Continuous spectrum
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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]
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 continous spectra belongs to spectral theory, a branch of functional analysis.
[edit] Physical interpretation
A 'continuous spectrum' of values would require an infinite set of generators or excitors, with infinitely and continuously-varied energy states. However, we can discuss a continuous spectrum of possible values without any logical problem.
The continuous spectum is an 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 principal' 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 principals 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.
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.
[edit] 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.
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.
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. But given that a real atom can only ever be excited by a stimulus which is itself 'quantum-theory compliant', can any electron ever exist in reality at a continuous set of energy levels? It would depend on the starting state of the universe possessing a continuous spectral character.
[edit] See also
Related physical concepts:
Mathematically rigorous point of view:
