Schrodinger half spin states expectation values

[crowlma]

Homework Statement

What is the expectation value of [itex]\hat{S}_{x}[/itex] with respect to the state [itex]\chi = \begin{pmatrix}
1\\
0
\end{pmatrix}[/itex]?
[itex]\hat{S}_{x} = \frac{\bar{h}}{2}\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}[/itex]

Homework Equations

[itex]<\hat{S}_{x}> = ∫^{\infty}_{-\infty}(\chi^{T})^{*}\hat{S}_{x}\chi[/itex]

The Attempt at a Solution

So I have [itex](\chi^{T})^{*}[/itex] as equalling (1 0), giving me: [itex]\frac{\hbar}{2} ∫^{\infty}_{-\infty}\begin{pmatrix}
1&0
\end{pmatrix}\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}\begin{pmatrix}
1\\
0
\end{pmatrix} [/itex] which simplifies to [itex]\frac{\hbar}{2} ∫^{\infty}_{-\infty}\begin{pmatrix}
1&0
\end{pmatrix}\begin{pmatrix}
1\\
0
\end{pmatrix} = 0 [/itex]. 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 [itex](\chi^{T})^{*}\hat{S}_{x}\chi[/itex]. There is a mistake in the column vector in the last line as well, although you somehow came up with the correct answer.

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