= Solution
{c}
Write $h_0=-\hbar^2\nabla^2/(2m)+V(\mathbf r)$ for the one-particle <Hamiltonian operator>. The <second quantization> of the <kinetic energy>, external <potential energy> and repulsive <contact interaction> is
$$
\boxed{\hat H=\int d^3r\,\hat\psi^\dagger(\mathbf r)h_0\hat\psi(\mathbf r)
+\frac{U_0}{2}\int d^3r\,\hat\psi^\dagger(\mathbf r)^2\hat\psi(\mathbf r)^2.}
$$
The factor $1/2$ counts each unordered particle pair once. <Normal ordering> excludes self-interaction: the interaction acts on two distinct <bosons>. The fixed <particle number> sector has $\hat N=\int d^3r\,\hat\psi^\dagger\hat\psi=N$. In three dimensions, this <contact interaction> is understood as a regulated low-<energy> interaction with a matched coupling, as in <renormalization of a three-dimensional dilute-gas contact coupling>; the finite-mode calculation below uses its projected matrix elements.
Back to article page
