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The square of the norm of the state
|psi> = a |up> + b |down>
|a|^2 + |b|^2
In the given case with a = sqrt(2/3) and b = i sqrt(1/3) this is clearly equal to 1.
You can see this also by writing the square of the norm as <psi|psi>. We have:
<psi| = a* <up| + b* <down|
<psi|psi> = |a|^2 <up|up> + |b|^2 <down|down> + a*b ...
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