Saturday, September 13, 2014

Pseudo-Review: Theoretical Solid State Physics, Volume 1 - Perfect Lattices in Equilibrium

The reason this is just a pseudo-review is that while the subject is very important and very interesting, this textbook is very much a typical physics textbook, so there's not much to say about it. It's clear and covers the field as it existed in1973 very well. There is lots of theory and plenty of empirics, so the student can get a good view of how things are done as well as why they are done that way. It's a bit dated, there's nothing on now important topics like quantum heterostructures, but a student that can grasp this book will grasp those as well. The book presumes a strong knowledge of physics - especially quantum and statistical mechanics. This is about all I can think in terms of actual reviewing, so now I will turn to some general observations.

A Minimal Riemann Surface, But Why?

As was mentioned in a previous post, solid state physics is an inherently quantum field. In fact, much of the mathematics of modern solid state physics comes from quantum field theory. In solid state physics, one can make the models of quantum field theory amusingly literal, resulting in such wacky creatures as spin ice. Reading about such things reminds me of the old anime Outlaw Star. In this sci-fi cartoon, there was an episode that was a riff on the old Hal Clement novel Mission of Gravity in which a magnetic monopole plays a role in the main character's escape from a prison plant. I hope this means that reading and thinking about this helps me write the "Is Every Possible World The Same Temperature?" series of posts.

But such distractions aside, the format of this book had a lot of little things that bothered me. This is very much a physicists book, where important theorems are used, not proven. One example is in what is essentially the founding theorem of perfect lattice part of solid state physics. There is some wavefunction \( \Psi(\vec{x},t) \) that describes the overall behavior of a crystal. We start by assuming that the system is in equilibrium, so that \( \Psi(\vec{x},t) = \Psi(\vec{x},t) \). We do this because: 1. Less variables to deal with and 2. This situation is quite applicable to the real world.

We go on to assume the crystal is large enough that border effects can be ignored and perfectly regular. That is, there is some lattice vectors \(\vec{v_1}, \vec{v_2}, \vec{v_3}\). We can examine the set of transforms \(T_{\vec{v_i}}\) such that: $$T_{\vec{v_i}}(\Psi(\vec{x})) = \Psi(\vec{x}+\vec{v_i}).$$If you continue along the crystal enough, you'll land in a location that is indistinguishable from where you started. For instance, if you start on an atom, you'll end up on another atom of with the same surroundings. This assumption comes from crystallography, and is applicable to some systems but not others. If true enough, there exists an N such that $$T_{\vec{v_i}}^N(\Psi(\vec{x})) = \Psi(\vec{x}+N \vec{v_i}) = \Psi(\vec{x}).$$This means that the transform \(T_{\vec{v_i}}\) is a root of unity, and we can use Lagrange's Theorem to bring in the powerful tools of group theory. Lagrange's Theorem and Fourier Analysis together give the theory of Bloch Waves.

This is all simple enough, even obvious once you've seen group theory for a bit. But I still would have preferred the book to present it more like this, with Lagrange's Theorem and everything showing it's face. I would like to find a book that is like Khinchin's Mathematical Foundations of Statistical Mechanics but for solid state stuff. In essence, what I really wanted the book to give me insight into math from a physics perspective, rather than what the book is - a book to give insight into physics using math. Oh well.

Writing this post has made me decide that the next post in the "Is Every Possible World The Same Temperture?" series must be on what quantum mechanics even is. Then when I want to make a statement about quantum physics, I can do it by dropping a pointer. I hope to build a maximally uninterpreted bare theory that all the future posts will try to get behind, hopefully all succeeding and all failing. See you soon!

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