Lets chose only one of the possible energies and try to plot $\psi_I$, $\psi_{II}$ and $\psi_{III}$. I choose energy which is equal to $0.37\, W_p$$0.17\, W_p$ and calculate constants $\mathcal K$ and $\mathcal L$ (i fixed the definition for $\mathcal K$):
Here is the GNUPLOT script used to draw 2nd graphsgraph: