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Rahmuss
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[SOLVED] Expectation Values of Spin Operators
b) Find the expectation values of S_{x}, S_{y}, and S_{z}
From part a)
X = A \begin{pmatrix}3i \\ 4 \end{pmatrix}
Which was found to be: A = \frac{1}{5}
S_{x} = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}
S_{y} = \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix}
S_{z} = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}
I have it setup as:
\left\langle S_{x}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{x}X \Rightarrow
\int^{\infty}_{-\infty}X^{*} \frac{\hbar}{2} \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}\frac{3i}{5} \\ \frac{4}{5} \end{pmatrix}\Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar}{2}\begin{pmatrix}\frac{-3i}{5} \\ \frac{4}{5} \end{pmatrix} \frac{\hbar}{2} \begin{pmatrix}\frac{3i}{5} \\ \frac{4}{5} \end{pmatrix} \Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{-12i}{25} + \frac{12i}{25} \right] \Rightarrow 0
\left\langle S_{y}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{y}X \Rightarrow
\int^{\infty}_{-\infty}X^{*} \frac{\hbar}{2} \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix}\frac{3i}{5} \\ \frac{4}{5} \end{pmatrix}\Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar}{2} \begin{pmatrix}\frac{-3i}{5} \\ \frac{4}{5} \end{pmatrix} \begin{pmatrix}\frac{4i}{5} \\ \frac{-3}{5} \end{pmatrix} \frac{\hbar}{2} \Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{12i}{25} - \frac{12i}{25} \right] \Rightarrow 0
\left\langle S_{z}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{z}X \Rightarrow
\int^{\infty}_{-\infty}X^{*} \frac{\hbar}{2} \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}\frac{3i}{5} \\ \frac{4}{5} \end{pmatrix}\Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar}{2} \begin{pmatrix}\frac{-3i}{5} \\ \frac{4}{5} \end{pmatrix} \begin{pmatrix}\frac{3i}{5} \\ \frac{-4}{5} \end{pmatrix} \frac{\hbar}{2} \Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{9}{25} - \frac{16}{25} \right] \Rightarrow \frac{-7\hbar^{2}}{100}
The first two seem like they're fine; but the last one doesn't seem right. Now if it was:
\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{9}{25} + \frac{16}{25} \right] \Rightarrow \frac{\hbar^{2}}{4}
Then that would at least seem to be in the right direction. So what am I missing?
Homework Statement
b) Find the expectation values of S_{x}, S_{y}, and S_{z}
Homework Equations
From part a)
X = A \begin{pmatrix}3i \\ 4 \end{pmatrix}
Which was found to be: A = \frac{1}{5}
S_{x} = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}
S_{y} = \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix}
S_{z} = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}
The Attempt at a Solution
I have it setup as:
\left\langle S_{x}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{x}X \Rightarrow
\int^{\infty}_{-\infty}X^{*} \frac{\hbar}{2} \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}\frac{3i}{5} \\ \frac{4}{5} \end{pmatrix}\Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar}{2}\begin{pmatrix}\frac{-3i}{5} \\ \frac{4}{5} \end{pmatrix} \frac{\hbar}{2} \begin{pmatrix}\frac{3i}{5} \\ \frac{4}{5} \end{pmatrix} \Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{-12i}{25} + \frac{12i}{25} \right] \Rightarrow 0
\left\langle S_{y}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{y}X \Rightarrow
\int^{\infty}_{-\infty}X^{*} \frac{\hbar}{2} \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix} \begin{pmatrix}\frac{3i}{5} \\ \frac{4}{5} \end{pmatrix}\Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar}{2} \begin{pmatrix}\frac{-3i}{5} \\ \frac{4}{5} \end{pmatrix} \begin{pmatrix}\frac{4i}{5} \\ \frac{-3}{5} \end{pmatrix} \frac{\hbar}{2} \Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{12i}{25} - \frac{12i}{25} \right] \Rightarrow 0
\left\langle S_{z}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{z}X \Rightarrow
\int^{\infty}_{-\infty}X^{*} \frac{\hbar}{2} \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}\frac{3i}{5} \\ \frac{4}{5} \end{pmatrix}\Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar}{2} \begin{pmatrix}\frac{-3i}{5} \\ \frac{4}{5} \end{pmatrix} \begin{pmatrix}\frac{3i}{5} \\ \frac{-4}{5} \end{pmatrix} \frac{\hbar}{2} \Rightarrow
\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{9}{25} - \frac{16}{25} \right] \Rightarrow \frac{-7\hbar^{2}}{100}
The first two seem like they're fine; but the last one doesn't seem right. Now if it was:
\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{9}{25} + \frac{16}{25} \right] \Rightarrow \frac{\hbar^{2}}{4}
Then that would at least seem to be in the right direction. So what am I missing?
