Electrons correlations through a study of the Helium atom
What are electron correlations? How do you compute the ground state energy of a multi-electronic system, when you wish to go beyond the single-particle approximation (like in Hartree-Fock theory)? The most general (and in many ways also the most brute force!) approach is called Configuration Interaction. It is in theory a well formulated route to attack the electron correlation problem exactly. In this summary we will illustrate these concepts on the simple case of a Helium atom (arguably the simplest possible electron correlation problem).
Mathematics of a single particle in a periodic potential
The purpose of this summary is to gather a wealth of useful approximations and mathematical results on the quantum mechanics of a single particle (here an electron) in a periodic potential. In particular, we go over the concept of Born Von-Karman periodic boundary conditions, and later on the result by Felix Bloch that bears his name, namely "Bloch's Theorem".
|The Quantum Mechanics of a single particle in a periodic potential|
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A detailed analysis of the finite Kronig-Penney model
I decided in that summary to give an in-depth mathematical analysis for a simplified model of a finite crystal. Very general results are outlined and the important consequences of the choice of boundary conditions are put forth. The analysis, especially the physical analysis, could be pushed further but I chose to focus mostly on the mathematics of the model.
Mathematical expression for the gravitational force between two real spherical masses
remember that physics class on the motion of planets. And the assertion from your teacher that the mathematical expression for the gravitational force between the Sun and the Earth is the simple one of Newton, even though you learned that this latter expression is in theory only true for point masses and not actual non-zero sized spheres... Well here is a short summary on how to prove that outrageous assertion. Have fun!
|mathematical expression for the gravitational force between two non-zero sized spheres|
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A very brief introduction to the goals of (Computational) Quantum Mechanics
The purpose of this very short introduction is to present the fundamental equation in Quantum Mechanics, namely the many-body (or often in practice the many-electron) Schrodinger equation. Then I expose briefly what level of accuracy chemists and physicists would like to get from theoretical / computational Quantum Mechanics in order for the theory to be experimentally useful. Finally I introduce atomic units as a way to simplify the many-body Schrodinger equation and make the fundamental constants of physics, like the Planck constant, disappear.
|Introduction to the goals of computational quantum mechanics|
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The Born-Oppenheimer approximation
My purpose in that little summary is to outline a motivation for introducing the Born-Oppenheimer or frozen nuclei approximation. This fundamental result is used implicitly throughout computational quantum mechanics like quantum chemistry or solid-state physics, and is very rarely explained. Rather than going at length about justifying that approximation, I illustrate the concept and give an idea of the relative error incurred in using that approximation on a simple hydrogen atom model.
A study of Hydrogenoid atoms
In that little section, I spend some time introducing the notion of hydrogenoid atoms, what they are, what is the most general description of them, and then goes on to compute their exact spectra. This summary is pretty self-contained and I take the deliberate approach to start from scratch (from the general Schrodinger equation for the problem at hand) and walk the reader step-by-step into the mathematical solution to the problem.