- Bound state
physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. In quantum mechanics(where the number of particles is conserved), a bound state is a state in the Hilbert spacethat corresponds to two or more particles whose interaction energyis negative, and therefore these particles cannot be separated unless energyis spent. The energy spectrumof 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 through in order to decay. This is true for some radioactive nucleiand for some electretmaterials able to carry electric charge for rather long periods.)
In general, a
stablebound 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.
quantum field theory, a stablebound state of n particles with masses m1, ..., mn shows up as a pole in the S-matrixwith a center of mass energy which is less than m1+...+mn. An unstablebound state (see resonance) shows up as a pole with a complex center of mass energy.
protonand an electroncan 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- is formed. Only the lowest energy bound state, the ground stateis stable. The other excited states are unstable and will decay into bound states with less energy by emitting a photon.
* A nucleus is a bound state of
protons and neutrons ( nucleons).
positronium"atom" is an unstable bound state of an electronand a positron. It decays into photons.
protonitself 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.
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.
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