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It is a delightful fact that one can get both the fundamental equation of classical mechanics (Newton’s Second Law) and the fundamental equation of quantum mechanics (Schrödinger’s equation) by solving very simple variational problems based on the familiar conservation of mechanical energy equation In the present note I want to briefly set out the relevant calculationsContinue reading “Simple variational setups yielding Newton’s second law and Schrödinger’s equation”

A particle of mass with position vector and velocity (with respect to some specified origin) has a linear momentum vector and angular momentum vector where is the vector product operation. The magnitude of the angular momentum vector is where is the angle between and . The direction of is given by the right-hand rule whenContinue reading “Changing variables in the square of the angular momentum operator”

In three-dimensional rectangular coordinates, the partial differential equation known as Laplace’s equation takes the form This equation is applicable to a wide range of problems in physics but it is often necessary to make a change of variables from rectangular to spherical polar coordinates in order to better match the spherical symmetry of particular contexts.Continue reading “Matrix calculus approach to changing variables in Laplace’s equation”

In mathematics, a singularity is a point at which a mathematical object, e.g., a function, is not defined or behaves badly in some way. Singularities can be isolated, e.g., removable singularities, poles and essential singularities, or nonisolated, e.g., branch cuts. For teaching purposes, I want to delve into some of the mathematical aspects of isolatedContinue reading “On classifying singularities, with a quick look at a Schwarzschild black hole”

For teaching purposes, I was trying to find different ways of proving the familiar result that the complex square root function is discontinuous everywhere on the negative real axis. As I was working on alternative proofs, it became very clear to me how sensitive all the proofs were to the particular definition of the principalContinue reading “Different branch cuts for the principal argument, log and square root functions”

Sir William Rowan Hamilton famously discovered the key rules for quaternion algebra while walking with his wife past a bridge in Dublin in 1843. A plaque now commemorates this event. I needed to use the quaternion rotation operator recently and while digging around the literature on this topic I noticed that a lot of itContinue reading “A note on the quaternion rotation operator”

For the purposes of some work I was doing involving ternary numbers, I became interested in finding a quick and easily programmable method for converting prime reciprocals into ternary representation. By trial and error I found a Euclidean-like division algorithm which works well, as illustrated by the following application to the calculation of the ternaryContinue reading “A division algorithm for converting prime reciprocals into ternary numbers”

In a series of papers from 1905, his `annus mirabilis’, Albert Einstein analysed Brownian motion and derived the following partial differential equation (known as the diffusion equation) from physical principles to describe the process: This equation has as a solution the probability density function (see Einstein, A., 1956, Investigations on the Theory of Brownian Movement,Continue reading “A variant of Einstein’s partial differential equation for Brownian motion”

Chebyshev’s inequality and the Borel-Cantelli lemma are seemingly disparate results from probability theory but they combine beautifully in demonstrating a curious property of Brownian motion: that it has finite quadratic variation even though it has unbounded linear variation. Not only do the proofs of Chebyshev’s inequality and the Borel-Cantelli lemma have some interesting features themselves,Continue reading “Applying Chebyshev’s inequality and the Borel-Cantelli lemma to Brownian motion”

The literature involving fractional Brownian motion has expanded over the past three decades or so. Fractional Brownian motion actually has a long history, having been first introduced in the 1940s by the great Andrey Kolmogorov, and then reintroduced and further developed in a seminal paper by Mandelbrot and Van Ness in 1968 (for an interestingContinue reading “Brownian motion and fractional Brownian motion as self-similar stochastic processes”