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and then [itex]\psi^* \psi = <1,-1|1,-1>=2[/itex]

which gives [itex]\psi= \frac{1}{\sqrt{2}} |1,-1>[/itex] for the normalised ket.

but what about [itex]\psi=|1,-1>+2|0,0>+|-1,1>[/itex]

i get [itex]\psi^*=<1,-1| +2<0,0| + <-1,1|[/itex]

now I am guessing that seeing as i want to normalise the whole wavefunction [itex]\psi[/itex] i can't just normalise the kets individually so multiplying every term by every other term i get non-zero contributions giving

[itex]<1,-1|1,-1>+<-1,1|-1,1>+<1,-1|-1,1>+<-1,1|1,-1>=0[/itex] which is impossible

however if i can normalise them seperately then i would get for my normalised wavefunction

[itex]\psi=\frac{1}{\sqrt{2}} |1,-1> +2|0,0> + \frac{1}{\sqrt{2}} |-1,1>[/itex]

so which is right (if either) and why?