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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. 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". 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 {\displaystyle \{m_{k}\}_{k=1}^{n}} corresponds to a pole in the S-matrix with a center-of-mass energy less than ∑ k m k {\displaystyle \textstyle \sum _{k}m_{k}}. An unstable bound state shows up as a pole with a complex center-of-mass energy.