1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Spin 3/2 along an arbitrary direction

  1. Mar 29, 2016 #1
    1. The problem statement, all variables and given/known data

    Consider a particle of spin 3/2. Find the matrix for the component of the spin along a unit vector
    with arbitrary direction n. Find its eigenvalues and eigenvectors.

    2. Relevant equations

    I know that the general spin operator is
    \begin{equation}
    \widehat{S} = a\cdot \widehat{S}_x + b\cdot \widehat{S}_y + c\cdot \widehat{S}_z
    \end{equation}
    and that
    \begin{equation}
    \hat{n} = (\sin{\phi}\cos{\theta},\sin{\phi}\sin{\theta},\cos{\phi})
    \end{equation}

    3. The attempt at a solution

    Now
    \begin{equation}
    \widehat{S}_x \stackrel{\cdot}{=} \frac{\hbar}{2}\begin{pmatrix}0 & \sqrt3 & 0 & 0 \\ \sqrt3 & 0 & 2 & 0 \\ 0 & 2 & 0 & \sqrt3 \\ 0 & 0 & \sqrt3 & 0\end{pmatrix}\, ,\
    \widehat{S}_y \stackrel{\cdot}{=} \frac{\hbar}{2i}\begin{pmatrix}0 & \sqrt3 & 0 & 0 \\ -\sqrt3 & 0 & 2 & 0 \\ 0 & -2 & 0 & \sqrt3 \\ 0 & 0 & -\sqrt3 & 0\end{pmatrix}\, ,\
    \widehat{S}_z \stackrel{\cdot}{=} \hbar\begin{pmatrix}3/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 0 \\ 0 & 0 & -1/2 & 0 \\ 0 & 0 & 0 & -3/2\end{pmatrix}\ .
    \end{equation}

    So then we get
    \begin{equation}
    \hat{n}\cdot \widehat{S} \stackrel{\cdot}{=} \frac{\hbar}{2}(\sin{\phi}\cos{\theta})\begin{pmatrix}0 & \sqrt3 & 0 & 0 \\ \sqrt3 & 0 & 2 & 0 \\ 0 & 2 & 0 & \sqrt3 \\ 0 & 0 & \sqrt3 & 0\end{pmatrix}\ + \frac{\hbar}{2i}(\sin{\phi}\sin{\theta})\begin{pmatrix}0 & \sqrt3 & 0 & 0 \\ -\sqrt3 & 0 & 2 & 0 \\ 0 & -2 & 0 & \sqrt3 \\ 0 & 0 & -\sqrt3 & 0\end{pmatrix}\ + \hbar(\cos{\phi})\begin{pmatrix}3/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 0 \\ 0 & 0 & -1/2 & 0 \\ 0 & 0 & 0 & -3/2\end{pmatrix}\ .
    \end{equation}

    Now, I know I should determine the secular equation (characteristic equation) after adding the matrices and find the eigenvectors and eigenvalues from there, but I keep getting long and complicated equations that don't seem to factor. Is there any shorter method or any other way to do it?
     
  2. jcsd
  3. Mar 29, 2016 #2

    blue_leaf77

    User Avatar
    Science Advisor
    Homework Helper

    A less space-consuming calculation is done by using the braket notation instead of matrices. In the basis of the eigenstates of ##S_z##, the braket version of ##S## is
    $$
    S = \sum_{m=-3/2}^{3/2} \sum_{m'=-3/2}^{3/2} |m\rangle \langle m| S |m'\rangle \langle m'|
    $$
    Write the eigenvalue problem ##S|u_1\rangle = \lambda |u_1\rangle## where ##|u_1\rangle = a|3/2\rangle + b|1/2\rangle + c|-1/2\rangle + d|-3/2\rangle ##. You are then to find ##\lambda##'s and ##a##, ##b##, ##c##, and ##d## for each one value of ##\lambda##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted