Uncertainty Restrictions on Orbital Angular Momentum

The general uncertainty relation of quantum mechanics forbids the simultaneous and completely accurate measurements of two observables whose operators do not commute. As will be shown here, the one-dimensional orbital angular momentum operators do not mutually commute. It follows then that, for example, one cannot suggest a system in which there is non-zero orbital angular momentum in only one dimension of orientation, as the knowledge of the orbital angular momentum in said dimension would imply that it is impossible to know that there is zero orbital angular momentum in the other two dimensions. Certain restrictions can thus be derived from the general uncertainty relation for orbital angular momentum.

Quantum Harmonic Oscillator Wave Function

The wave function of a quantum harmonic oscillator varies depending on the energy level of the particle being described. Using the number operator, the wave function of a ground state harmonic oscillator can be found. Repetitively applying the raising operator to the ground state wave function then allows the derivation of the general formula describing wave functions of higher energy levels.

Equivalency of Hermite Polynomial Definitions

Hermite polynomials are a set of polynomials which arise frequently in probability calculations and in quantum physics when finding the wave functions of harmonic oscillators. Often, Hermite polynomials are defined by two seemingly different mathematical definitions. Here, it is proved that these definitions in fact yield equivalent results.

Gaussian Integral

The Gaussian integral, otherwise known as the Euler-Poisson or probability integral, is the definite integral of the Gaussian function, $e^{x^2}$, over $(-\infty,\infty)$. This particular definite integral arises often when performing statistical calculations and when normalizing quantum mechanic wave functions. Here, the value of the Gaussian integral is derived through double integration in polar coordinates, namely shell integration.

Quantum Harmonic Oscillator Number & Ladder Operators

A quantum harmonic oscillator is similar in many ways to a classical one; it is simply a particle that undergoes repetitive motion, bound by a potential with an equilibrium point. The major difference between quantum and classical harmonic oscillators, however, is that in quantum systems, particles can occupy only certain discrete energy levels. Here, the number operator and ladder operators — operators concerned with energy levels — of quantum harmonic oscillators are derived.

Zero-Point Energy

The zero-point energy of a quantum mechanical system is the energy of the system in its ground state (when the system has the lowest possible energy). Take for example, a harmonically oscillating particle in a potential well. While classically, it may be possible to suppose that the particle could have zero energy if its kinetic energy and the energy potential at its position are both zero, quantum mechanically speaking, such a system is impossible. The Heisenberg uncertainty relation forbids such a scenario where both the particle's position and momentum are known. Here, the Heisenberg uncertainty relation and the Schrodinger equation are used to derive the zero-point energy of a quantum harmonic oscillator.

Rectangular Potential Barrier

Rectangular potential barriers, also called square potential barriers, are formed by energy potentials which create wall-like barricades for particles. Essentially, a potential barrier is a potential step except the energy potential returns to zero at some finite positive $x$-position, $a$, and remains zero beyond that point. Here, we'll derive the wave function of a particle facing a rectangular potential barrier, then find the transmission and reflection coefficients of the particle upon encountering the barrier.

Hyperbolic Functions

Hyperbolic functions are related to the unit hyperbola, given by $x^2 - y^2 = 1$, analogous to the way trigonometric functions are related to the unit circle. Both trigonometric and hyperbolic functions can be used to parameterize their respective unit conics. However, while the unit circle's central angle, the argument taken by trigonometric functions, is indeed what one might consider to be an "angle" in the usual meaning of the word, the hyperbolic angle taken by hyperbolic functions is perhaps less intuitively defined.

Potential Step

Potential steps are created by energy potentials which form step-like barricades for particles. Before the potential step, the energy potential is uniformly zero, but at the step, the energy potential rises instantaneously to a finite value and remains constant at that value for all positions beyond the step. Here, we'll derive the wave function of a particle facing a potential step, then find the transmission and reflection coefficients of the particle upon encountering the step.

Probability Current

The probability current, also known as the probability flux, of a wave function at a certain point describes the rate of flow at which probability passes through that point, analogous to the way electrical current describes the rate of flow at which electrical charge passes through a point in a medium. Probability currents are used, for example, when calculating reflection and transmission coefficients for particles encountering potential steps or potential barriers.

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?