Bound state

In quantum physics, a bound state is a special quantum state of a particle subject to a potential such that the particle has a tendency to remain localised 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. In general, the energy spectrum of the set of bound states is discrete, unlike 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."^[1] Examples include certain radionuclides and electrets.

In relativistic quantum field theory, a stable bound state of $n$ particles with masses $\{m_{k}\}_{k=1}^{n}$ corresponds to a pole in the S-matrix with a center-of-mass energy less than $\sum _{k}{m_{k}}$ . An unstable bound state shows up as a pole with a complex center-of-mass energy.

Examples

An overview of the various families of elementary and composite particles, and the theories describing their interactions

A proton and an electron 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 states are unstable and will decay into stable (but not other unstable) bound states with less energy by emitting a photon.
A positronium "atom" is an unstable bound state of an electron and a positron. It decays into photons.
Any state in the Quantum harmonic oscillator is bound, but has positive energy. Note that $\lim _{x\to \pm \infty }{V_{\text{QHO}}(x)}=\infty$ ,so the below does not apply.
A nucleus is a bound state of protons and neutrons (nucleons).
The proton itself is a bound state of three quarks (two up and one down; one red, one green and one blue). 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 can form a bound pair in an optical lattice.^[2]^[3]^[4] The JCH Hamiltonian also supports two-polariton bound states when the photon-atom interaction is sufficiently strong.^[5]

Definition

Let $H$ be a complex separable Hilbert space, $U=\lbrace U(t)\mid t\in {\mathbb {R}}\rbrace$ be a one-parameter group of unitary operators on $H$ and $\rho =\rho (t_{0})$ be a statistical operator on $H$ . Let $A$ be an observable on $H$ and $\mu (A,\rho )$ be the induced probability distribution of $A$ with respect to $ρ$ on the Borel σ-algebra of $\mathbb {R}$ . Then the evolution of $ρ$ induced by $U$ is bound with respect to $A$ if $\lim _{R\rightarrow \infty }{\sup _{t\geq t_{0}}{\mu (A,\rho (t))(\mathbb {R} _{>R})}}=0$ , where ${\mathbb {R}}_{{>R}}=\lbrace x\in {\mathbb {R}}\mid x>R\rbrace$ .

More informally, a bound state is contained within a bounded portion of the spectrum of $A$ . For a concrete example: let $H=L^{2}({\mathbb {R}})$ and let $A$ be position. Given compactly-supported $\rho =\rho (0)\in H$ and $[-1,1]\subseteq {\mathrm {Supp}}(\rho )$ .

If the state evolution of $ρ$ "moves this wave package constantly to the right", e.g. if $[t-1,t+1]\in {\mathrm {Supp}}(\rho (t))$ for all $t\geq 0$ , then $ρ$ 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 $ρ$ "just moves $ρ$ inside a bounded domain", then $ρ$ is bound with respect to position.

Properties

Let $A$ have measure-space codomain $(X;\mu )$ . A quantum particle is in a bound state if it is never found “too far away from any finite region $R\subseteq X$ ,” i.e. using a wavefunction representation,

${\begin{aligned}0&=\lim _{R\to \infty }{\mathbb {P} ({\text{particle measured inside }}X\setminus R)}\\&=\lim _{R\to \infty }{\int _{X\setminus R}|\psi (x)|^{2}\,d\mu (x)}\end{aligned}}$

Consequently, $\int _{X}{|\psi (x)|^{2}\,d\mu (x)}$ is finite. In other words, a state is a bound state if and only if it is finitely normalizable.

As finitely normalizable states must lie within the discrete part of the spectrum, bound states must lie within the discrete part. However, as Neumann and Wigner pointed out, a bound state can have its energy located in the continuum spectrum.^[6] In that case, bound states still are part of the discrete portion of the spectrum, but appear as Dirac masses in the spectral measure.

Position-bound states

Consider the one-particle Schrödinger. If a state has energy $E<\operatorname {max} {\left(\lim _{x\to \infty }{V(x)},\lim _{x\to -\infty }{V(x)}\right)}$ , then the wavefunction $ψ$ satisfies, for some $X>0$

{\frac {\psi ^{\prime \prime }}{\psi }}={\frac {2m}{\hbar ^{2}}}(V(x)-E)>0{\text{ for }}x>X

so that $ψ$ is exponentially suppressed at large $x$ . Hence, negative energy-states are bound if V vanishes at infinity.

References

↑ Sakurai, Jun (1995). "7.8". In Tuan, San. Modern Quantum Mechanics (Revised ed.). Reading, Mass: Addison-Wesley. pp. 418–9. ISBN 0-201-53929-2. Suppose the barrier were infinitely high...we expect bound states, with energy E>0....They are stationary states with infinite lifetime. In the more realistic case of a finite barrier, the particle can be trapped inside, but it cannot be trapped forever. Such a trapped state has a finite lifetime due to quantum-mechanical tunneling....Let us call such a state quasi-bound state because it would be an honest bound state if the barrier were infinitely high. (Formatting in original.)
↑ K. Winkler; G. Thalhammer; F. Lang; R. Grimm; J. H. Denschlag; A. J. Daley; A. Kantian; H. P. Buchler; P. Zoller (2006). "Repulsively bound atom pairs in an optical lattice". Nature. 441: 853. Bibcode:2006Natur.441..853W. PMID 16778884. arXiv:cond-mat/0605196 . doi:10.1038/nature04918.
↑ Javanainen, Juha; Odong Otim; Sanders, Jerome C. (Apr 2010). "Dimer of two bosons in a one-dimensional optical lattice". Phys. Rev. A. American Physical Society. 81 (4): 043609. Bibcode:2010PhRvA..81d3609J. arXiv:1004.5118 . doi:10.1103/PhysRevA.81.043609.
↑ M. Valiente & D. Petrosyan (2008). "Two-particle states in the Hubbard model". J. Phys. B: At. Mol. Opt. Phys. 41: 161002. Bibcode:2008JPhB...41p1002V. doi:10.1088/0953-4075/41/16/161002.
↑ Max T. C. Wong & C. K. Law (May 2011). "Two-polariton bound states in the Jaynes-Cummings-Hubbard model". Phys. Rev. A. American Physical Society. 83 (5): 055802. Bibcode:2011PhRvA..83e5802W. arXiv:1101.1366 . doi:10.1103/PhysRevA.83.055802.
↑ von Neumann, John; Wigner, Eugene (1929). "Über merkwürdige diskrete Eigenwerte". Physikalische Zeitschrift. 30: 465–467.

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Bound state

Examples

Definition

Properties

Position-bound states

See also

References