The radial Schroedinger equation for the hydrogen atom can be written as $$ -\frac{\hbar^2}{2m}\frac{\partial^2 u(r)}{\partial r^2}- \left(\frac{ke^2}{r}-\frac{\hbar^2l(l+1)}{2mr^2}\right)u(r)=Eu(r), $$ or with dimensionless variables $$ -\frac{1}{2}\frac{\partial^2 u(\rho)}{\partial \rho^2}- \frac{u(\rho)}{\rho}+\frac{l(l+1)}{2\rho^2}u(\rho)-\lambda u(\rho)=0, \tag{3} $$ with the hamiltonian $$ H=-\frac{1}{2}\frac{\partial^2 }{\partial \rho^2}- \frac{1}{\rho}+\frac{l(l+1)}{2\rho^2}. $$ Use variational parameter \( \alpha \) in the trial wave function $$ u_T^{\alpha}(\rho)=\alpha\rho e^{-\alpha\rho}. \tag{4} $$