A nice way to double-check that these relations are correct is to do a Taylor expansion of the translation operator acting on a position-space wavefunction. Expanding the exponential to all orders, the translation operator generates exactly the full Taylor expansion of a test function: ψ ψ ( r − − x ) = T ^ ^ ( x ) ψ ψ ( r ) = exp ⁡ ⁡ ( − − i x ⋅ ⋅ p ^ ^ ℏ ℏ ) ψ ψ ( r ) = ( ∑ ∑ n = 0 ∞ ∞ 1 n ! ( − − i ℏ ℏ x ⋅ ⋅ p ^ ^ ) n ) ψ ψ ( r ) = ( ∑ ∑ n = 0 ∞ ∞ 1 n ! ( − − x ⋅ ⋅ ∇ ∇ ) n ) ψ ψ ( r ) = ψ ψ ( r ) − − x ⋅ ⋅ ∇ ∇ ψ ψ ( r ) + 1 2 ! ( x ⋅ ⋅ ∇ ∇ ) 2 ψ ψ ( r ) − − … … So every translation operator generates exactly the expected translation on a test function if the function is analytic in some domain of the complex plane.