# Bound state

In physics, a bound state describes a system where a particle is 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 either an external potential, or may be the result of the presence of another particle.

In quantum mechanics (where the number of particles is conserved), a bound state is a state in Hilbert space that corresponds to two or more particles whose interaction energy is less than the total energy of each separate particle, and therefore these particles cannot be separated unless energy is spent. The energy spectrum of 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 nuclei and for some electret materials able to carry electric charge for rather long periods.)

In general, a stable bound state is said to exist in a given potential of some dimension if stationary square-integrable wavefunctions exist (normalized in the range of the potential). The energies of these wavefunctions are negative.

In relativistic quantum field theory, a stable bound state of n particles with masses m1, … , mn shows up as a pole in the S-matrix with a center of mass energy which is less than m1 + … + mn. An unstable bound state (see resonance) 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

## In mathematical quantum physics

Let H be a complex separable Hilbert space, $U = \lbrace U(t) \mid t \in \mathbb{R} \rbrace$ be a one-parametric 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 let $\mu(A,\rho)$ be the induced probability distribution of A with respect to ρ on the Borel σ-algebra on $\mathbb{R}$. Then the evolution of ρ induced by U is said to be bound with respect to A if $\lim_{R \rightarrow \infty} \sum_{t \geq t_0} \mu(A,\rho(t))(\mathbb{R}_{> R}) = 0$, where $\mathbb{R}_{>R} = \lbrace x \in \mathbb{R} \mid x > R \rbrace$.

Example: Let $H = L^2(\mathbb{R})$ and let A be the position observable. Let $\rho = \rho(0) \in H$ have compact support 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 a bound state with respect to the position.
• If $\rho$ does not change in time, i.e. $\rho(t) = \rho$ for all $t \geq 0$, then $\rho$ is a bound state with respect to position.
• More generally: If the state evolution of ρ "just moves ρ inside a bounded domain", then ρ is also a bound state with respect to position.

It should be emphasized that a bound state can have its energy located in the continuum spectrum. This fact was first pointed out by John von Neumann and Eugene Wigner in 1929. [5] This exotic type of bound state has been realized in several simple models. [6] [7]