Schrodinger half spin states expectation values

[crowlma]

Homework Statement

What is the expectation value of \hat{S}_{x} with respect to the state \chi = \begin{pmatrix}<br /> 1\\<br /> 0<br /> \end{pmatrix}?
\hat{S}_{x} = \frac{\bar{h}}{2}\begin{pmatrix}<br /> 0&amp;1\\<br /> 1&amp;0<br /> \end{pmatrix}

Homework Equations

&lt;\hat{S}_{x}&gt; = ∫^{\infty}_{-\infty}(\chi^{T})^{*}\hat{S}_{x}\chi

The Attempt at a Solution

So I have (\chi^{T})^{*} as equalling (1 0), giving me: \frac{\hbar}{2} ∫^{\infty}_{-\infty}\begin{pmatrix}<br /> 1&amp;0<br /> \end{pmatrix}\begin{pmatrix}<br /> 0&amp;1\\<br /> 1&amp;0<br /> \end{pmatrix}\begin{pmatrix}<br /> 1\\<br /> 0<br /> \end{pmatrix} which simplifies to \frac{\hbar}{2} ∫^{\infty}_{-\infty}\begin{pmatrix}<br /> 1&amp;0<br /> \end{pmatrix}\begin{pmatrix}<br /> 1\\<br /> 0<br /> \end{pmatrix} = 0. Is this right?

 

[MisterX]

This is a discrete system and it doesn't make sense to have an integral there. Also we wouldn't need a sum or integral since that is already implied by (\chi^{T})^{*}\hat{S}_{x}\chi. There is a mistake in the column vector in the last line as well, although you somehow came up with the correct answer.

\langle \hat{S}_x \rangle = (\chi^{T})^{*}\hat{S}_{x}\chi