= Contact interaction
{disambiguate=quantum mechanics}
{title2=$H_{\rm int}=\tfrac g2\int\hat\psi^\dagger(x)^2\hat\psi(x)^2\,dx$}
= Contact interaction
{synonym}
In a one-dimensional identical-particle model, a contact interaction has two-body potential $U(x,x')=g\delta(x-x')$, using the <Dirac delta distribution>. For <bosons>, its <second quantization> is the displayed normally ordered <Hamiltonian operator>. Projection onto <orthonormal> modes gives $H_{\rm int}=(g/2)\sum_{ijkl}I_{ijkl}a_i^\dagger a_j^\dagger a_ka_l$, where $I_{ijkl}=\int\varphi_i^*\varphi_j^*\varphi_k\varphi_l\,dx$. The overlap integrals determine which mode-changing processes survive. A finite-mode model describes that projection; discarded modes can still matter to the full physical system.
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