Why do we introduce the Hamiltonian in the form $$ \hat{H} = \hat{H}_0 + \hat{H}_I? $$ There are many reasons for this. Let us look at some of them, using the harmonic oscillator in three dimensions as our starting point. For the harmonic oscillator we know that $$ \hat{h}_0(x_i)\psi_{\alpha}(x_i)=\varepsilon_{\alpha}\psi_{\alpha}(x_i), $$

where the eigenvalues are \( \varepsilon_{\alpha} \) and the eigenfunctions are \( \psi_{\alpha}(x_i) \). The subscript \( \alpha \) represents quantum numbers like the orbital angular momentum \( l_{\alpha} \), its projection \( m_{l_{\alpha}} \) and the principal quantum number \( n_{\alpha}=0,1,2,\dots \).

The eigenvalues are $$ \varepsilon_{\alpha} = \hbar\omega \left(2n_{\alpha}+l_{\alpha}+\frac{3}{2}\right). $$