A perturbative method for resolving contact interactions in quantum mechanics.

This article is an investigation of the effectiveness of quantum mechanics when a finite region of space is omitted from analysis; this is done as part of a method to resolve the nature of short-ranged interactions without explicitly modeling them. This is accomplished with an artificial boundary behind which obscured short-ranged physical effects may operate. This may be necessary for two reasons: (1) there are phenomena that operate over a short (but non-zero) range that cannot be reliably modeled with a potential function and/or (2) the entire Hamiltonian being used is expected to lose its predictive power when applied at short distances. Omitting a finite volume of the space from analysis implies that the strict unitarity requirement of quantum mechanics must be relaxed, since particles can actually propagate beyond the boundary. Strict orthogonality of eigenmodes and hermiticity of the Hamiltonian must also be relaxed in this method; however, all of these canonical relations are obeyed when averaged over sufficiently long times. A free function of integration that depends on momentum is interpreted as a function encoding information needed to match a long-distance wavefunction to an appropriate state function on the other side of the boundary. What is achieved appears to be an effective long wavelength theory, at least for stationary systems. As examples, the quantum defect theory of the one-dimensional Coulomb interaction is recovered, as well as a new perspective of the inverse-square potential. Potential applications of this method may include three-dimensional atomic systems and two-dimensional systems, such as graphene.