We want to manipulate this equation further to make it as similar to that in (a) as possible. We define a 'frequency' $$\omega_r^2=\frac{1}{4}\frac{mk}{\hbar^2} \alpha^4,$$ and fix the constant $\alpha$ by requiring $$\frac{m\alpha \beta e^2}{\hbar^2}=1$$ or $$\alpha = \frac{\hbar^2}{m\beta e^2}.$$