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 Schrodinger 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?
