Contact interaction (quantum mechanics, $H_{\mathrm{int}}=\frac{g}{2}\int {\hat{\psi }}^{†}(x{)}^2\hat{\psi }(x{)}^{2}dx$)

In a one-dimensional identical-particle model, a contact interaction has two-body potential $U(x,x^{\prime })=g\delta (x-x^{\prime })$, using the Dirac delta distribution. For bosons, its second quantization is the displayed normally ordered Hamiltonian operator. Projection onto orthonormal modes gives $H_{\mathrm{int}}=(g/2){\sum }_{ijkl}{I}_{ijkl}{a}_i^{†}{a}_j^{†}{a}_k{a}_l$, where $I_{ijkl}=\int {\phi }_i^{\ast }{\phi }_j^{\ast }{\phi }_k{\phi }_{l}dx$. 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.

For a periodic box of volume $V$, expanding $\psi (\mathbf{r})=V^{-1/2}{\sum }_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{r}}{a}_{\mathbf{k}}$ in $H_{\mathrm{int}}=\frac{u}{2}\int d^{3}r{\psi }^{†}{\psi }^{†}\psi \psi$ gives the displayed normal ordering. The spatial integral is $V{\delta }_{{\mathbf{k}}_1+{\mathbf{k}}_2,{\mathbf{k}}_3+{\mathbf{k}}_4}$ by Fourier orthogonality, enforcing momentum conservation and leaving the factor $V^{-1}$. The coupling $u$ in three dimensions has units of energy times volume.

A low-energy contact coupling is defined through a physical scattering length. Ultraviolet pieces of zero-point energy cannot be interpreted independently of the matching or counterterms fixing that coupling.

Project a bosonic field on a ring onto angular-momentum modes zero and one. The kinetic level spacing is $\mathcal{E}={\hslash }^2/(2m{R}^2)$. The contact-interaction integral enforces angular momentum conservation, leaving two same-mode terms and four mixed terms. The displayed projected Quantum Hamiltonian follows by normal ordering. Couplings to discarded modes belong to the full model and can limit the two-mode approximation.

For distinct box modes ${\phi }_1=\sqrt{2/L}\sin (p\pi x/L)$ and ${\phi }_2=\sqrt{2/L}\sin (q\pi x/L)$, the displayed contact interaction overlaps produce self terms $3g{\sum }_j{n}_j(n_j-1)/(4L)$, a density term $2g{n}_1{n}_2/L$, and pair conversion $g(a_1^{†2}{a}_2^2+a_2^{†2}{a}_1^2)/(2L)$. Pair conversion preserves total particle number but prevents diagonalization by individual occupation numbers. The additional overlaps are $I_{1112}=-{\delta }_{q,3p}/(2L)$ and $I_{1222}=-{\delta }_{p,3q}/(2L)$, following from ${\sin }^{3}u=(3\sin u-\sin 3u)/4$. Third-harmonic mode pairs therefore also permit single-particle transfer assisted by the other particles. For the first two box modes these additional terms vanish.

Two distinct normalized plane waves on a periodic one-dimensional ring of length $L$ give the displayed projected contact interaction. Momentum conservation excludes pair conversion between these two modes. Four mixed operator orderings give the coefficient of $n_1{n}_2$. At fixed particle number $N$, the energy is $g[N(N-1)+2n_1(N-n_1)]/(2L)$. For repulsive coupling, its minima are the two endpoint Fock states with all particles in one mode; a one-particle energy difference selects the lower mode. This ground-state statement concerns the two-mode projection.