Sine waveA sine wave, sinusoidal wave, or sinusoid is a mathematical curve defined in terms of the sine trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in mathematics, as well as in physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is: where: A, amplitude, the peak deviation of the function from zero. f, ordinary frequency, the number of oscillations (cycles) that occur each second of time.
Legendre polynomialsIn mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions.
Associated Legendre polynomialsIn mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation or equivalently where the indices l and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on only if l and m are integers with 0 ≤ m ≤ l, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial.
Specific angular momentumIn celestial mechanics, the specific relative angular momentum (often denoted or ) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative position and relative linear momentum, divided by the mass of the body in question. Specific relative angular momentum plays a pivotal role in the analysis of the two-body problem, as it remains constant for a given orbit under ideal conditions. "Specific" in this context indicates angular momentum per unit mass.
Spherical basisIn pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers.
Wave function collapseIn quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an observation, and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation.
Periodic functionA periodic function or cyclic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic. A function f is said to be periodic if, for some nonzero constant P, it is the case that for all values of x in the domain.
HydrogenHydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, and highly combustible. Hydrogen is the most abundant chemical substance in the universe, constituting roughly 75% of all normal matter. Stars such as the Sun are mainly composed of hydrogen in the plasma state. Most of the hydrogen on Earth exists in molecular forms such as water and organic compounds.
Harmonic analysisHarmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on the real line, or by Fourier series for periodic functions. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis.
AtomAn atom is a particle that consists of a nucleus of protons and neutrons surrounded by a cloud of electrons. The atom is the basic particle of the chemical elements, and the chemical elements are distinguished from each other by the number of protons that are in their atoms. For example, any atom that contains 11 protons is sodium, and any atom that contains 29 protons is copper. The number of neutrons defines the isotope of the element. Atoms are extremely small, typically around 100 picometers across.
WaveIn physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a traveling wave; by contrast, a pair of superimposed periodic waves traveling in opposite directions makes a standing wave.
Schrödinger equationThe Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. The equation is named after Erwin Schrödinger, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics.
