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For a qubit defined as: [tex]\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)[/tex]

Since [tex]|0\rangle[/tex] and [tex]|1\rangle[/tex] are the eigenstates of [tex]\sigma_z[/tex] then measuring sigma_z will yield either [tex]|0\rangle[/tex] or [tex]|1\rangle[/tex]. Measuring [tex]\sigma_x[/tex] on the same qubit will give one of the eigenstates of [tex]\sigma_x[/tex], which are [tex]\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)[/tex] and [tex]\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)[/tex].

Only problem is I don't see how you can obtain these general/algebraic states? When I make a measurement on the qubit ie. ([tex](\frac{1}{\sqrt{2}}(\langle 0| +\langle1|))\sigma_x(\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle))[/tex] I just obtain a number? How can I keep this algebraic structure and prove the above?