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The wave function of an initially very localized free particle. In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively). Wave functions are complex ...
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics.
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 only difference between a homogeneous and an inhomogeneous (partial) differential equation is that in the homogeneous form we only allow 0 to stand on the right side ((,) =), while the inhomogeneous one is much more general, as in (,) could be any function as long as it's continuous and can be continuously differentiated twice.
When the equation is applied to waves, k is known as the wave number. 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]
Green's functions may be categorized, by the type of boundary conditions satisfied, by a Green's function number. Also, Green's functions in general are distributions, not necessarily functions of a real variable. Green's functions are also useful tools in solving wave equations and diffusion equations.
Consider the Dirichlet problem for the wave equation describing a string attached between walls with one end attached permanently and the other moving with the constant velocity i.e. the d'Alembert equation on the triangular region of the Cartesian product of the space and the time:
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form: