The Big PictureToday (Mon Week 2), we opened class with a bit more logistics, discussing where we'd hold lectures (Mon and Wed in 60101, Fri in McCollough 122), as well as what textbooks we'd use:
It sounds like we'll be covering all sorts of topics in this class. I'm curious if we'll manage to tie it all together coherently, or if it's just going to be a ‘‘taste’’ of many different topics. Quantum Statistical MechanicsLast time in class, we sort of dived right away into solving the 1D Ising Model, but we didn't get talk much about the big picture of what exactly we were doing or why we were doing it. To help us understand why we wanted to calculate certain quantities, Prof. Kivelson explained to us the general framework of quantum statistical mechanics – the way to start with the description of a physical system and end with macroscopic observables that we can measure. So, what steps are involved in understanding the thermodynamic behavior of a system? Formal Procedure of Quantum Statistical Mechanics
Say we have a quantum mechanical system with Hamiltonian , and we wish to find its thermodynamic properties in the canonical ensemble.
From these quantities, we can also compute the ensemble averages of any quantum mechanical observable by computing where again we can write either as an abstract trace against the observable or as a concrete sum over energy states. Using the same method, we can also calculate correlation functions such as , which could represent a variety of things:
I realize these explanations are pretty dry and unenlightening; if time allows, I'll come back later and elaborate a bit more. Game Plan for Solving Ising ModelNow that we knew why we wanted to calculate certain things, we went on to explicitly solve the 1D Ising Model. Remember that the first step is to compute the partition function, which requires us to sum over all the energy states. This is a bit of a hassle for us, since for a 1D Ising Model with sites, we need to sum over all the possible states. As we take the thermodynamic limit , there will just be too many terms to add up, so as Prof. Kivelson says, we'll need to be clever. Here is our game plan: Solving the 1D Ising Model by Being Clever
We went through everything again in class on Wednesday. Here are my detailed notes! Leave a Comment Below!Comment Box is loading comments...
