Should I mention in my statement of purpose that I am applying because of my girlfriend
I can't see any scenario in which that would help and can see many in which it would hurt you. The first question asked would be "Is this guy serious???" People are looking for people dedicated to the academic process, research, learning, etc. Personal reasons such as this, and many others, shouldn't be mentioned. Take about why it is that you want to work with this faculty and what  your long term academic goals are.