GerritJan Linker

Completeness relation or closure relation The completeness relation or also called closure relation is often used in the algebra in quantum mechanics: Σ_{i} s_{i}><s_{i} = 1 = 1 Suppose we have a complete set of orthonormal functions (states) s_{i}> An arbitrary function Ψ can be written as a linear combination: Ψ>=Σ_{i}c_{i}s_{i}> Multiplying both sides by <s_{j} gives: <s_{j}Ψ> = <s_{j}Σ_{i}c_{i}s_{i}> <=> <s_{j}Ψ> = Σ_{i}c_{i}<s_{j}s_{i}> = Σ_{i}c_{i}δ_{ji} = c_{j} <=> <s_{j}Ψ> = c_{j} Thus, Ψ> = Σ_{i}c_{i}s_{i}> = Σ_{i}<s_{i}Ψ>s_{i}> = Σ_{i}s_{i}><s_{i}Ψ> <=> Ψ> = (Σ_{i}s_{i}><s_{i}) Ψ> thus, Σ_{i}s_{i}><s_{i} = 1 = 1
