This article discusses the most common techniques used in the representation of crystalline structures as graphs. This includes the determination of edges and of the associated features, which encode physical information.

An overview of the BCS model, particularly from a mean-field approach. The formation of Cooper pairs is discussed and the mean-field Hamiltonian is treated up to the Bogoliubov transformation.

Special Relativity really doesn't need to be too hard as it can be described using only algebra (assuming no acceleration). For conceptual understanding, the theory can be broken down into 4 fundamental concepts: velocity addition, time dilation, length contraction, and the rear-clock ahead effect.

Here, a recently proposed framework for quantum neural networks is tested for a simple architecture on a randomly generated set of data. Explicit forms of all the mathematical objects relevant to the example 2x3x2 model are also provided as well as some results.

Here, a brief and intuitive justification is given for the second-quantized form of the tight-binding Hamiltonian. Starting from the free electron's Hamiltonian we build to the position space second-quantized Hamiltonian for a lattice with localized electrons.

A basic example of perturbation theory is given to motivate the relevant concepts. Specifically, we show a simple method to approximate the solution of a certain quintic polynomial.

Here an extremely simplified spin Hamiltonian of the NV-Center in diamond is presented to give some intuition for the use of such systems as qubits.

Here we derive the common form of the standard Lagrangian, namely the difference of kinetic and potential energy. This is done by assuming D'Alembert's principle and extending it to finite periods of time via Hamilton's Principle.

The Klein-Gordon equations describe a relativistic quantum field, but not one we really care about in most cases. Generally, though, it's briefly mentioned in the introduction of the Dirac equation or used as a first example of a quantum field. Here, we derive the form of the equations in two ways and then discuss their applicability.

SSH is one of the most common protocols for remote server connection and control over untrusted networks. Coupled with SCP for secure file transfer in similar circumstances, these are must know tools for most working professionals interfacing with remote devices.

Here the basic model of a spring is compared to the dynamical behaviour of generic systems around points of equilibrium. This is used to justify the commonly applied simple harmonic approximation.

Special Relativity really doesn't need to be too hard as it can be described using only algebra (assuming no acceleration). For conceptual understanding, the theory can be broken down into 4 fundamental concepts: velocity addition, time dilation, length contraction, and the rear-clock ahead effect.

Here, a sagging floor supported by wooden joists is compared to the deflection of a simply supported beam under uniform load. This comparison is used to construct a simple model for the a sagging floor that may be applied by taking two simple measurements of the floor in question. Finally, a difference in length for the sagging floor (relative a flat floor) is calculated.

Here, a brief and intuitive justification is given for the second-quantized form of the tight-binding Hamiltonian. Starting from the free electron's Hamiltonian we build to the position space second-quantized Hamiltonian for a lattice with localized electrons.

The relation for calculating torque is often cast as a cross product, namely that between the position vector relative to the axis and the force at that position. However, in the realm of Clifford Algebras, we may replace this cross product with the exterior product which gives 2-blades as opposed to orthogonal vectors.

The implications of functions being vectors are explored. Namely, the concept of basis sets, linear operators, and self-adjoint operators are extended to the realm of one-dimensional complex-valued functions.