Undergraduate Research

Undergraduate Research is a course I first started teaching in the Fall semester of 2022. I am currently teaching a second time in Fall semester 2023. The goal of this course is to bring students quickly from a level of undergraduate Quantum Mechanics 1 to a level that where they have some basic idea of what is done in many current research papers in condensed matter theory. Since only a small amount of the class should be spent on introductions to different topics and most on actual research, it is a very compressed course that has a lot of its "fluff" removed to ensure that it can be done in a short amount of time of 3 weeks. Similarly, homework problems are chosen to be not very demanding but such that they convey the most important points.

Anyone who wants can feel free to use my notes or part of them. I am most happy about any potential feedback. (At this stage still many typos were found so be careful!  For convenience I also provide handwritten versions of the typeset lectures in case there is typos but even the handwritten notes might includes some mistakes)

A great thanks to Ibraheem Faisal Al-Yousef who was kind enough to typeset the first 4 lectures in LaTeX.

Note: Files marked in  green can be downloaded. Files marked in red are not available.

Lecture downloads:

Lecture 1 (typeset)     Lecture 1 (handwritten)

This lecture covers an introduction to second quantization and identical particles

Lecture 2 (typeset)     Lecture 2 (handwritten)

This lecture covers how to rewrite ordinary Hamiltonians in second quantization and introduces an easy to understand version of exact diagonalization.

Lecture 3 (typeset)     Lecture 3 (handwritten)

This lecture motivates interest in quadratic (in field operators) Hamiltonians as class of "easily" solvable Hamiltonians. Both unitary diagonalization and (in the case of Bogolibov-type bosonic Hamiltonians) paraunitary diagonalization schemes are presented

Lecture 4 (typeset)     Lecture 4 (handwritten)

This lecture describes how to find effectie quadratic Hamiltonians. It considers 3 cases. First, the case of weak quartic/interaction term. Second, the case of a Hamiltonian where parts of the quartic term commute with the Hamiltonian that allows the Hamiltonian to break into non-interacting sectors. Third, a brief survey of some examples for mean-field theories.

Lecture 5 (typeset)     Lecture 5 (handwritten)

An information entropy approach to quantum statistical mechanics is motivated and different ensembles (microcanonical and canonical)for the density matrix are derived.

Lecture 6 (typeset)     Lecture 6 (handwritten)

This lecture finishes the elementary treatment of quantum statistical mechanics by deriving the grand canonical ensemble and particle occupation numbers (Bose-Einstein and Fermi-Dirac distributions) for free bosons and fermions.

Lecture 7 (typeset)     Lecture 7 (handwritten)

This lecture introduces time evolution according to a time independent Hamiltonian and briefly mentions how the density matrix is related to the time evolution operator via a Wick rotation. Time evolution (and by extension statistical mechanics) is difficult for interacting systems, which makes a perturbative treatment necessary. We introduce the interaction picture, time ordered exponentials, the Dyson series and the Trotter-Suzuki expansion.

Lecture 8 (typeset)     Lecture 8 (handwritten)

We note that enforcing symmetries and exact properties exactly can improve approximations. Motivated by this we derive the Magnus expansion for a time evolution operator that enforces unitarity and thereby is more reliable than a Dyson series. We next, derive time evolution equations for operators (Heisenberg picture) and Neumann equation for the density matrix and point out the reason for the difference in sign that appear.