Scientific Research

Structure of the Bogoliubov-Valatin Canonical Basis Set

Abstract: We discuss the mathematical properties of the Bogoliubov-Valatin basis set of quasiparticle wave functions for a fermion system, with particular emphasis on the properties of the canonical basis set. The properties of the canonical basis set, apart from their definition, are largely unknown. In particular, what is the physically required size of the canonical wave function set in order to correctly describe superfluid systems. While the cardinality of the set of quasiparticle wave functions for an isolated system in vacuum is 𝔠=|ℝ^3|, the basis set for a finite system in a finite volume is countable, with cardinality ℵ_0=|ℤ|=|ℕ|. However, the size of the canonical basis set for an isolated system in a finite volume or for a periodic system is typically much smaller than the size of the entire basis set, and it is determined by the level of the spatial resolution. We show how one can get insight into the character of the canonical wave functions and we justify the minimum number of canonical wave functions needed for a given system.

• arXiv

Wolfram Function Repository

I am the principle author of the HardSphereSimulation function in the Wolfram Function Repository, which can be downloaded by anyone who uses Mathematica. It allows one to run simulations of hard sphere systems, which can be valuable for modeling the behavior of simple liquids and gases, among other things.

• Documentation Page.

• Blog post documenting early explorations of the hard sphere gas for the Wolfram Physics Project.

I am the author of the HydrogenWavefunction function in the Wolfram Function Repository, which can be downloaded by anyone who uses Mathematica. This function gives the functional form of the hydrogen atom position-space wavefunctions, which are one of the few exactly solvable, useful systems in quantum mechanics.

• Documentation Page.

I am the author of the PeriodicBoxDisplacement and PeriodicBoxDistance functions in the Wolfram Function Repository, which can be downloaded by anyone who uses Mathematica. These function give the (shortest) displacement vector/distance between two points in a box with periodic boundary conditions, respectively. They can be used for all sorts of simulations! (I personally have used them extensively in my hard sphere simulations.)

• PeriodicBoxDisplacement: Documentation Page.

• PeriodicBoxDistance: Documentation Page.

Misc

I participated in a Wolfram Research livestream about the role of networks in chemistry, which you can check out here.