The following mathematical properties of the harmonic oscillator are handy.

• First of all we have a complete basis of orthogonal eigenvectors. These have well-know expressions and can be easily be encoded.
• With a complete basis \( \psi_{\alpha}(x_i) \), we can construct a new basis \( \phi_{\tau}(x_i) \) by expanding in terms of a harmonic oscillator basis, that is

$$ \phi_{\tau}(x_i)=\sum_{\alpha} C_{\tau\alpha}\psi_{\alpha}(x_i), $$ where \( C_{\tau\alpha} \) represents the overlap between the two basis sets.

• As we will see later, the harmonic oscillator basis allows us to compute in an expedient way matrix elements of the interactions between two nucleons. Using the above expansion we can in turn represent nuclear forces in terms of new basis, for example the Woods-Saxon basis to be discussed later here.