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excitation energies ω$_{n}$ = ( Z $^{2}$/ 2)(1 − 1 /n $^{2}$) ( n = 1 , 2 , .. ). The corresponding eigenfunctions are given by Laguerre polynomials: 

As a final example let us consider the case of two electrons repelling each other with the “soft” Coulomb potential e 2 √ ( $^{x$_{1}$}$−x$_{2}$ ) $^{2}$+ a 2 in a one-dimensional parabolic trap of natural frequency ω$_{0}$ (the cutoff a > 0 serves to eliminate the pathological behavior of the interaction at x$_{1}$ = x$_{2}$ ). This is a model that can be solved nu- merically thanks to the separation of center of mass and 

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## C. Two-electron systems 

relative variable, and analytically in the limit of strong correlation. The hamiltonian is 

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where X = x$_{1}$ + x$_{2}$ 2 and P = p$_{1}$ + p$_{2}$ are, respectively, the coordinate and the momentum of the center of mass and x = x$_{1}$ − x$_{2}$ , p = $^{p$_{1}$}$−p$_{2}$ 2 are the coordinate and the momentum in the relative channel. Notice that for a fixed strength e 2 of the interaction we can go from the weakly correlated to the strongly correlated regime by varying the value of ω$_{0}$ : ω$_{0}$ → ∞ corresponds to the non- interacting limit, and ω$_{0}$ → 0 to the strongly correlated limit. 

The ground-state wave function is 

where ψ$_{0}$ ( X ) and φ$_{0}$ ( x ) are, respectively, the ground- state wave functions of the center of mass and of the relative hamiltonian. The general excited state is 

where ψ$_{n}$ ( X ) is the wave function of the n-th excited state of the center of mass hamiltonian and φ$_{m}$ ( x ) is the wave function of the m -th excited state of the relative hamiltonian. 

The ground-state of the system is a spin singlet ( S = 0) and for this reason in the following we consider only singlet states, which are connected to the ground-state by the current density operator. The relative wave function for such states is symmetric: φ$_{m}$ ( − x ) = φ$_{m}$ ( x ). This wave function has 2 m nodes, of which m with x > 0 and m (symmetrically placed) with x < 0. The center of mass wave function, 

can be either symmetric or antisymmetric, depending on the parity of n , and has n nodes. Here ℓ$_{cm}$ = √ ℏ / 2 mω$_{0}$ . The ground-state has n = 0 , m = 0 and all the other states are characterized by positive values of the integers n and m . 2 

In the non interacting limit ( ω$_{0}$ → ∞ , or e → 0) the relative wave function is 

where ℓ$_{0}$ = √ 2 ℏ /mω$_{0}$ . The ground-state density is a gaussian centered at the origin. The excitation energies, expressed in units of ω$_{0}$ , are sums of the excitation ener- gies of two identical harmonic oscillators: 

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