Spring 2006 5 Qm

A proton (spin - 1/2 ) is in the spin state

(1)
\begin{align} \chi=\frac{1}{4}\left( \begin{array}{c} 2 - 3 \imath \\ \imath \sqrt{3} \\ \end{array} \right) \end{align}

where the spinor corresponds to the representation where, e.g., the state with spin up along the z-axis is

(2)
\begin{align} \chi_+ = \left( \begin{array}{c} 1\\ 0\\ \end{array} \right) \end{align}

(a) What is the expectation value for the angular momentum operator in the z-direction?

(b) What is the expectation value for the angular momentum operator in the x-direction?

(c) What is the expectation value for the angular momentum operator in the y-direction?

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