Quantum-Mechanical Many-Body Problem with Hard-Sphere Interaction

The system under consideration is an $N$-particle quantum-mechanical system enclosed in a volume $V$, in which the particles interact via two-body hard-sphere potentials, with hard-sphere diameter $a$. The two-body hard-sphere problem is first discussed by a generalization of Fermi's pseudopotential by means of which the problem is formulated entirely in terms of the scattering phase shifts. It is then shown that a pseudopotential for the $N$-body problem can be introduced, and leads to an expansion of the energy levels of the system in powers of $a$. The first order energy levels of a Bose and a Fermi system are calculated. For the Bose system, the first order energy formula exhibits an "energy gap" above the ground state, leading to properties of the system not dissimilar to that of a superfluid. The ground state energy for a Bose system is calculated to order $a^3$ and that for the Fermi system, to order $a^2$. The physical interpretation and validity of these results are discussed.