The system: two particles (fermions normally) in a harmonic oscillator trap in two dimensions

The Hamiltonian of the quantum dot is given by

$$\hat{H} = \hat{H}_0 + \hat{V},$$

where $\hat{H}_0$ is the many-body HO Hamiltonian, and $\hat{V}$ is the inter-electron Coulomb interactions. In dimensionless units,

$$\hat{V}= \sum_{i < j}^N \frac{1}{r_{ij}},$$

with $r_{ij}=\sqrt{\mathbf{r}_i^2 - \mathbf{r}_j^2}$.

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