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The example in the book is solving for the probabilities of getting +h(bar)/2 and -h(bar)/2 if we are to measure the spin angular momentum Sx.

I was able to follow the derivation up to the point where they obtained the eigenspinors:

X+ = [1/sqrt2 1/sqrt2]' and X- = [1/sqrt2 -1/sqrt2]'

But I don't get how they go from those to formulating the spinor:

X = [(a+b)/sqrt2]X+ + [(a-b)/sqrt2]X-

Any guidance would be much appreciated - thanks in advance.