The derivation of the Robertson inequality for operators A ^ ^ and B ^ ^ requires A ^ ^ B ^ ^ ψ ψ and B ^ ^ A ^ ^ ψ ψ to be defined. There are quantum systems where these conditions are not valid. One example is a quantum particle on a ring, where the wave function depends on an angular variable θ θ in the interval [ 0 , 2 π π ] . Define "position" and "momentum" operators A ^ ^ and B ^ ^ by A ^ ^ ψ ψ ( θ θ ) = θ θ ψ ψ ( θ θ ) , θ θ ∈ ∈ [ 0 , 2 π π ] , and B ^ ^ ψ ψ = − − i ℏ ℏ d ψ ψ d θ θ , with periodic boundary conditions on B ^ ^ . The definition of A ^ ^ depends the θ θ range from 0 to 2 π π . These operators satisfy the usual commutation relations for position and momentum operators, [ A ^ ^ , B ^ ^ ] = i ℏ ℏ . More precisely, A ^ ^ B ^ ^ ψ ψ − − B ^ ^ A ^ ^ ψ ψ = i ℏ ℏ ψ ψ whenever both A ^ ^ B ^ ^ ψ ψ and B ^ ^ A ^ ^ ψ ψ are defined, and the space of such ψ ψ is a dense subspace of the quantum Hilbert space.