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The square wave in mathematics has many definitions, which are equivalent except at the discontinuities: It can be defined as simply the sign function of a sinusoid: = () = () = () = (), which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities.
In this case, the wave functions are square integrable. One can initially take the function space as the space of square integrable functions, usually denoted L 2. The displayed functions are solutions to the Schrödinger equation. Obviously, not every function in L 2 satisfies the Schrödinger equation for the hydrogen atom.
Consequently, the wave function also became a four-component function, governed by the Dirac equation that, in free space, read (+ (= )) =. This has again the form of the Schrödinger equation, with the time derivative of the wave function being given by a Hamiltonian operator acting upon the wave function.
The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle. In optics, the Helmholtz equation is the wave equation for the electric field. [1] The equation is named after Hermann von Helmholtz, who studied it in 1860. [2]
Furthermore, the square of the amplitude (determining the probability density) is inversely proportional to p(x), reflecting the length of time the classical particle spends near x. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an Airy function .
Functional approximation of square wave using 5 harmonics Functional approximation of square wave using 25 harmonics Functional approximation of square wave using 125 harmonics. The Gibbs phenomenon is a behavior of the Fourier series of a function with a jump discontinuity and is described as the following:
A modulated wave resulting from adding two sine waves of identical amplitude and nearly identical wavelength and frequency. A common situation resulting in an envelope function in both space x and time t is the superposition of two waves of almost the same wavelength and frequency: [2]
A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum).