Bound state
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] 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.
[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.
Define a one-parameter group of unitary operators
[dubious – discuss][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[10] such that In general, a quantum state is a bound state if and only if it is finitely normalizable for all times
[11] Furthermore, a bound state lies within the pure point part of the spectrum of
[12] More informally, "boundedness" results foremost from the choice of domain of definition and characteristics of the state rather than the observable.
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.
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.
, taking limit of x going to infinity on both sides, the wavefunctions vanish and gives
which proves that the energy eigenfunction of a 1D bound state is unique.
Thus every 1D bound state can be represented by completely real eigenfunctions.
Note that real function representation of wavefunctions from this proof applies for all non-degenerate states in general.
bound wavefunction ordered according to increasing energy has exactly
Due to the form of Schrödinger's time independent equations, it is not possible for a physical wavefunction to have
[16] 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 = ℏ/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,
Note, however, that, if the Higgs interaction did not break electroweak symmetry at the electroweak scale, then the SU(2) weak interaction would become confining.
