Infinite Square & Box Potential Wells

In quantum mechanics, energy wells are are formed by energy potentials that hinder a particle's movement within a certain area. An example of a particle stuck in a potential well is an electron caged in a negatively charged box. The only way for the electron to escape is if it has enough kinetic energy to trade off for potential energy as it approaches the negatively charged electric field. Infinite square potential wells are one-dimensional energy wells which restrict particles inside within its infinitely high potential walls. Inside the well, the energy potential is uniformly zero, but to leave the square well, an infinite amount of energy is required. Infinite box wells are simply three-dimensional rectangular cages formed by square wells in each of the three dimensions.

Uncertainty Principle

The uncertainty principle, simply put, states that certain measurements cannot be simultaneously taken of a system such that there is absolute certainty in every measurement. Perhaps the most well known of all uncertainty relations is Heisenberg's between momentum and position measurements. To derive the relations that dictate the general uncertainty relation and the Heisenberg uncertainty relation, we'll need to do some playing around using the matrix interpretation of quantum mechanics.

Hamiltonian & Schrodinger Equation

The Hamiltonian is an operator which gives the total energy of a system by adding together the system's kinetic energy and potential energy. Schrodinger's time-independent equation is a simple mathematical equivocation of this relation between Hamiltonians and total energy. Schrodinger's time-independent equation, more difficultly derived, shows how the the total energy of a system can also be found using operations which rely on the time evolution of wave functions.

Momentum Operator

Quantum mechanical wave functions are analogous to electromagnetic radiation waves in many aspects. Much like how waves define the properties of electromagnetic radiation, wave functions can also define a system's properties. To find the momentum of a system, we must use the momentum operator, a Hermitian operator which returns the momentum of the system like so: $\hat{p} \psi= \mathbf{p} \psi$ where $\psi$ is the wave function of the system, $\hat{p}$ is the momentum operator, and $\mathbf{p}$ is the momentum.

Fourier Transform & Normalizing Constants

Fourier transformation is an operation in which a function is transformed between position space and momentum space or between time domain and frequency domain. (Momentum space or $k$-space, is actually defined by wavenumbers which are directly related to momentum. The wavenumber of a wave is the unit angle of oscillation the wave undergoes per unit of distance traveled by the wave.) This post will only focus on the transformations between position and momentum space, but the logic employed here will also apply for the other possible Fourier transformations as well.

Dirac Delta Function & Sinc Representation

The Dirac delta function is a function that comes in useful when deriving the equations for Fourier transforms and evaluating other physics calculations. It is defined as a function, $\delta(x)$, which is equal to $0$ for all $x$ except at $x=0$, and if integrated over all $x$, gives a value of $1$. In other words, the Dirac delta function is an infinitely thin, infinitely tall spike at $x=0$ with an area of $1$.

Euler's Formula

Just to be clear, this is the proof for the formula $e^{ix} = \cos{x}+i\sin{x}$, not the other weird equation with the same name involving polyhedrons. Let's start by considering the expression $\frac{1}{1+x^2}$. If you were asked to integrate this expression with respect to $x$, $tan^{-1}{x}+C$ is probably the first thing that comes to mind, but what if you wanted to be a rebel and integrate it another way?

Parabolic Mirrors & Focal Points

At one point or another in your dealings with high school physics, you've probably been told that concave parabolic mirrors are the only concave mirrors that have focal points when parallel light rays are reflected. Just considering the shape of a parabola, this seems to make a lot of sense, but what about other mirrors? Would quartic functions such as $y=x^4$ or sextic functions like $y=x^6$ also generate mirrors with focal points?