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A bound state is a composite of two or more fundamental building blocks, such as particles, atoms, or bodies, that behaves as a single object and in which energy is required to split them. In
quantum physics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, a bound state is a
quantum state In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system ...
of a
particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
subject to a
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
such that the particle has a tendency to remain localized in one or more regions of space. The potential may be external or it may be the result of the presence of another particle; in the latter case, one can equivalently define a bound state as a state representing two or more particles whose interaction energy exceeds the total energy of each separate particle. One consequence is that, given a potential vanishing at infinity, negative-energy states must be bound. The energy spectrum of the set of bound states are most commonly discrete, unlike scattering states of free particles, which have a continuous spectrum. Although not bound states in the strict sense, metastable states with a net positive interaction energy, but long decay time, are often considered unstable bound states as well and are called "quasi-bound states". Examples include radionuclides and Rydberg atoms. In relativistic
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, a stable bound state of particles with masses \_^n corresponds to a pole in the
S-matrix In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
with a center-of-mass energy less than \textstyle\sum_k m_k. An
unstable In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability ...
bound state shows up as a pole with a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
center-of-mass energy.


Examples

*A
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
and an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
can move separately; when they do, 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 – is formed. Only the lowest-energy bound state, the ground state, is stable. Other
excited state In quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Add ...
s are unstable and will decay into stable (but not other unstable) bound states with less energy by emitting a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
. *A
positronium Positronium (Ps) is a system consisting of an electron and its antimatter, anti-particle, a positron, bound together into an exotic atom, specifically an onium. Unlike hydrogen, the system has no protons. The system is unstable: the two part ...
"atom" is an unstable bound state of an
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
and a
positron The positron or antielectron is the particle with an electric charge of +1''elementary charge, e'', a Spin (physics), spin of 1/2 (the same as the electron), and the same Electron rest mass, mass as an electron. It is the antiparticle (antimatt ...
. It decays into
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
s. *Any state in the
quantum harmonic oscillator The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, ...
is bound, but has positive energy. Note that \lim_ = \infty , so the below does not apply. *A nucleus is a bound state of
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s and
neutron The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s (
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number. Until the 1960s, nucleons were thought to be ele ...
s). *The
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
itself is a bound state of three
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...
s (two up and one down; one red, one
green Green is the color between cyan and yellow on the visible spectrum. It is evoked by light which has a dominant wavelength of roughly 495570 nm. In subtractive color systems, used in painting and color printing, it is created by a com ...
and one
blue Blue is one of the three primary colours in the RYB color model, RYB colour model (traditional colour theory), as well as in the RGB color model, RGB (additive) colour model. It lies between Violet (color), violet and cyan on the optical spe ...
). However, unlike the case of the hydrogen atom, the individual quarks can never be isolated. See confinement. *The Hubbard and Jaynes–Cummings–Hubbard (JCH) models support similar bound states. In the Hubbard model, two repulsive bosonic
atoms Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
can form a bound pair in an optical lattice. The JCH Hamiltonian also supports two- polariton bound states when the photon-atom interaction is sufficiently strong.


Definition

Let -finite measure space (X, \mathcal A, \mu) be a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
associated with separable
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
H. Define a one-parameter group of unitary operators (U_t)_ , a density operator \rho = \rho(t_0) and an
observable In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum ...
T on H. Let \mu(T,\rho) be the induced probability distribution of T with respect to \rho. Then the evolution :\rho(t_0)\mapsto _t(\rho)t_0) = \rho(t_0 +t) is bound with respect to T if :\lim_ = 0 , where \mathbb_ = \lbrace x \in \mathbb \mid x > R \rbrace . A quantum particle is in a bound state if at no point in time it is found “too far away" from any finite region R\subset X. Using a
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
representation, for example, this means :\begin 0 &= \lim_ \\ &= \lim_, \end such that :\int_X < \infty. In general, a quantum state is a bound state ''if and only if'' it is finitely normalizable for all times t\in\mathbb and remains spatially localized. Furthermore, a bound state lies within the pure point part of the spectrum of T ''if and only if'' it is an
eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
of T. More informally, "boundedness" results foremost from the choice of
domain of definition In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain of ...
and characteristics of the state rather than the observable.See
Expectation value (quantum mechanics) In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as ...
for an example.
For a concrete example: let H := L^2(\mathbb) and let T be the position operator. Given compactly supported \rho = \rho(0) \in H and 1,1\subseteq \mathrm(\rho). *If the state evolution of \rho "moves this wave package to the right", e.g., if -1,t+1\in \mathrm(\rho(t)) for all t \geq 0, then \rho is not bound state with respect to position. *If \rho does not change in time, i.e., \rho(t) = \rho for all t \geq 0, then \rho is bound with respect to position. *More generally: If the state evolution of \rho "just moves \rho inside a bounded domain", then \rho is bound with respect to position.


