Bound state

In quantum physics, a bound state is a special quantum state of a particle 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 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. In quantum physics, a bound state is a special quantum state of a particle 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 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 extstyle sum _{k}m_{k}} . An unstable bound state shows up as a pole with a complex center-of-mass energy. Let H be a complex separable Hilbert space, U = { U ( t ) ∣ t ∈ R } {displaystyle U=lbrace U(t)mid tin mathbb {R} brace } be a one-parameter group of unitary operators on H and ρ = ρ ( t 0 ) {displaystyle ho = ho (t_{0})} be a statistical operator on H. Let A be an observable on H and μ ( A , ρ ) {displaystyle mu (A, ho )} be the induced probability distribution of A with respect to ρ on the Borel σ-algebra of R {displaystyle mathbb {R} } . Then the evolution of ρ induced by U is bound with respect to A if lim R → ∞ sup t ≥ t 0 μ ( A , ρ ( t ) ) ( R > R ) = 0 {displaystyle lim _{R ightarrow infty }{sup _{tgeq t_{0}}{mu (A, ho (t))(mathbb {R} _{>R})}}=0} , where R > R = { x ∈ R ∣ x > R } {displaystyle mathbb {R} _{>R}=lbrace xin mathbb {R} mid x>R brace } . More informally, a bound state is contained within a bounded portion of the spectrum of A. For a concrete example: let H = L 2 ( R ) {displaystyle H=L^{2}(mathbb {R} )} and let A be position. Given compactly-supported ρ = ρ ( 0 ) ∈ H {displaystyle ho = ho (0)in H} and [ − 1 , 1 ] ⊆ S u p p ( ρ ) {displaystyle subseteq mathrm {Supp} ( ho )} . Let A have measure-space codomain ( X ; μ ) {displaystyle (X;mu )} . A quantum particle is in a bound state if it is never found “too far away from any finite region R ⊆ X {displaystyle Rsubseteq X} ,” i.e. using a wavefunction representation, 0 = lim R → ∞ P ( particle measured inside  X ∖ R ) = lim R → ∞ ∫ X ∖ R | ψ ( x ) | 2 d μ ( x ) {displaystyle {egin{aligned}0&=lim _{R o infty }{mathbb {P} ({ ext{particle measured inside }}Xsetminus R)}\&=lim _{R o infty }{int _{Xsetminus R}|psi (x)|^{2},dmu (x)}end{aligned}}} Consequently, ∫ X | ψ ( x ) | 2 d μ ( x ) {displaystyle int _{X}{|psi (x)|^{2},dmu (x)}} is finite. In other words, a state is a bound state if and only if it is finitely normalizable. As finitely normalizable states must lie within the discrete part of the spectrum, bound states must lie within the discrete part. However, as Neumann and Wigner pointed out, a bound state can have its energy located in the continuum spectrum. In that case, bound states still are part of the discrete portion of the spectrum, but appear as Dirac masses in the spectral measure.