# A day in the life of a Theoretical Physicist

Following up on trends I see continuously on YouTube and other platforms, I thought I might talk a little bit about what a day in the life of a theoretical physicist looks like based on my own experience, given that not many people might know much about this. Firstly, I must point out that I am not technically a “theoretical physicist” yet, since I am still in the process of completing my PhD in the next couple of months. There are a lot of people who prefer not to call themselves “physicists” until they have acquired the title of “Dr.”…

# The essence of Monte Carlo

Monte Carlo methods (MC) are a series of mathematical methods through which we can gain insight into the statistical properties of a system by running repeated independent trials and stacking them up against each other. These methods find plenty of uses in areas like engineering, physics, finance, data science, machine learning, and more. However, MC methods can be very challenging to understand, and lack of understanding often leads to the wrong implementation, which ultimately leads to either erroneous results or erroneous interpretations of those results. …

# The physics of single-slit diffraction

Diffraction is a physical phenomenon in which a wave (could be an acoustic wave, water wave, electromagnetic wave, etc) bends around the edges of some kind of obstacle. When we look at the passage of light through a narrow slit for example, we note how most of the light in the original beam is blocked, and the projected pattern on a back-screen resembles the shape of the aperture through which the light beam travels. However, if the aperture is smaller than the wavelength of the incoming light, an interesting phenomenon arises: the light interferes with itself, forming a diffraction pattern…

# Solving the Korteweg-de Vries equation

The Korteweg-de Vries equation (KdV for short) is a type of nonlinear partial differential equation (PDE) which is commonly used to model the behaviour of waves in shallow water. Despite it being nonlinear, it is a PDE that possesses a special set of analytic solutions known as “travelling wave solutions”. Here, we will look at one of those analytic solutions and how it is derived using coordinate transformations.

We shall begin with the standard KdV equation in 1 spatial dimension

where the solution u(x,t) represents the vertical displacement of water molecules at each position x and time t along a…

# Travelling wave solutions to partial differential equations

In the world of mathematics, partial differential equations (abbreviated as PDEs) are used for modelling natural phenomena, from very basic transport and diffusion processes (e.g. kinetic transport of gas molecules inside a closed system), to more complicated physical processes (such as fluid dynamics around complicated geometries). These equations are often unsolvable analytically, and so we often employ powerful computers and specialized software in order to find approximate solutions. However, there are special cases in which analytic solutions to these PDEs do exist, and these special analytic solutions can tell us a lot about the behaviour of such a physical system…

# Linear regression with polynomials

In my previous two stories “Linear regression with straight lines” and “Linear regression with quadratic equations”, I explained how to find the optimum parameters of specific functions so as to minimize the sum squared error between a fit function y(x) and a data set S(x), using the method of least squares analysis (LSA). Now, I will extend these methods to a polynomial of any degree.

# Mathematical Derivation

Suppose we have a polynomial of order n, meaning that the highest power of x is n as in the equation

since the numbered parameters b0, b1, and so on start at n = 0…

# Linear regression with quadratic equations

Linear regression is the technique by which we mathematically find a “line of best fit” (which is not necessarily a straight line) for a particular set of data. This technique is widely used in science, engineering, business, research, and more; in order to find relationships between different variables and make predictions about their future behaviour. In my previous article “Linear regression with straight lines”, we looked at the mathematics of fitting a straight line to a data set. The limitations of doing this are clear: not every relationship between a set of variables is linear (in fact, most relationships aren’t)…

# The theory of Antenna Radiation

Antennas are everywhere. Whether it be in the form of large towers sitting atop rooftops on buildings, satellite dishes, or even in the form of very tiny printed circuits on your smartphone; antennas play a crucial role in transmitting information across the globe and into outer space without the need for wires or cables spanning ridiculous distances. So how does an antenna work, exactly? To answer this question, we will look at how electromagnetic waves can be produced in a simple model known as the Hertzian dipole antenna.

*Note: this article assumes you are familiar with some vector calculus and…

# Linear regression with straight lines

Linear regression is a widely used mathematical technique in which a certain function (often a straight line or polynomial) is used to fit a set of data or raw measurements, showing a correlation between 2 or more variables. The idea is that we choose a certain function which represents a measure of the error between the “line of best fit” and the actual data, and then we use techniques from Calculus to minimize that error by choosing the most optimum parameters for that line of best fit. …

# Deriving the speed of light from Maxwell’s equations

The speed of light in a vacuum is approximately 3,000,000 m/s (slightly less, in fact, but we will stick to this simpler number for now). There have been numerous experiments that have allowed us to measure this quantity very accurately, but there is a rather elegant way to arrive at the same number by using nothing more than mathematics and a few other less-complicated-to-measure numbers in physics. …

## Oscar Nieves

Oscar is a physicist, educator and STEM enthusiast. He is currently finishing a PhD in Theoretical Physics with a focus on photonics and stochastic dynamics.

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