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Expectation Values of Spin Operators

  1. Dec 3, 2007 #1
    [SOLVED] Expectation Values of Spin Operators

    1. The problem statement, all variables and given/known data
    b) Find the expectation values of [tex]S_{x}, S_{y}, and S_{z}[/tex]


    2. Relevant equations
    From part a)
    [tex]X = A \begin{pmatrix}3i \\ 4 \end{pmatrix}[/tex]

    Which was found to be: [tex]A = \frac{1}{5}[/tex]

    [tex]S_{x} = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}[/tex]

    [tex]S_{y} = \begin{pmatrix}0 & -i \\ i & 0 \end{pmatrix}[/tex]

    [tex]S_{z} = \begin{pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix}[/tex]

    3. The attempt at a solution
    I have it setup as:

    [tex]\left\langle S_{x}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{x}X \Rightarrow[/tex]

    [tex]\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[/tex]

    [tex]\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[/tex]

    [tex]\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{-12i}{25} + \frac{12i}{25} \right] \Rightarrow 0[/tex]

    [tex]\left\langle S_{y}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{y}X \Rightarrow[/tex]

    [tex]\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[/tex]

    [tex]\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[/tex]

    [tex]\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{12i}{25} - \frac{12i}{25} \right] \Rightarrow 0[/tex]

    [tex]\left\langle S_{z}\right\rangle = \int^{\infty}_{-\infty}X^{*}S_{z}X \Rightarrow[/tex]

    [tex]\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[/tex]

    [tex]\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[/tex]

    [tex]\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{9}{25} - \frac{16}{25} \right] \Rightarrow \frac{-7\hbar^{2}}{100}[/tex]

    The first two seem like they're fine; but the last one doesn't seem right. Now if it was:

    [tex]\int^{\infty}_{-\infty}\frac{\hbar^{2}}{4}\left[\frac{9}{25} + \frac{16}{25} \right] \Rightarrow \frac{\hbar^{2}}{4}[/tex]

    Then that would at least seem to be in the right direction. So what am I missing?
     
  2. jcsd
  3. Dec 3, 2007 #2

    Gokul43201

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    Where do you get your second factor of [itex]\hbar /2[/itex] from? Also, you need to throw away the integrals and write the bra as a row vector (not a column vector).
     
  4. Dec 3, 2007 #3
    Oh, you're right, it's just [tex]S_{x}[/tex], not [tex]S_{x}^{2}[/tex]. Thanks. And I'll change the vectors (on my homework); but is the rest correct then?
     
  5. Dec 4, 2007 #4

    Gokul43201

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    Yes, but there are no integrals involved when you use matrices.
     
  6. Dec 12, 2010 #5
    Re: [SOLVED] Expectation Values of Spin Operators

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