You don't assume that |\psi>=\alpha |1>+ \beta |2>, there is a reason for that. The completeness relation gives,
I=|1><1|+|2><2| [look up completeness relation if you don't know about it.]
which means, |\psi>=I|\psi>=|1><1|\psi>+|2><2|\psi>
or, |\psi>=\alpha |1>+ \beta |2> ,
where...