The smallest value of the IPR corresponds to a fully delocalized state, ψ ψ ( x ) = 1 / N for a system of size N , where the IPR yields ∑ ∑ x | ψ ψ ( x ) | 4 = N / ( N 1 / 2 ) 4 = 1 / N . Values of the IPR close to 1 correspond to localized states (pure states in the analogy), as can be seen with the perfectly localized state ψ ψ ( x ) = δ δ x , x 0 , where the IPR yields ∑ ∑ x | ψ ψ ( x ) | 4 = 1 . In one dimension IPR is directly proportional to the inverse of the localization length, i.e., the size of the region over which a state is localized. Localized and delocalized (extended) states in the framework of condensed matter physics then correspond to insulating and metallic states, respectively, if one imagines an electron on a lattice not being able to move in the crystal (localized wave function, IPR is close to one) or being able to move (extended state, IPR is close to zero).