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Separation of variables makes the angular factor a spherical harmonic. Applying the Quantum Hamiltonian givesConsequently $R{Y}_{lm}$ is an energy eigenfunction provided the radial factor satisfiesThis is the radial Schrödinger equation for a central potential, here with the attractive Coulomb potential. An arbitrary $R$ does not give an eigenfunction; regularity, decay and wavefunction normalization select admissible bound quantum states.

For a $2p$ atomic orbital, $l=1$. The radial factor obeysSubstitution into the radial Schrödinger equation for a central potential givesThus the energy eigenvalue isThe normalization constant cancels; $E_{\mathrm{h}}$ is the Hartree energy.

Introduce the reduced radial wavefunction $u(r)=r\psi (r)$. Multiplying the radial Schrödinger equation for a central potential by $-2mr/{\hslash }^2$ givesFor the decaying bound state, take $\alpha >0$ and put $u=f{e}^{-\alpha r}$. Differentiation yieldsSince $2mE/{\hslash }^2=-{\alpha }^2$, the two constant multiples of $f$ cancel. Dividing by the nonzero exponential givesThis separates the large-radius exponential decay from the remaining Coulomb bound state equation.