This is a deliberately incomplete and informal collection of theoretical physics ideas that I find especially compelling for one reason or another. These notes are written for pleasure and understanding, not as comprehensive lecture notes on any particular topic, and they may contain mistakes. It is my hope that this collection will grow with time.
A note about the notes: for the time being at least, the accompanying PDF notes I post here are deliberately highly compact. This page is partly a pedagogical experiment. I found during my PhD that if I really needed to understand a topic deeply, finding a high quality source and copying it out by hand was remarkably effective at making the material stick. However, obviously this can get extremely time consuming, and sources which are both compact and high quality are especially valuable. The notes here are created, aspirationally, in this spirit: they are meant to contain enough of the essential argument to stand on their own, while remaining short enough to copy out by hand in full, for anyone who wishes to deeply internalize any material which speaks to them. If I post them here, I have done so myself.
Conical Degeneracy and Berry Phase
Many physical systems have a region where two states or modes come close in energy while all other states remain well separated. Near such a crossing, the essential physics can often be reduced to a 2x2 Hamiltonian. Examples include two electronic states of a molecule as the nuclei move (e.g., in the Jahn-Teller effect), a spin-1/2 in a magnetic field, the two polarization modes of light near a special propagation direction in a biaxial crystal, and the electronic states near a Dirac point in graphene. In each case one parameter can change the relative energies of the two states while another mixes them. The details differ, but the local two-state geometry is the same.
One reason I like this topic is that it gives a particularly clean example of how interesting structure can emerge simply because a physical system depends on external parameters. As those parameters are varied, eigenvalues can approach, avoid one another, or become exactly degenerate. The corresponding eigenvectors can acquire nontrivial geometry of their own. This same general theme appears elsewhere in physics and math––for example, in the bifurcations that occur when a parameter in an ordinary differential equation is varied.
Potential Energy Surfaces
Potential-energy surfaces (PESs) arise when the energy of a quantum state is evaluated as a function of a set of slowly varying or collective coordinates. In molecules they organize equilibrium shapes, vibrations, and chemical reactions; in materials, atomic diffusion, adsorption, and surface chemistry; and in nuclear physics, collective deformation and fission. The common construction is simple: treat those coordinates as parameters, solve the remaining quantum problem, and obtain the energies E_n over the resulting parameter space. Minima, barriers, valleys, avoided crossings, and intersections then acquire direct physical meaning.
I encountered potential energy surfaces during my PhD research on nuclear fission, where it is a common practice in the fission theory community to evaluate the nuclear energy as a function of various kinds of collective deformations; people then often try to infer fission dynamics from such surfaces, an approximation which is sometimes useful and sometimes badly misleading. The same basic construction also appears in molecular physics, where it is far more widely known. The notion of a PES naturally leads one to considerations of adiabatic vs. non-adiabatic motion, the latter being an exceptionally interesting class of quantum many-body dynamics, of which nuclear fission is a concrete example.