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Schrodinger half spin states expectation values

  1. Nov 6, 2013 #1
    1. The problem statement, all variables and given/known data

    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]


    2. Relevant equations

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

    3. 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?
     
  2. jcsd
  3. Nov 7, 2013 #2
    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]
     
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