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Spin Matrices for Spin 1

  1. Nov 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Construct the spin matrices (Sx,Sy,Sz) for a particle of spin 1. Determine the action of Sz, S+, and S- on each of these states.

    2. Relevant equations
    s=1 m=-1, 0, 1
    Sz=hm |sm>
    S+= h [2-m(m+1)]^1/2 |s m+1>
    S-= h [2-m(m-1)]^1/2 |s m-1>
    *"h" is actually h-bar

    3. The attempt at a solution
    I've been trying to follow the same method as for spin 1/2, where |1/2 1/2> is a vector (1 0) and |1/2 -1/2> is (0 1), but I don't understand how going between notation for vectors yields these results, and thus I don't know how to get the vector components for the spin 1 case.
     
  2. jcsd
  3. Nov 27, 2009 #2
    Use the following equations to construct the matrix:

    [tex]
    \langle m'|S_x|m\rangle=(\delta_{m,m'+1}+\delta_{m+1,m'})\frac{\hbar}{2}\sqrt{s(s+1)-m'm}
    [/tex]

    [tex]
    \langle m'|S_y|m\rangle=(\delta_{m,m'+1}-\delta_{m+1,m'})\frac{\hbar}{2i}\sqrt{s(s+1)-m'm}
    [/tex]

    [tex]
    \langle m'|S_z|m\rangle=\delta_{mm'}m\hbar
    [/tex]

    with [itex]s=1[/itex], you know that [itex]m=-1,0,1[/itex].
     
  4. Nov 27, 2009 #3
    Eh, maybe I'm a little more confused than I thought. Can you be a little more...descriptive, maybe? I'm not seeing how those equations apply.
     
  5. Nov 27, 2009 #4
    The spin matrices--for spin 1--look like this:

    [tex]
    \hat{S}_x=\left(\begin{array}{ccc} \langle 1|S_x|1\rangle & \langle 1|S_x|0\rangle & \langle 1|S_x|-1\rangle \\ \langle 0|S_x|1\rangle & \langle 0|S_x|0\rangle & \langle 0|S_x|-1\rangle \\ \langle -1|S_x|1\rangle & \langle -1|S_x|0\rangle & \langle -1|S_x|-1\rangle
    \end{array}\right)
    [/tex]

    so each 1,0 and -1 are the [itex]m[/itex] and [itex]m'[/itex] values. The delta's are the Kronecker delta:

    [tex]
    \delta_{mn}= \left< \begin{array}{ll} 1 & m=n \\ 0 & m\neq n\end{array}\right.
    [/tex]

    It should just be matching the m's and delta's to get values for each component.

    EDIT: For a quick example:

    [tex]
    \langle 1|S_x|0\rangle=(1+0)\frac{\hbar}{2}\sqrt{1(1+1)-1\cdot0}=\sqrt{2}\frac{\hbar}{2}
    [/tex]
     
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