What does a spin 1/2 wave function actually look like

I have only ever seen the wavefunction for a spin 1/2 particle written in the basis set |α> |β>. I was interested in how a wavefunction |ψ> = a|α> + b|β> might be rewritten in a continous basis and hence would need to know what the actual functions of |α> & |β> were.



Thanks