The Bloch spectrum is a concept in
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, ...
in the field of theoretical
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
; this concept addresses certain
energy spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
considerations. Let ''H'' be the one-dimensional
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 the ...
operator
:
where ''U
α'' is a
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
of period ''α''. The Bloch spectrum
["An upper bound on the allowed bands of the Bloch spectrum of one-dimensional Schrödinger operators with periodic potentials", N. N. Meiman, ''Journal of Mathematical Physics'', March, 1983, volume 24, issue 3, pp. 539–540 ] of ''H'' is defined as the set of values ''E'' for which all the solutions of (''H'' − ''E'')φ = 0 are bounded on the whole
real axis
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poi ...
. The Bloch spectrum consists of the half-line ''E''
0 < ''E'' from which certain closed intervals
'E''2''j''−1, ''E''2''j''(''j'' = 1, 2, ...) are omitted. These are forbidden bands (or gaps) so the (''E''
2''j''−2, ''E''
2''j''−1) are allowed bands.
References
Quantum mechanics
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