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Elementary quantum spin in Sakurai

  1. Apr 4, 2015 #1
    1. The problem statement, all variables and given/known data

    I am currently working on a seemingly straightforward eigenvalue problem appearing as problem 1.8 in Sakurai's Modern QM. He asks us to find an eigenket [itex]\vert\vec S\cdot\hat n;+\rangle[/itex] with [tex]\vec S\cdot\hat n\vert\vec S\cdot\hat n;+\rangle = \frac\hbar 2\vert\vec S\cdot\hat n;+\rangle[/tex] where the unit vector n is defined by polar angle alpha and azimuthal angle beta.

    2. Relevant equations

    The definitions of the Pauli sigma matrices, along with the formula [tex]\hat n = (\sin\alpha\cos\beta,\sin\alpha\sin\beta,\cos\alpha)[/tex] for conversion to Cartesian coordinates.

    3. The attempt at a solution

    [tex]\vec S\cdot\hat n = \frac\hbar 2\sin\alpha\cos\beta\begin{pmatrix}0&1\\1&0\end{pmatrix}+\sin\alpha\sin\beta\begin{pmatrix}0&-i\\i&0\end{pmatrix}+\cos\alpha\begin{pmatrix}1&0\\0&-1\end{pmatrix} = \frac\hbar 2\begin{pmatrix}\cos\alpha&\sin\alpha e^{-i\beta}\\\sin\alpha e^{i\beta}&-\cos\alpha\end{pmatrix}.[/tex] Thus we have the [itex]\lambda=1[/itex] eigenvalue problem for the matrix [tex]\begin{pmatrix}\cos\alpha&e^{-i\beta}\sin\alpha\\e^{i\beta}\sin\alpha&-\cos\alpha\end{pmatrix}.[/tex] This becomes the system of equations [tex]\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}x\cos\alpha+ye^{-i\beta}\sin\alpha-x\\-y\cos\alpha+xe^{i\beta}\sin\alpha-y\end{pmatrix}.[/tex] Thus beta appears to be completely arbitrary, and alpha = n*pi, however this appears to have no solutions as we get cosines equaling 2 which is nonsense. The characteristic polynomial predicts a lambda=1 eigenvalue - where am I going wrong?
  2. jcsd
  3. Apr 4, 2015 #2


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    This looks good to me.

    I don't understand your matrix on the right hand side. How did you get the ##-x## and the ##-y## terms on the far right of the matrix?

    ##\alpha## and ##\beta## are fixed by the choice of ##\hat{n}##. You need to find ##x## and ##y##.
    Last edited: Apr 4, 2015
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