Properties

As finitely normalizable states must lie within the pure point part of the spectrum, bound states must lie within the pure point part. However, as Neumann and Wigner pointed out, it is possible for the energy of a bound state to be located in the continuous part of the spectrum. This phenomenon is referred to as bound state in the continuum.


Position-bound states

Consider the one-particle Schrödinger equation. If a state has energy E < \max, then the wavefunction satisfies, for some X > 0 :\frac=\frac(V(x)-E) > 0\textx > X so that is exponentially suppressed at large . This behaviour is well-studied for smoothly varying potentials in the WKB approximation for wavefunction, where an oscillatory behaviour is observed if the right hand side of the equation is negative and growing/decaying behaviour if it is positive. Hence, negative energy-states are bound if V(x) vanishes at infinity.


Non-degeneracy in one-dimensional bound states

One-dimensional bound states can be shown to be non-degenerate in energy for well-behaved wavefunctions that decay to zero at infinities. This need not hold true for wavefunctions in higher dimensions. Due to the property of non-degenerate states, one-dimensional bound states can always be expressed as real wavefunctions.


Node theorem

Node theorem states that n\text bound wavefunction ordered according to increasing energy has exactly n-1 nodes, i.e., points x=a where \psi(a)=0 \neq \psi'(a). Due to the form of Schrödinger's time independent equations, it is not possible for a physical wavefunction to have \psi(a) = 0 = \psi'(a) since it corresponds to \psi(x)=0 solution.


Requirements

A
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
with mass mediating a weakly coupled interaction produces an Yukawa-like interaction potential, :V(r) = \pm\frac e^, where \alpha_\chi=g^2/4\pi, is the gauge coupling constant, and is the reduced Compton wavelength. A
scalar boson A scalar boson is a boson whose spin equals zero. A ''boson'' is a particle whose wave function is symmetric under particle exchange and therefore follows Bose–Einstein statistics. The spin–statistics theorem implies that all bosons have a ...
produces a universally attractive potential, whereas a vector attracts particles to antiparticles but repels like pairs. For two particles of mass and , the Bohr radius of the system becomes :a_0=\frac and yields the dimensionless number :D=\frac = \alpha_\chi\frac = \alpha_\chi\frac. In order for the first bound state to exist at all, D\gtrsim 0.8. Because the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
is massless, is infinite for
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
. For the
weak interaction In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
, the Z boson's mass is , which prevents the formation of bound states between most particles, as it is the
proton A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
's mass and the
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
's mass. Note, however, that, if the Higgs interaction did not break electroweak symmetry at the electroweak scale, then the SU(2)
weak interaction In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
would become confining.


See also

* Bethe–Salpeter equation * Bound state in the continuum * Composite field * Cooper pair * Exciton * Resonance (particle physics) * Levinson's theorem


Remarks


References

Quantum field theory Quantum states


Further reading

* {{cite book , last1=Blanchard , first1=Philippe , last2=Brüning , first2=Edward , title=Mathematical Methods in Physics: Distributions, Hilbert Space Operators, Variational Methods, and Applications in Quantum Physics , date=2015 , publisher=Springer International Publishing , location=Switzerland , isbn=978-3-319-14044-5 , page=431 , edition=2nd , language=English , chapter=Some Applications of the Spectral Representation