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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:

enter image description here

However, I'm not sure how to proceed with this. Any ideas how to plot the nature of the graph manually ? Thank you.

1 Answers1

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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.

rob
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  • One point to remember is that $\cos^{-1}(\cos(x))\neq x$ unless and until $0\leq x \leq \pi$. But yes, we can manipulate accordingly. –  Aug 28 '17 at 13:38