A List of Problems in Many Electron Systems
I have long grappled with the best way to engage with physics. One the one hand, I could spend time writing about topics in physics that I enjoy. On the other hand, I could spend time actually doing physics problems out of a textbook, and even attempt to do experiments myself. In the latter case, I almost always end up with a deeper understanding of subject matter because I can actually create examples and solve them. However, solving small problems takes a while and one can easily lose sight of the bigger picture. I think that writing about physics from a general point of view helps alleviate this somewhat and balances it out.
In physics, building up a model takes time and requires more understanding to actually come up with anything interesting. Moreover, things tend to be slightly more expensive to test. In the words of Gwern:
“Half the challenge of fighting procrastination is the pain of starting—I find when I actually get into the swing of working on even dull tasks, it’s not so bad. So this suggests a solution: never start. Merely have perpetual drafts, which one tweaks from time to time. And the rest takes care of itself”
From https://www.gwern.net/About#target-audience
In this spirit, I've compiled a list of problems that I've made or adapted form textbooks, that have helped me understand the topic of the quantum mechanics of large systems. This list is a Notion database, so it updates whilst I do work and solve problems. The database contains a live list of problems that I am compiling, along with their authors and of-the-hat applications.
The Importance of the Path Integral
Many of the exercises below involve manipulation of a path integral to compute correlated values. A lot of these computations are made easier through added source terms and then 'completing the square'. For example, consider a Gaussian integral with a linear source term - we can compute this quite easily by completing the square in the integrand and then pulling out a constant scaling term.
The principle holds true also for path integrals, and associated functions we wish to compute. Is the path integral correct? Well, we can check quantities against the usual method of computing correlation functions.