Expectation Values of Spin Operators

[Rahmuss]

[SOLVED] Expectation Values of Spin Operators

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?

 

[Gokul43201]

Where do you get your second factor of $\hbar /2$ from? Also, you need to throw away the integrals and write the bra as a row vector (not a column vector).

 

[Rahmuss]

Oh, you're right, it's just $$S_{x}$$, not $$S_{x}^{2}$$. Thanks. And I'll change the vectors (on my homework); but is the rest correct then?

 

[Gokul43201]

Yes, but there are no integrals involved when you use matrices.

 

[shankar3274]

how is the wave function defined?
also don't confuse with matrices and integrals