We have thus $$ -\frac{d^2}{d\rho^2} u(\rho) + \frac{mk}{\hbar^2} \alpha^4\rho^2u(\rho) = \frac{2m\alpha^2}{\hbar^2}E u(\rho) . $$ The constant \( \alpha \) can now be fixed so that $$ \frac{mk}{\hbar^2} \alpha^4 = 1, $$ or $$ \alpha = \left(\frac{\hbar^2}{mk}\right)^{1/4}. $$