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.