I'm trying to plot the nature of a graph from a certain equation manually.
The equation is $$P\frac{\sin(\alpha a)}{\alpha a}+\cos(\alpha a)=\cos(ka)$$ where $P$ is a constant.
Here, $\alpha^2=\frac{2m}{\hbar^2}E$ where $\frac{2m}{\hbar^2}$ is a constant, say $C$. Hence, $\alpha^2=CE$.
I need to show that the graph between $E$ and $k$ looks something like:
However, I'm not sure how to proceed with this. Any ideas how to plot the nature of the graph manually ? Thank you.
Without loss of generality, set $a=1$ and solve for $k$:
$$ k = \cos^{-1} \left( P \frac{\sin \alpha}\alpha + \cos \alpha \right) $$
Your $k$ will be defined only when the term in parentheses has absolute value less than one. A plot of $\alpha^2$ versus $k$ gives you the right side of your figure; the left side is a reflection due to the symmetry of the cosine function.