If ψ ψ ( r ) and ϕ ϕ ( r ) are two position-space wavefunctions, then the inner product of ψ ψ with ϕ ϕ is: ∫ ∫ d 3 r ψ ψ ∗ ∗ ( r ) ϕ ϕ ( r ) while the inner product of T ^ ^ ( a ) ψ ψ with T ^ ^ ( a ) ϕ ϕ is: ∫ ∫ d 3 r ψ ψ ∗ ∗ ( r − − a ) ϕ ϕ ( r − − a ) By change of variables, these two inner products are exactly the same. Therefore, the translation operators are unitary, and in particular: ( T ^ ^ ( x ) ) † † = ( T ^ ^ ( x ) ) − − 1 .