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In physics, continuous spectrum refers to a range of values which may be graphed to fill a range with closely-spaced or overlapping intervals. The term is derived from the use of the word spectrum to describe the 'ghost-like' rainbow which appears when white light is shone through a clear scattering medium, such as water droplets or a prism. [1] Image File history File links Broom_icon. ...
Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the fundamental laws of the universe and their precise formulation in a mathematical framework. ...
The idea of a continuous spectrum can be viewed as "a continuous set of eigenvalues" — an apparent contradiction in terms. Eigenvectors occur discretely. The mathematics of continuous spectra belongs to spectral theory, a branch of functional analysis. In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. ...
Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...
Physical interpretation
A 'continuous spectrum' of values would require an infinite set of generators or exciters, with infinitely and continuously-varied energy states. However, we can discuss a continuous spectrum of possible values without any logical problem.
The continuous spectrum is a theoretical circumstance which can only be described in a similarly abstract way, and never humanly verified to exist in reality, or enumerated even on paper. But we can verify that some phenomenon has an 'in principle' continuous spectrum of possible values, by first forming a coherent theoretical basis for the phenomenon to possess a continuous spectrum of possible states, and then testing selected values to whatever precision we are able. If we find no exceptions under a rigorous examination, we might decide to label the phenomenon as consistent with the idea of a 'continuous spectrum'.
But this concept is something which, even if it exists, can never be fully verified to do so, due to the infinite nature of the task in question. We can only ever partially perceive any infinity that we do not embody, and how can we enumerate any infinity which we do embody within finite time?
Continuous spectra, in short, are illusionary, or imaginary. The name is chosen well.
It is simple to prove a spectrum is 'discontinuous', and so a spectrum which has been examined rigorously and not found to be discontinuous, might be described as 'continuous' for convenience. However if applying commonly-accepted principles behind the scientific method, we should never assume that a real phenomenon is actually continuous, as we can never fully verify it to be so. Scientific method is a body of techniques for investigating phenomena and acquiring new knowledge, as well as for correcting and integrating previous knowledge. ...
An approximation made of a large (not necessarily infinite) set of discrete energy states, perceived through an aggregating detector may have a convincing appearance of being a 'continuous spectrum' when averaged over sufficient time, and so may be referred to by scientists as this.
A spectrum may be described as being 'continuous in the region' X to Y, where X and Y are values between which the spectrum is seen to be continuous, and outside which, spectral values are either untested, or are known to be absent.
Quantum mechanical interpretations with respect to Hamiltonians of scattering values Experimentally, computing the spectra or cross sections associated with scattering experiments (like for instance high resolution electron energy loss spectroscopy) usually requires the computation of the non quantized or continuous spectrum (density of states) of the Hamiltonian. This is particularly true when broad resonances or strong background scattering is observed. The branch of quantum mechanics concerned with these scattering events is referred to as scattering theory. Extremely high resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of matter and its properties by investigating light, sound, or particles that are emitted, absorbed or scattered by the matter under investigation. ...
In nuclear and particle physics, the concept of a cross section is used to express the likelihood of interaction between particles. ...
In particle physics, scattering is a class of phenomena by which particles are deflected by collisions with other particles. ...
High resolution electron energy loss spectroscopy (HREELS) is a kind of surface vibrational spectroscopy. ...
The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...
This article is about resonance in physics. ...
Scattering theory is a branch of physics and especially of quantum mechanics whose aim is the study of scattering events. ...
The position operator usually has a continuous spectrum, much like the momentum operator in an infinite space. But the momentum in a compact space, the angular momentum, and the Hamiltonian of various physical systems, specially bound states, tend to have a discrete (quantized) spectrum -- that is where the name quantum mechanics comes from. However computing the spectra or cross sections associated with scattering experiments (like for instance high resolution electron energy loss spectroscopy) usually requires the computation of the non quantized or continuous spectrum (density of states) of the Hamiltonian. This is particularly true when broad resonances or strong background scattering is observed. The branch of quantum mechanics concerned with these scattering events is referred to as scattering theory. The formal scattering theory has a strong overlap with the theory of continuous spectra. In mathematical formulations of quantum mechanics, an operator is a linear transformation from a Hilbert space to itself. ...
In classical mechanics, momentum (pl. ...
In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
This gyroscope remains upright while spinning due to its angular momentum. ...
The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...
In physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. ...
Fig. ...
Extremely high resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of matter and its properties by investigating light, sound, or particles that are emitted, absorbed or scattered by the matter under investigation. ...
In nuclear and particle physics, the concept of a cross section is used to express the likelihood of interaction between particles. ...
In particle physics, scattering is a class of phenomena by which particles are deflected by collisions with other particles. ...
High resolution electron energy loss spectroscopy (HREELS) is a kind of surface vibrational spectroscopy. ...
This article is about resonance in physics. ...
Scattering theory is a branch of physics and especially of quantum mechanics whose aim is the study of scattering events. ...
The quantum harmonic oscillator or the hydrogen atom are examples of physical systems in which the Hamiltonian has a discrete spectrum. In the case of the hydrogen atom, it has both continuous as well as discrete part of the spectrum; the continuous part represents the ionized atom. The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...
Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. ...
See also Related physical concepts: Mathematically rigorous point of view: High resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines). ...
A materials emission spectrum is the amount of electromagnetic radiation of each frequency it emits when it is heated (or more generally when it is excited). ...
A materials absorption spectrum shows the fraction of incident electromagnetic radiation absorbed by the material over a range of frequencies. ...
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