The translation operator is denoted T (ε), where ε represents the length of the translation. It satisfies the following identity: T ( ε ε ) | ψ ψ ⟩ ⟩ = ∫ ∫ d x T ( ε ε ) | x ⟩ ⟩ ⟨ ⟨ x | ψ ψ ⟩ ⟩ that becomes ∫ ∫ d x | x + ε ε ⟩ ⟩ ⟨ ⟨ x | ψ ψ ⟩ ⟩ = ∫ ∫ d x | x ⟩ ⟩ ⟨ ⟨ x − − ε ε | ψ ψ ⟩ ⟩ = ∫ ∫ d x | x ⟩ ⟩ ψ ψ ( x − − ε ε )