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Angular momentum operator

Spin

Spin one-half

{title2=$\mathbf n\mathbin{\cdot}\mathbf S$}

For $\mathbf n=(\sin\theta,0,\cos\theta)$, the normalized eigenstates of $\mathbf n\cdot\mathbf S$ are
$$
|\uparrow_\theta\rangle
=\cos\frac\theta2|\uparrow\rangle
+\sin\frac\theta2|\downarrow\rangle,
\qquad
|\downarrow_\theta\rangle
=-\sin\frac\theta2|\uparrow\rangle
+\cos\frac\theta2|\downarrow\rangle,
$$
with eigenvalues $+\hbar/2$ and $-\hbar/2$.


Spin one-half along an axis

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Spin one-half along an axis ( $n \cdot S$ )

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Spin one-half along an axis (n⋅S)

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