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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, a bound state is a composite of two or more building blocks (particle
Elementary particle

In particle physics, an elementary particle or fundamental particle is a wiktionary:particle not known to have substructure; that is, it is not known to be made up of smaller particles....
s or bodies
Physical body

In physics, a physical body is a collection of masses, taken to be one. For example, a cricket ball can be considered an object but the ball also consists of many particles ....
) that behaves as a single object. In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 (where the number of particles is conserved), a bound state is a state in the Hilbert space
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
 that corresponds to two or more particles whose interaction energy
Interaction energy

In physics, interaction energy is the contribution to the total energy that is caused by an interaction between the objects being considered.The interaction energy usually depends on the relative position of the objects....
 is negative, and therefore these particles cannot be separated unless energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 is spent. The energy spectrum
Energy spectrum

An energy spectrum is a distribution of energy among a large assemblage of particles. It is a statistical representation of the wave energy as a function of the wave frequency, and an empirical estimator of the spectral function....
 of a bound state is discrete, unlike the continuous spectrum of isolated particles.






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In physics
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
, a bound state is a composite of two or more building blocks (particle
Elementary particle

In particle physics, an elementary particle or fundamental particle is a wiktionary:particle not known to have substructure; that is, it is not known to be made up of smaller particles....
s or bodies
Physical body

In physics, a physical body is a collection of masses, taken to be one. For example, a cricket ball can be considered an object but the ball also consists of many particles ....
) that behaves as a single object. In quantum mechanics
Quantum mechanics

Quantum mechanics is a set of principles underlying the most fundamental known description of all physical systems at the microscopic scale . Notable amongst these principles are both a dual wave-like and particle-like behavior of matter and radiation, and prediction of probabilities in situations where classical physics predicts certaintie...
 (where the number of particles is conserved), a bound state is a state in the Hilbert space
Hilbert space

The mathematics concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces....
 that corresponds to two or more particles whose interaction energy
Interaction energy

In physics, interaction energy is the contribution to the total energy that is caused by an interaction between the objects being considered.The interaction energy usually depends on the relative position of the objects....
 is negative, and therefore these particles cannot be separated unless energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 is spent. The energy spectrum
Energy spectrum

An energy spectrum is a distribution of energy among a large assemblage of particles. It is a statistical representation of the wave energy as a function of the wave frequency, and an empirical estimator of the spectral function....
 of a bound state is discrete, unlike the continuous spectrum of isolated particles. (Actually, it is possible to have unstable bound states with a positive interaction energy provided that there is an "energy barrier" that has to be tunnelled
Quantum tunnelling

In quantum mechanics, wave-mechanical tunneling is an evanescent wave that occurs because the behaviour of particles is governed by Schroedinger equation....
 through in order to decay. This is true for some radioactive nuclei and for some electret
Electret

Electret is a dielectric material that has a quasi-permanent electric charge or dipole polarization . An electret generates internal and external electric fields, and is the electrostatic equivalent of a permanent magnet....
 materials able to carry electric charge for rather long periods.)

In general, a stable
Stable

File:H?ststall Elfviks g?rd dec 2008.jpgA stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stall s for individual animals....
 bound state is said to exist in a given potential of some dimension if stationary wavefunctions exist (normalized in the range of the potential). The energies of these wavefunctions are negative.

In relativistic
Theory of relativity

File:spacetime curvature.pngThe theory of relativity, or simply relativity, generally refers specifically to two theories of Albert Einstein: special relativity and general relativity....
 quantum field theory
Quantum field theory

Quantum field theory or QFT provides a theoretical framework for constructing quantum mechanics models of systems classically described by field or of Many-body problem....
, a stable
Stable

File:H?ststall Elfviks g?rd dec 2008.jpgA stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stall s for individual animals....
 bound state of n particles with masses m1, ..., mn shows up as a pole
Pole (complex analysis)

In complex analysis, a mathematical discipline, a pole of a meromorphic function is a certain type of mathematical singularity that behaves like the singularity of at ....
 in the S-matrix with a center of mass energy which is less than m1+...+mn. An unstable bound state (see resonance
Resonance

In physics, resonance is the tendency of a system to oscillate at maximum amplitude at certain Frequency, known as the system's resonance frequencies ....
) shows up as a pole with a complex
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 center of mass energy.

