I'm teaching myself quantum mechanics and am learning about bra-ket notation. There is a particular operator used, the ket-bra (e.g. |X><X|). To understand it, I'm trying to come up with an orthonormal basis for C^2 as a simple case (i.e., the 2-dimensional vector space over the field of complex numbers). That is, I want two vectors, each with two components, each component a complex number, that span C^2 and are orthonormal. I've tried some combinations like (1,0) (0,i) and such, but no luck. Right now my TI-89 is chugging away looking at a few thousand possible vector combinations, but there has to be a better way. Can anybody suggest how I would find two such vectors? Thanks.