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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.
The case = is called the ground state, its energy is called the zero-point energy, and the wave function is a Gaussian. [22] The harmonic oscillator, like the particle in a box, illustrates the generic feature of the Schrödinger equation that the energies of bound eigenstates are discretized. [11]: 352
In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called field operators (or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that ...
Therefore, problems in quantum mechanics analyze the system's wave function. Using mathematical formulations, such as the Schrödinger equation, the time evolution of a known wave function can be deduced. The square of the absolute value of this wave function is directly related to the probability distribution of the particle positions, which ...
For example, a beam of electrons can be diffracted just like a beam of light or a water wave. The concept that matter behaves like a wave was proposed by French physicist Louis de Broglie ( / d ə ˈ b r ɔɪ / ) in 1924, and so matter waves are also known as de Broglie waves .
Defining equation (physical chemistry) List of electromagnetism equations; List of equations in classical mechanics; List of equations in fluid mechanics; List of equations in gravitation; List of equations in nuclear and particle physics; List of equations in wave theory; List of photonics equations; List of relativistic equations
For a quantum particle with a wave function | moving in a one-dimensional potential (), the time-independent Schrödinger equation can be written as + = Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy at most, so that the degree of degeneracy never exceeds two.
The phase velocity is the rate at which the phase of the wave propagates in space. The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.