Is there any non-trivial many-body system for which the exact solution to Schrödinger's equation is known? (By non-trivial, I mean a system with particle-particle interactions.) Perhaps something like positronium, or two electrons in a box.
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One of my favorite non-trivial, exact many-body ground states is the solution of a very specific spin-1 magnetic insulator in 1D, with a hamiltonian $$H_{AKLT}=\sum_{\langle ij\rangle}\vec{S}_i \cdot \vec{S}_j + \frac{1}{3}(\vec{S}_i \cdot \vec{S}_j)^2$$ It turns out that you can construct the ground state by looking at the spin-1 operators as a projection onto the triplet subspace of two spin-1/2 operators, where the spin-1/2 objects form nearest-neighbor singlet bonds in a very special way. (More details can be found http://en.wikipedia.org/wiki/AKLT_Model) This exact ground state informs our understanding of the spin-1 Heisenberg model (i.e., without the biquadratic interaction), and the "fractionalized" spin-1/2 "edge states" that this state predicts for a magnet with open boundary conditions have been observed in experiments (see again the wiki article and its references) |
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For a Hamiltonian, for which Laughlins wavefunction is the exact ground state, see F. D. M. Haldane , Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States, Phys. Rev. Lett. 51, 605–608 (1983), http://prl.aps.org/abstract/PRL/v51/i7/p605_1 . Like the Hamiltonian in the answer of wsc, this Hamiltonian is a sum of projections, which represent the interactions. And in both cases, the ground state is the state, which is annihilated by all these projections. |
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The exact ground state for N structureless bosons interacting with contact interactions $V(x_1-x_2) = g \delta(x_1-x_2)$ is known. In free space (also with infinite width periodic boundary conditions) for $g<0$ this is $$ \psi_{\rm ground} \propto \exp\left(\frac{m g}{2 \hbar^2} \sum_{1 \le j < k \le N} |x_j-x_k| \right)$$ Which is a state that is localised with pair correlations but has a center-of-mass which is free (described by a plane wave). See Bethe Ansatz for more detail. |
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The most elegant example I have found is Hooke's atom, also called harmonium. It consists of two electrons that are trapped in a harmonic well: $$ H=-\nabla^2_1-\nabla^2_2+\frac{1}{\left|\mathbf{r}_1-\mathbf{r}_2\right|}+\frac{1}{2}k\left(\mathbf{r}_1^2+\mathbf{r}_2^2\right) $$ For certain values of the spring constant k, this Hamiltonian can be solved exactly. For example, when k = ¼, the ground state is: $$ \Psi\left(\mathbf{r}_1,\mathbf{r}_2\right)=\frac{1}{2\sqrt{8\pi^{5/2}+5\pi^3}}\left(1+\frac{1}{2}\left|\mathbf{r}_1-\mathbf{r}_2\right|\right)\textrm{exp}\left(-\frac{1}{4}\left(r_1^2+r_2^2\right)\right) $$ Source: Wikipedia |
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