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 $$ \alpha = \frac{\hbar^2}{m\beta e^2}. $$