of $V_o$ for the bound states to be $1,2,3,\dots$ or none?

 

[Vanadium 50]

What if I want to find...

I'm aware of the fact that solutions can only be calculated graphically

Doesn't that answer your question?

There is a relationship between a, m and V0 and the number of bound states. I doubt it has an analytic form, but the way you get it asking that the Nth state be just on the edge of being bound. If I did it right, for large N,

V_0 \approx (N-1)^2 \frac{h^2}{32ma^2}By the way, graphing is not the only way to find the solutions. You can also do it numerically.