where a † † a is the number operator. So the exponential of the phase operator displaces the number operator in the same fashion as the momentum operator acts as the generator of translations in quantum mechanics : exp ⁡ ⁡ ( i p ^ ^ x o ℏ ℏ ) x ^ ^ exp ⁡ ⁡ ( − − i p ^ ^ x o ℏ ℏ ) = x ^ ^ + x 0 .