# Quantum Observations

• ### Quasicrystalline Art

September 5, 2019

• ### Pulsing a two-band model to discover topology

February 20, 2017

In systems with an anomalous quantum Hall effect, the quantized Hall conductivity comes from the integral of a Chern number over some manifold. Usually, this integral is derived via Kubo formula. However, there is geometry involved in how the state evolves, and in fact, we can use the dynamics of the current following a weak pulse in order to find the DC conductivity. The route is easy enough: Say we have a conductivity which when written with respect to time is $\sigma_{xy}(t-t')$, and without loss of generality we apply a pulse $E_x(t) = A_x\delta(t)$, then we can find the current response in the perpendicular direction

• ### Subtleties in linear response theory

December 22, 2014

In linear response theory, we consider some small perturbation to a Hamiltonian and look at the response of some observable to that perturbation. In the case considered here, the perturbation is an electric field, and the response is current. The linear response that characterizes these quantities is called the conductance.

• ### Current in Single Particle Quantum Mechanics

January 14, 2014

For simplicity, I will only use one-dimension in this post, but this can be generalized to higher dimensions rather easily.

Many textbooks on Quantum Mechanics mention current density can be derived from the continuity equation and probability. The usual method for figuring this out is to assume you have some Hamiltonian $H = p^2/2m + V(x)$ where $p$ is the momentum and $x$ is the position. In this way the current density is written in terms of the wave function $\psi(x,t)$ as

• ### The delta-function Potential Lattice

June 14, 2013

I was messing around with some simple problems, and found this simple but illustrative problem. It starts you off in basic quantum mechanics and introduces concepts in a very straightforward way to both get to a more condensed matter perspective while even showing some interesting effects that have experimental consequences (energy bands and band gaps opening).

• ### Use GmailTeX to compose and view emails with LaTeX

May 16, 2013

I’ve received numerous emails with pseudo-LaTeX in them, and I’ve composed many emails as well. My normal solution is to convert LaTeX to unicode with the helpful Mac application Unicodeit. However, this method is incomplete and misses some of the more complicated LaTeX.

• ### Spin-orbit coupled Hamiltonian

April 28, 2013

For applications in many parts of condensed matter physics and cold atoms physics, we use what is known as the Rashba spin-orbit coupled Hamiltonian. This Hamiltoninan is so-named because it couples momentum $\mathbf{p}$ to the spin $\mathbf{S}=\frac12\sigma$ where $\sigma = (\sigma_x,\sigma_y,\sigma_z)$ are the Pauli matrices and $\mathbf{p}=(p_x,p_y,p_z)$ is a vector of momentum operators:

Blog - Justin H. Wilson