In nuclear physics the one-body part \( u_{\mathrm{ext}}(x_i) \) is often approximated by a harmonic oscillator potential. However, as we also noted with the Woods-Saxon potential there is no external confining potential in nuclei.

What many people do then, is to add and subtract a harmonic oscillator potential, with $$ \hat{u}_{\mathrm{ext}}(x_i)=\hat{u}_{\mathrm{ho}}(x_i)= \frac{1}{2}m\omega^2 r_i^2, $$ where \( \omega \) is the oscillator frequency. This leads to $$ \hat{H} = \hat{H_0} + \hat{H_I} = \sum_{i=1}^A \hat{h}_0(x_i) + \sum_{i < j=1}^A \hat{v}(x_{ij})-\sum_{i=1}^A\hat{u}_{\mathrm{ho}}(x_i), $$ with $$ H_0=\sum_{i=1}^A \hat{h}_0(x_i) = \sum_{i=1}^A\left(\hat{t}(x_i) + \hat{u}_{\mathrm{ho}}(x_i)\right). $$ Many practitioners use this as the standard Hamiltonian when doing nuclear structure calculations. This is ok if the number of nucleons is large, but still with this Hamiltonian, we do not obey translational invariance. How can we cure this?