= Effective two-dimensional contact coupling
{title2=$g_{2D}=g_{3D}\int|\chi(z)|^4dz$}
Factorizing a normalized condensate field as $\Psi_{3D}(x,y,z)=\psi(x,y)\chi(z)$ with $\int|\chi|^2dz=1$ reduces its contact interaction energy to $(g_{2D}/2)\int|\psi|^4dxdy$, with the displayed coupling. Thus $g_{2D}$ has units of energy times area, whereas $g_{3D}=4\pi\hbar^2a/m$ has units of energy times volume for a physical <scattering length> $a$. For a normalized axial Gaussian $\chi(z)=\pi^{-1/4}a_z^{-1/2}e^{-z^2/(2a_z^2)}$, direct Gaussian integration gives $g_{2D}=g_{3D}/(\sqrt{2\pi}a_z)$. A model with density normalized per unit axial length can instead retain $g_{3D}$ with a line-density normalization.
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