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The simplest approach is to focus on the description in terms of plane matter waves for a free particle, that is a wave function described by =, where is a position in real space, is the wave vector in units of inverse meters, ω is the angular frequency with units of inverse time and is time.
The term is also used, even more specifically, to mean a "monochromatic" or sinusoidal plane wave: a travelling plane wave whose profile () is a sinusoidal function. That is, (,) = (() +) The parameter , which may be a scalar or a vector, is called the amplitude of the wave; the scalar coefficient is its "spatial frequency"; and the scalar is its "phase shift".
Then the group velocity of the plane wave is defined as = = =, which agrees with the formula for the classical velocity of the particle. The group velocity is the (approximate) speed at which the whole wave packet propagates, while the phase velocity is the speed at which the individual peaks in the wave packet move. [ 5 ]
A plane wave is an important mathematical idealization where the disturbance is identical along any (infinite) plane normal to a specific direction of travel. Mathematically, the simplest wave is a sinusoidal plane wave in which at any point the field experiences simple harmonic motion at one frequency.
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, who discovered the theorem in 1929. [1] Mathematically, they are written [2]
A superposition of 1D plane waves (blue) each traveling at a different phase velocity (traced by blue dots) results in a Gaussian wave packet (red) that propagates at the group velocity (traced by the red line). The group velocity of a collection of waves is defined as =.
In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves: = = (+) (^ ^), where . i is the imaginary unit,; k is a wave vector of length k,
Comparing our exact solution with the usual monochromatic electromagnetic plane wave as treated in special relativity (i.e., as a wave in flat spacetime, neglecting the gravitational effects of the energy of the electromagnetic field), one sees that the striking new feature in general relativity is the expansion and collapse cycles experienced ...