Challenge Level

Curve sketching is an essential art in the application of mathematics to science. A good sketch of a curve does not need to be accurately plotted to scale, but will encode all of the key information about the curve: turning points, maximum or minimum values, asymptotes, roots and a sense of the scale of the function.

Sketch $V(r)$ against $r$ for each of these tricky curves, treating $a, b$ and $c$ as unknown constants in each case. As you make your plots, ask yourself: do different shapes of curve emerge for different ranges of the constants, or will the graphs look similar (i.e. same numbers of turning points, regions etc.) for the various choices?

1. An approximation for the potential energy of a system of two atoms separated by a distance $r$

$$V(r) = a\left[\left(\frac{b}{r}\right)^{12}-\left(\frac{b}{r}\right)^6\right]$$

2. A radial probability density function for an electron orbit

$$V(r) = ar^2e^{-\frac{r}{b}}$$

3. Potential energy for the vibrational modes of ammonium

$$

V(r)=ar^2+be^{-cr^2}

$$