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You will likely be familiar with the notion of a hydrogen atom, having seen something about its discrete energy levels and orbitals at some point or another. This is an example of a quantum system. In this talk, we explore what transpires when taking a quantum system and placing it into a three-dimensional container having some prescribed geometry. In the limit where the container is large (relative to the natural lengthscale of the quantum system), its influence over the quantum system is negligible; yet, as the container is made small (comparable to the aforementioned lengthscale), geometric information intrinsic to the container plays an important role in determining the energy and orbital structure of the system. We describe how to do (numerically-assisted) perturbation theory in this small-container limit and then match it to the large-box regime, using a combination of these asymptotics and direct simulations to tell the story of geometrically confined quantum systems. Much of our focus will be on linear Schrödinger equations governing single-particle quantum systems; however, time permitting, we will briefly discuss how to do similar things to study geometrically confined nonlinear Schrödinger equations, with geometric confinement of Bose-Einstein condensates being a primary motivation. Geometric confinement of an attractive Bose-Einstein condensate can, for instance, modify the collapse threshold and enhance stability, with the particular choice of confining geometry shifting the boundary of instability, staving off the collapse which is prevalent in three-dimensional attractive condensates.
Further Information
Dr Rob Van Gorder’s research focuses on how physical phenomena can be described, predicted, and controlled using applied mathematics. He works across mathematical modelling, analytical and asymptotic methods, and numerical simulation, applying this combination to a wide range of physical systems.
His interests in fluid dynamics centre on fundamental flow structures—such as vortices, bubbles, waves, and boundary layers—and how they evolve, persist, or break apart. He also studies spatial instabilities and pattern formation, investigating how mechanisms such as Turing and Benjamin–Feir instabilities extend to heterogeneous or non-autonomous systems arising in chemistry, physics, biology, and epidemiology.
In theoretical physics, Dr Van Gorder works on quantum mechanics, quantum fluids, and nonlinear waves, including the dynamics of Bose–Einstein condensates, quantised vortices in superfluid helium, and confined quantum systems. Across these areas, he aims to understand how nonlinear and quantum systems behave under realistic constraints and external forcing.
His recent publications include work on pattern formation and diffusive instabilities in Proceedings of the Royal Society A.