Examples

  • A proton
    Proton

    The proton is a subatomic particle with an electric charge of +1 elementary charge. It is found in the nucleus of each atom but is also stable by itself and has a second identity as the hydrogen ion, H+....
     and an electron
    Electron

    The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
     can move separately; the total center-of-mass energy is positive, and such a pair of particles can be described as an ionized atom. Once the electron starts to "orbit" the proton, the energy becomes negative, and a bound state - namely the hydrogen atom
    Hydrogen atom

    A hydrogen atom is an atom of the chemical element hydrogen. The Electric charge neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force....
     - is formed. Only the lowest energy bound state, the ground state is stable. The other excited state
    Excited state

    Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....
    s are unstable and will decay into bound states with less energy by emitting a photon
    Photon

    In physics, the photon is an elementary particle, the quantum of the electromagnetic field and the basic unit of light and all other forms of electromagnetic radiation....
    .
  • A nucleus
    Atomic nucleus

    The nucleus of an atom is the very dense region, consisting of nucleons , at the center of an atom. Although the size of the nucleus varies considerably according to the mass of the atom, the size of the entire atom is comparatively constant....
     is a bound state of proton
    Proton

    The proton is a subatomic particle with an electric charge of +1 elementary charge. It is found in the nucleus of each atom but is also stable by itself and has a second identity as the hydrogen ion, H+....
    s and neutron
    Neutron

    The neutron is a subatomic particle with no net electric charge and a mass slightly larger than that of a proton.Neutrons are usually found in atomic nucleus....
    s (nucleon
    Nucleon

    In physics, a nucleon is a collective name for two baryons: the neutron and the proton. They are constituents of the atomic nucleus and until the 1960s were thought to be elementary particles....
    s).
  • A positronium
    Positronium

    Positronium is a system consisting of an electron and its antimatter, a positron, bound together into an "exotic atom". The orbit of the two particles and the set of energy levels is similar to that of the hydrogen atom ....
     "atom" is an unstable bound state
    Resonance

    In physics, resonance is the tendency of a system to oscillate at maximum amplitude at certain Frequency, known as the system's resonance frequencies ....
     of an electron
    Electron

    The electron is a subatomic particle that carries a negative electric charge. It has elementary particle and is believed to be a point particle....
     and a positron
    Positron

    The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1, a spin of 1/2, and the same mass as an electron....
    . It decays into photon
    Photon

    In physics, the photon is an elementary particle, the quantum of the electromagnetic field and the basic unit of light and all other forms of electromagnetic radiation....
    s.
  • The proton
    Proton

    The proton is a subatomic particle with an electric charge of +1 elementary charge. It is found in the nucleus of each atom but is also stable by itself and has a second identity as the hydrogen ion, H+....
     itself is a bound state of three quark
    Quark

    Quarks are a type of elementary particle and major constituents of matter. They are the only particles in the Standard Model to experience all four fundamental interaction, which are also known as fundamental interactions....
    s (two up
    Up quark

    The up quark is a particle described by the Standard Model theory of physics. It is a first-generation quark with a charge of +elementary charge....
     and one down
    Down quark

    The down quark is a first-generation quark with a charge of - elementary charge. It is the second-lightest of all the six flavour of quarks, the lightest being the up quark....
    ; one red
    Quantum chromodynamics

    Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
    , one green
    Quantum chromodynamics

    Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
     and one blue
    Quantum chromodynamics

    Quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons ....
    ). However, unlike the case of the hydrogen atom, the individual quarks can never be isolated. See confinement
    Colour confinement

    Color confinement, often called just confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed....
    .


In mathematical quantum physics

Let be a complex separable Hilbert space, be a one-parametric group of unitary operators on and be a statistical operator on . Let be an observable on and let be the induced probability distribution of with respect to on the Borel -algebra on . Then the evolution of induced by is said to be bound with respect to if , where .

Example: Let and let be the position observable. Let have compact support and .

  • If the state evolution of "moves this wave package constantly to the right", e.g. if for all , then is not a bound state with respect to the position.


  • If does not change in time, i.e. for all , then is a bound state with respect to position.


  • More generally: If the state evolution of "just moves inside a bounded domain", then is also a bound state with respect to position.


See also


  • Composite field
    Composite field

    In quantum field theory, a composite field is a field defined in terms of other more "elementary" fields. It might describe a composite particle or it might not....
  • Resonance
    Resonance

    In physics, resonance is the tendency of a system to oscillate at maximum amplitude at certain Frequency, known as the system's resonance frequencies ....