Bound state

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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. [1]

Contents

In quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. [2] 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". [3] Examples include radionuclides and Rydberg atoms. [4]

In relativistic quantum field theory, a stable bound state of n particles with masses corresponds to a pole in the S-matrix with a center-of-mass energy less than . 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 Particle overview.svg
An overview of the various families of elementary and composite particles, and the theories describing their interactions

Definition

Let σ-finite measure space be a probability space associated with separable complex Hilbert space . Define a one-parameter group of unitary operators , a density operator and an observable on . Let be the induced probability distribution of with respect to . Then the evolution

is bound with respect to if

,

where .[ dubious ] [9]

A quantum particle is in a bound state if at no point in time it is found “too far away" from any finite region . Using a wave function representation, for example, this means

such that

In general, a quantum state is a bound state if and only if it is finitely normalizable for all times . [10] Therefore, bound states must lie within the pure point part of the spectrum of . [11]

More informally, the above characteristics are a consequence of the chosen domain of definition of rather than that of . [nb 1] For a concrete example: let and let be the position operator. Given compactly supported and .

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. [12] [13]

Position-bound states

Consider the one-particle Schrödinger equation. If a state has energy , then the wavefunction ψ satisfies, for some

so that ψ is exponentially suppressed at large x. 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. [14] Hence, negative energy-states are bound if V vanishes at infinity.

Non-Degeneracy in One dimensional bound states

1D 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 wavefunction 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-th bound wavefunction ordered according to increasing energy has exactly n-1 nodes, ie. points where . Due to the form of Schrödinger's time independent equations, it is not possible for a physical wavefunction to have since it corresponds to solution. [15]

Requirements

A boson with mass mχ mediating a weakly coupled interaction produces an Yukawa-like interaction potential,

,

where , g is the gauge coupling constant, and ƛi = /mic is the reduced Compton wavelength. A scalar boson produces a universally attractive potential, whereas a vector attracts particles to antiparticles but repels like pairs. For two particles of mass m1 and m2, the Bohr radius of the system becomes

and yields the dimensionless number

.

In order for the first bound state to exist at all, . Because the photon is massless, D is infinite for electromagnetism. For the weak interaction, the Z boson's mass is 91.1876±0.0021 GeV/c2, which prevents the formation of bound states between most particles, as it is 97.2 times the proton's mass and 178,000 times the electron's mass.

Note however that if the Higgs interaction didn't break electroweak symmetry at the electroweak scale, then the SU(2) weak interaction would become confining. [16]

See also

Remarks

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References

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Further reading