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 so 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 physics 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.
Cooper Pairs and Collective Fermionic Pairing
Bosons can exhibit coordinated collective quantum mechanical motion as individual particles when the temperature is lowered below a certain threshold, which is the phenomenon of Bose-Einstein condensation. Fermions cannot do this because of the Pauli exclusion principle; however, in the presence of short-range attractive interactions, fermions can form Cooper pairs: collective two-particle states built from time-reversed single-particle orbitals. In the idealized case of a degenerate shell containing many fermions in the presence of such an attractive interaction, these Cooper pairs themselves can form a condensate. Therefore, fermionic pairing is one of the simplest solvable examples of coordinated collective motion in a fermionic many-body system.
This essential idea underlies superconductivity in metals, pairing gaps and odd-even effects in atomic nuclei, as well as superfluidity in ultracold Fermi gases and neutron matter in the inner crust of neutron stars. Pairing was central to my graduate work on the theory of nuclear fission: as an actinide nucleus undergoes a large change of shape, pairing allows particles to reorganize among evolving single-particle orbitals and strongly influences the resulting collective dynamics. This note develops the elementary physics behind this remarkable form of fermionic collectivity.
The BCS Wave Function (Part 2)
How the pairing story is modified in the more realistic case when the single-particle levels are no longer degenerate. This case is more interesting physically as there is a competition between the single particle energies, which want the system to form a Fermi sea with a sharp Fermi surface, and the pairing energy, which wants to spread out occupation by forming a condensate of Cooper pairs. The result is a smearing of the Fermi surface, which has a variety of interesting consequences.
Eigenstate Thermalization Hypothesis
How is it that an isolated quantum many-body system can relax to equilibrium if it is prepared in a generic nonequilibrium initial state? Time evolution in quantum mechanics is unitary and hence reversible, whereas thermal equilibrium is characterized by time averages of observables reproducing phase space averages, and hence thermal equilibrium necessarily means the destruction of any memory of the initial conditions of the system. This note reviews the basic claims of the resolution to this paradox, namely the Eigenstate Thermalization Hypothesis.
Optical Potentials
Optical potentials are a tool one uses in nuclear physics when considering problems like nucleon-nucleus scattering. If one is concerned with elastic scattering only, which is just one of the many possible outcomes that can occur in this collision process, the problem can be distilled to a one-body quantum mechanical scattering problem, with the projectile particle moving in an effective mean field potential known as an optical potential. However, one must account in some way for all the other possible reactions (single-particle/collective excitations, particle exchange, compound nucleus formation, etc.), and the optical potential accomplishes this by way of being both energy-dependent and complex valued, which leads to an apparent non-conservation of probability. The latter is a feature, not a bug, and represents quantum mechanical amplitude escaping into other channels which are deliberately being excluded from the calculation. I have encountered these optical potentials––which, by the way, are named by analogy with optical materials with complex refractive indices, expressing the possibility of both light propagation and light absorption––during my postdoctoral work.