[
]
In function approximation
In condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
an energy eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
for a mobile charge carrier in a crystal can be expressed as a Bloch wave:
:
where ''n'' is the index for the band (for example, conduction or valence band) r is a spatial location, and k is a wavevector. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of the wavefunction ''u''''n'',k describing the behavior of the wavefunction close to the cores of the atoms of the lattice. The envelope is restricted to k-values within a range limited by the Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
of the crystal, and that limits how rapidly it can vary with location r.
In determining the behavior of the carriers using quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the ''envelope approximation'' usually is used in which the Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
is simplified to refer only to the behavior of the envelope, and boundary conditions are applied to the envelope function directly, rather than to the complete wavefunction.[ For example, the wavefunction of a carrier trapped near an impurity is governed by an envelope function ''F'' that governs a superposition of Bloch functions:
:
where the Fourier components of the envelope ''F''(k) are found from the approximate Schrödinger equation.][ In some applications, the periodic part ''u''k is replaced by its value near the band edge, say k=k0, and then:][
:
]
In diffraction patterns
Diffraction patterns from multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by:[
:
where α is the diffraction angle, ''d'' is the slit width, and λ is the wavelength. For multiple slits, the pattern is ][
:
where ''q'' is the number of slits, and ''g'' is the grating constant. The first factor, the single-slit result ''I1'', modulates the more rapidly varying second factor that depends upon the number of slits and their spacing.
]
Estimation
An envelope detector is an electronic circuit that extracts the envelope from a signal.
In digital signal processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
, the envelope may be estimated employing the Hilbert transform or a moving RMS amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of ampl ...
.
See also
*
*Empirical mode decomposition
Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...
*Envelope (mathematics)
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of ...
* Envelope tracking
*Instantaneous phase
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''comple ...
*Modulation
In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the '' carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informat ...
* Mathematics of oscillation
* Peak envelope power
* Spectral envelope
References
{{citizendium, title=Envelope function
Interference
Interferometry
Waves
Radio modulation modes
Radio electronics