Question

The Stern-Gerlach experiment for the measurement of electron spins is represented by the observable $\hat{S} = \hat{\sigma} \cdot \hat{n} = \hat{\sigma}_x \cdot n_x + \hat{\sigma}_y \cdot n_y + \hat{\sigma}_z \cdot n_z$, where $\hat{\sigma}$ are the Pauli spin matrices $(\hat{\sigma}_x) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $(\hat{\sigma}_y) = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, $(\hat{\sigma}_z) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, and $\hat{n} = (\sin\vartheta \cos\varphi, \sin\vartheta \sin\varphi, \cos\vartheta)$ is the orientation of the Stern-Gerlach apparatus in spherical coordinates. Calculate the eigenvalues $s_{\pm}$ and normalized eigenstates $|s_{\pm}\rangle$ of $\hat{S}$ prepared by the apparatus. Interpret the special case $\vartheta = 0, \varphi = 0$. Useful identities: $\sin\vartheta = 2\sin(\vartheta/2)\cos(\vartheta/2)$; $\cos\vartheta = 2\cos^2(\vartheta/2) - 1 = 1 - 2\sin^2(\vartheta/2)$

          The Stern-Gerlach experiment for the measurement of electron spins is represented by the observable $\hat{S} = \hat{\sigma} \cdot \hat{n} = \hat{\sigma}_x \cdot n_x + \hat{\sigma}_y \cdot n_y + \hat{\sigma}_z \cdot n_z$, where $\hat{\sigma}$ are the Pauli spin matrices
$(\hat{\sigma}_x) = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $(\hat{\sigma}_y) = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, $(\hat{\sigma}_z) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$,
and $\hat{n} = (\sin\vartheta \cos\varphi, \sin\vartheta \sin\varphi, \cos\vartheta)$ is the orientation of the Stern-Gerlach apparatus in spherical coordinates. Calculate the eigenvalues $s_{\pm}$ and normalized eigenstates $|s_{\pm}\rangle$ of $\hat{S}$ prepared by the apparatus. Interpret the special case $\vartheta = 0, \varphi = 0$.
Useful identities:
$\sin\vartheta = 2\sin(\vartheta/2)\cos(\vartheta/2)$; $\cos\vartheta = 2\cos^2(\vartheta/2) - 1 = 1 - 2\sin^2(\vartheta/2)$

the stern gerlach experiment for the measurement of electron spins is represented by the observable hats hatsigma cdot hatn hatsigmax cdot nx hatsigmay cdot ny hatsigmaz cdot nz where 13757

Added by Jose M.

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