The quantum dynamics and energetics of an excess proton in water have been studied computationally. Comparison of a quantum mechanical treatment of the transferring proton and the water solvent is made with a classical treatment of the same system. The exchange of the proton between two water molecules is found to be an activationless quantum process. Analysis of the microscopic structure of the solvent around the proton transfer complex is also carried out, and the quantum infrared spectrum of the transferring proton is calculated and analyzed in terms of Zundel polarization. The Grotthus mechanism for proton migration in water is also examined within the context of the model. Grotthus behavior is suggested to depend critically on the dynamics of water molecules in the second solvation shell of the H5O2+ complex, as well as the inward fluctuations of the oxygen-oxygen distance of water molecules that hydrogen bond to the H5O2+ complex in the first solvation shell. The quantum effects on the nuclear dynamics are found to be significant.