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When the parameters of some physical systems are precisely tuned, the systems can enter a phase transition in which the behaviour of observables changes dramatically. In particular, the systems can become scale-free in the sense of losing any relationship to scales of measurement, i.e., the systems suddenly switch to behaving the same irrespective of theContinue reading “A scale-free probability distribution must be a power law”

When you think of the classical harmonic oscillator, think of a mass connected to a spring oscillating at a natural frequency which is independent of the initial position or velocity of the mass. The natural frequency will depend only on the stiffness of the spring and the size of the mass. When you think ofContinue reading “Classical and quantum harmonic oscillators”

In the present note I want to explore the decomposition of an arbitrary Lorentz transformation in the form , where and are orthogonal Lorentz matrices and is a simple Lorentz matrix (to be defined below). We will use throughout a metric tensor of the form .  Any matrix that preserves the quadratic form is calledContinue reading “Decomposition of Lorentz transformations using orthogonal matrices”

An apparent paradox in Einstein’s Special Theory of Relativity, known as a Thomas precession rotation in atomic physics, has been verified experimentally in a number of ways. However, somewhat surprisingly, it has not yet been demonstrated algebraically in a straightforward manner using Lorentz-matrix-algebra. Authors in the past have resorted instead to computer verifications, or toContinue reading “Proving the relativistic rotation paradox”

B-splines and collocation techniques have been applied to the solution of Schrödinger’s equation in quantum mechanics since the early 1970s, but one aspect that is noticeably missing from this literature is the use of Gaussian points (i.e., the zeros of Legendre polynomials) as the collocation points, which can significantly reduce approximation errors. Authors in theContinue reading “Solving Schrödinger’s equation by B-spline collocation”

A Lie group is a group which is also a smooth differentiable manifold. Every Lie group has an associated tangent space called a Lie algebra. As a vector space, the Lie algebra is often easier to study than the associated Lie group and can reveal most of what we need to know about the group.Continue reading “Overview of the Lie theory of rotations”

Certain arithmetical functions, known as Dirichlet characters mod , are used extensively in analytic number theory. Given an arbitrary group , a character of is generally a complex-valued function with domain such that has the multiplicative property for all , and such that for some . Dirichlet characters mod are certain characters defined for aContinue reading “Dirichlet character tables up to mod 11”

In a previous note I studied the mathematical setup of Noether’s Theorem and its proof. I briefly illustrated the mathematical machinery by considering invariance under translations in time, giving the law of conservation of energy, and invariance under translations in space, giving the law of conservation of linear momentum. I briefly mentioned that invariance underContinue reading “Invariance under rotations in space and conservation of angular momentum”

In the present post I want to record some notes I made on the mathematical nuances involved in a proof of Noether’s theorem and the mathematical relevance of the theorem to some simple conservation laws in classical physics, namely, the conservation of energy and the conservation of linear momentum. Noether’s Theorem has important applications inContinue reading “Proving Noether’s theorem”

A geodesic can be defined as an extremal path between two points on a manifold in the sense that it minimises or maximises some criterion of interest (e.g., minimises distance travelled, maximises proper time, etc). Such a path will satisfy some geodesic equations equivalent to the Euler-Lagrange equations of the calculus of variations. A geodesicContinue reading “Alternative approaches to formulating geodesic equations on manifolds”