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.