A phase space representation is defined for quantum mechanical state functions. General requirements of quantum mechanics and some additional, very reasonable restrictions lead to a representation that is uniquely determined apart from a scaling. With one choice of this Scaling, the representation is equivalent to that of Bargmann, although different aspects are emphasized here. Equivalence between position and momentum representations leads to a phase space representation that is invariant to, or at most rescaled by, Fourier transform. Phase space equivalents of some common functions are presented including Cartesian Gaussian basis functions and the hydrogen atom ground state function.