Effective approach to non-relativistic quantum mechanics

Tue, 03/11/2015 - 12:00
David Jacobs (UCT)

Abstract: The boundary conditions on non-relativistic wavefunctions are often unconstrained by the basic precepts of quantum mechanics. The physical implications of this boundary condition freedom has remained largely unappreciated, despite the wealth of literature that exists on the subject, namely self-adjoint extensions. They must, however, be incorporated into the effective description of any quantum mechanical system in order to capture all possible short-distance physics that doesn't decouple in the long wavelength limit. A physically-intuitive method will be presented that employs artificial boundaries inserted at an intermediate (but arbitrary) scale, on which the most general boundary conditions are satisfied. Requiring measurable quantities (e.g. spectra and cross sections) to be independent of this artificial boundary, renormalization group-type equations are derived that determine how the boundary conditions must ``flow" with the scale of the artificial boundary. As examples, I will discuss this applies even to the free particle, harmonic oscillator, and Coulomb potentials. Connections are made to previously established results for contact potentials in certain atomic and condensed matter systems, and novel (I hope) predictions are made that could be verified in real laboratory systems.