Physicists sometimes like to use dramatic terminology to describe phenomena or ideas. For example, the
Ultraviolet Catastrophe was the phrase describing the failure of classical statistical physics to predict the correct form of thermal ("black body") radiation in the high frequency ("UV" at the time) limit. The obvious disagreement between what seemed like a solid calculation (based on
equipartition) and observation (that is, you are not bathed by gamma radiation from every thermal emitter around you) was a herald of the edge of validity of classical physics.
In condensed matter physics, there is another evocative phrase,
Anderson's Orthogonality Catastrophe. (
Original paper) Here's the scenario. Suppose we have a crystal lattice, within which are the electrons of an ordinary metal. In regular solid state physics/quantum mechanics, the idea is that the lattice provides a periodic potential energy for the electrons. We can solve the problem of a single electron in that periodic potential, and we find that the single-particle states (which look a lot like plane waves) form bands. The many-electron ground state is built up from products of those single-electron states (glossing over irrelevant details). The important thing to realize is that those single-particle states form a complete basis set - any arrangement of the electrons can be written as some linear combination of those states. (For students/nonexperts: This is analogous to Fourier series, where any reasonable function can be written as a linear combination of sines and cosines. Check out t
his great post about that.)
Now, imagine reaching in and replacing one of the atoms in the lattice with an impurity, an atom of a different type. What happens? Well, intuitively it seems like not much should happen; if there were 10
22 atoms in the lattice, it's hard to see how changing one of them could do much of anything. Indeed, if you compared the solutions to the single-particle problem before and after the change, you would find that the single-particle states are
almost identical. However, "almost" means "not quite". Suppose the overlap between the new and old single particle states was 0.9999999, where 1 = no change. For \( N \) electrons, that means that the new
many-particle ground state's overlap with the old many-particle ground state is goes something like \( 0.9999999^{N}\). For a thermodynamically large \(N\), that's basically zero. The new many-particle ground state is therefore
orthogonal to the old many-particle ground state. In other words, in terms of the old basis, it seems like adding one impurity (!) produces an infinite number of electron-hole excitations (!!) (since it takes an infinite number of terms to write the new many-particle ground state in terms of the old).
So, where does this fall apart? It doesn't! It's basically correct. The experimental signature of this ends up being apparent in
x-ray absorption spectra, in a variety of meso/nano experiments (
pdf), and in cold atoms (
pdf).
Any other good physics examples of overly dramatic language?