The Variational PrincipleWhen Prof. Kivelson walked into class today, he looked a bit taken by surprise. He just kind of stood there for a bit and stared at everyone who stared back at him. There was a funny look on his face, like, ‘‘Oh, you're expecting me to teach you something?’’ Well, yes, we would like you to teach us some statistical mechanics! In class, we spent way too much time proving the variational principle, and not enough time explaining what it actually means. Honestly, it's much more important to understand the logic behind a variational argument than to know how to prove it…so of all the sections on this page, the Motivation and Overview section is most important. (I don't even know if I'll get around to writing the rest of the sections…I have a life too, you know!) OutlineMotivation and SummaryOne of the key points today is that interacting systems are very difficult to solve in general. The variational principle is a useful tool to have in our pocket because it lets us leverage the Hamiltonians which we actually can solve to solve Hamiltonians which we can't. Key Point
The strategy of the variational principle is to use a problem we can solve to approximate a problem we can't. More preciesly, suppose we want to solve a hard system with a Hamiltonian But we don't care about the trial Hamiltonian – we care about the actual hamiltornian The variational principle tells us that: 1. Our best guess for the actual free energy is given by the variational free energy ![]() 2. No matter how good a guess your variational free energy ![]() 3. The best variational solution we can find is the one that gets as close as possible to the actual Hamiltonian. A quick comment about notation: When we write ![]() (Notice the subscript ‘‘tr’’ on the partition function and Hamiltonian here.) Minimizing the variational free energyThe key point of the variational principle is that our best guess is the one with the smallest variational free energy For instance, our family of trial Hamiltonians might be all possible 2D Ising models ![]() where we can pick the parameters In the picture below, I've illustrated my point. There's a whole bunch of different Here I've plotted how Okay I think I've nailed the point into the floor enough by now. Operationally, here are the steps to apply the variational principle: Steps
Hooray, we've learned the variational principle. I'm not sure if I'll get around to finishing up the rest of this page…for now just go on to the next page about non-interacting spins. Jenson's InequalityProving the Variational PrincipleConnection to Quantum Mechanics’ Variational PrincipleDiscussionLeave a Comment Below!Comment Box is loading comments...
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