(mid) and discrete spectrum lines of a
fluorescent lamp (bottom) In the
physical sciences, the spectrum of a
physical quantity (such as
energy) may be called
continuous if it is non-zero over the whole spectrum domain (such as
frequency or
wavelength) or
discrete if it attains non-zero values only in a
discrete set over the
independent variable, with
band gaps between pairs of
spectral bands or
spectral lines. The classical example of a continuous spectrum, from which the name is derived, is the part of the
spectrum of the light emitted by
excited atoms of
hydrogen that is due to free
electrons becoming bound to a hydrogen ion and emitting photons, which are smoothly spread over a wide range of wavelengths, in contrast to the
discrete lines due to electrons falling from some bound
quantum state to a state of lower energy. As in that classical example, the term is most often used when the range of values of a physical quantity may have both a continuous and a discrete part, whether at the same time or in different situations. In
quantum systems, continuous spectra (as in
bremsstrahlung and
thermal radiation) are usually associated with free particles, such as atoms in a gas, electrons in an
electron beam, or
conduction band electrons in a
metal. In particular, the
position and
momentum of a free particle has a continuous spectrum, but when the particle is confined to a limited space its spectrum becomes discrete. Often a continuous spectrum may be just a convenient model for a discrete spectrum whose values are too close to be distinguished, as in the
phonons in a
crystal. The continuous and discrete spectra of physical systems can be modeled in
functional analysis as different parts in the
decomposition of the spectrum of a
linear operator acting on a
function space, such as the
Hamiltonian operator. The classical example of a discrete spectrum (for which the term was first used) is the characteristic set of discrete
spectral lines seen in the
emission spectrum and
absorption spectrum of isolated
atoms of a
chemical element, which only absorb and emit light at particular
wavelengths. The technique of
spectroscopy is based on this phenomenon. Discrete spectra are seen in many other phenomena, such as vibrating
strings,
microwaves in a
metal cavity,
sound waves in a
pulsating star, and
resonances in high-energy
particle physics. The general phenomenon of discrete spectra in physical systems can be mathematically modeled with tools of
functional analysis, specifically by the
decomposition of the spectrum of a
linear operator acting on a
functional space.
In classical mechanics In
classical mechanics, discrete spectra are often associated to
waves and
oscillations in a bounded object or domain. Mathematically they can be identified with the
eigenvalues of
differential operators that describe the evolution of some continuous variable (such as
strain or
pressure) as a function of time and/or space. Discrete spectra are also produced by some
non-linear oscillators where the relevant quantity has a non-
sinusoidal
waveform. Notable examples are the sound produced by the
vocal cords of mammals. and the
stridulation organs of
crickets, whose spectrum shows a series of strong lines at frequencies that are integer multiples (
harmonics) of the
oscillation frequency. A related phenomenon is the appearance of strong harmonics when a sinusoidal signal (which has the ultimate "discrete spectrum", consisting of a single spectral line) is modified by a non-linear
filter; for example, when a
pure tone is played through an overloaded
amplifier, or when an intense
monochromatic laser beam goes through a
non-linear medium. In the latter case, if two arbitrary sinusoidal signals with frequencies
f and
g are processed together, the output signal will generally have spectral lines at frequencies , where
m and
n are any integers.
In quantum mechanics In
quantum mechanics, the discrete spectrum of an
observable refers to the
pure point spectrum of
eigenvalues of the
operator used to model that observable. Discrete spectra are usually associated with systems that are
bound in some sense (mathematically, confined to a
compact space). The
position and
momentum operators have continuous spectra in an infinite domain, but a discrete (quantized) spectrum in a compact domain and the same properties of spectra hold for
angular momentum,
Hamiltonians and other operators of quantum systems. The
quantum harmonic oscillator and the
hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the hydrogen atom the spectrum has both a continuous and a discrete part, the continuous part representing the
ionization. Hydrogen spectrum.svg|The discrete part of the emission spectrum of hydrogen Solar Spectrum.png|Spectrum of sunlight above the atmosphere (yellow) and at sea level (red), revealing an absorption spectrum with a discrete part (such as the line due to ) and a continuous part (such as the bands labeled ) Deuterium lamp 1.png|Spectrum of light emitted by a
deuterium lamp, showing a discrete part (tall sharp peaks) and a continuous part (smoothly varying between the peaks). The smaller peaks and valleys may be due to measurement errors rather than discrete spectral lines. == See also ==