We solve the full two-body problem using exact diagonalization. The wave function ansatz is then given by a product of Hartree single-particle states
$$ \begin{align} \vert\Phi_I\rangle &= \sum_{k = 0}^{N^L} \sum_{l = 0}^{N^R} C_{kl, I} \vert \phi^{L}_k \phi^{R}_l\rangle, \tag{8} \end{align} $$where no symmetry is assumed for the wavefunction as the particles are distinguishable, and the index \( I = (i, j) = iN^R + j \) is a compound index denoting the excited two-body state we are looking at.