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##\def \sqx{\frac{1}{\sqrt{2}}}##

If S

\begin{equation}|S_{xu}\rangle =\begin{pmatrix} \sqx \\ \sqx\end{pmatrix} = \sqx \left[ |S_{zu}\rangle + |S_{zd}\rangle\right]\end{equation}

How can the two z-vectors span the entire space? How can the spin in the x-direction be a linear combination of the spin vectors in only the z-direction?

If S

_{zu}represents spin-up in the z-direction and S_{zd}represents spin-down in the z-direction then the vector which represents spin-up in the**x-direction**is given by the superposition of the z states \begin{equation}|S_{zu}\rangle =\begin{bmatrix}1 \\ 0\end{bmatrix}, |S_{zd}\rangle =\begin{bmatrix}0 \\1\end{bmatrix} \end{equation} and\begin{equation}|S_{xu}\rangle =\begin{pmatrix} \sqx \\ \sqx\end{pmatrix} = \sqx \left[ |S_{zu}\rangle + |S_{zd}\rangle\right]\end{equation}

How can the two z-vectors span the entire space? How can the spin in the x-direction be a linear combination of the spin vectors in only the z-direction?

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