Continuous spectrum
Encyclopedia
The spectrum
Spectrum (functional analysis)
In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the...

 of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum.

If is a topological vector space and is a linear map, the spectrum of is the set of complex numbers such that is not invertible.
We divide the spectrum depending on why this is not invertible.

If is not injective, we say that is in the point spectrum of . Elements of the point spectrum are called eigenvalues of and non-zero elements of the null space of are known as eigenvectors of . Thus is an eigenvalue of if and only if there is a non-zero vector such that .

If does not have closed range, but the range is dense in , we say that is in the continuous spectrum of . The union of the point spectrum and the continuous spectrum is known as the set of generalized eigenvalues. Thus is a generalized eigenvalue of if and only if there is a sequence of vectors , bounded away from zero, such that .

Finally, if does not have closed range, and its range is not dense in , we say that is in the residual spectrum of .

Quantum mechanical interpretations

The position operator
Position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L2, the Hilbert space of complex-valued and square-integrable ...

 usually has a continuous spectrum, much like the momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

 operator in an infinite space. But the momentum in a compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

, the angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

, and the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

 of various physical systems, specially bound state
Bound state
In physics, a bound state describes a system where a particle is subject to a potential such that the particle has a tendency to remain localised in one or more regions of space...

s, tend to have a discrete (quantized) spectrum -- that is where the name quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 comes from. However computing the spectra
Spectroscopy
Spectroscopy is the study of the interaction between matter and radiated energy. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative...

 or cross section
Cross section (physics)
A cross section is the effective area which governs the probability of some scattering or absorption event. Together with particle density and path length, it can be used to predict the total scattering probability via the Beer-Lambert law....

s associated with scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...

 experiments (like for instance high resolution electron energy loss spectroscopy
High resolution electron energy loss spectroscopy
High Resolution Electron Energy Loss Spectroscopy is a tool used in surface science. The inelastic scattering of electrons from surfaces is utilized to study electronic excitations or vibrational modes of the surface or of molecules adsorbed to a surface...

) 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
Scattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...

. The formal scattering theory has a strong overlap with the theory of continuous spectra.

The quantum harmonic oscillator
Quantum harmonic oscillator
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics...

 and the hydrogen atom
Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

 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.

See also

Related physical concepts:
  • Astronomical spectroscopy
    Astronomical spectroscopy
    Astronomical spectroscopy is the technique of spectroscopy used in astronomy. The object of study is the spectrum of electromagnetic radiation, including visible light, which radiates from stars and other celestial objects...

     (examples of continuous spectra)
    • Thermal radiation
      Thermal radiation
      Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

    • Brehmsstrahlung
    • Synchrotron radiation
      Synchrotron radiation
      The electromagnetic radiation emitted when charged particles are accelerated radially is called synchrotron radiation. It is produced in synchrotrons using bending magnets, undulators and/or wigglers...

    • Inverse compton scattering
  • Non-continuous (line) spectra:
    • Emission spectrum
      Emission spectrum
      The emission spectrum of a chemical element or chemical compound is the spectrum of frequencies of electromagnetic radiation emitted by the element's atoms or the compound's molecules when they are returned to a lower energy state....

    • Absorption spectrum


Mathematically rigorous point of view:
  • Decomposition of spectrum (functional analysis)
    Decomposition of spectrum (functional analysis)
    In mathematics, especially functional analysis, the spectrum of an operator generalizes the notion of eigenvalues. Given an operator, it is sometimes useful to break up the spectrum into various parts...

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