Chapter 14: Problem 1
Verify for each direction $$ \mathbf{n}=\sin \theta \cos \phi \mathbf{e}_{x}+\sin \theta \sin \phi c_{y}+\cos \theta \boldsymbol{e}_{x} $$ the spin operator $$ \sigma_{n}=\left(\begin{array}{cc} \cos \theta & \sin \theta \mathrm{e}^{-1 \phi} \\ \sin \theta \mathrm{e}^{i \phi} & -\cos \theta \end{array}\right) $$ has cigenvalues \(\pm 1\). Show that up to phase, the cigenvectors can be expressed as $$ \left.\mid+n)=\left(\begin{array}{c} \cos \frac{1}{2} \theta \mathrm{c}^{-t \phi} \\ \sin \frac{1}{2} \theta \end{array}\right), \quad \mid-n\right)=\left(\begin{array}{c} -\sin \frac{1}{2} \theta \mathrm{e}^{-1 \phi} \\ \cos \frac{1}{2} \theta \end{array}\right) $$ and compute the expectation values for spin in the dircction of the various axes $$ \left\langle\sigma_{1}\right\rangle_{\text {tn }}=\left(\pm n\left|\sigma_{1}\right| \pm n\right) $$ For a bcam of particles in a pure state \(\mid+n\) ) show that after a measurcment of spin in the \(+x\) direction the probability that the spin is in this direction is \(\frac{1}{2}(1+\sin \theta \cos \phi)\).
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