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Eigenvalues and Eigenstates of Spin Operator

  1. Oct 29, 2011 #1
    I'm not exactly looking for help finding the eigenvalues of the spin operator, I'm mainly wondering if there is a better technique to do it.

    1. The problem statement, all variables and given/known data
    Find the eigenvalues and corresponding eigenstates of a spin 1/2 particle in an arbitrary direction (θ,[itex]\phi[/itex]) using the Pauli Matrices

    2. Relevant equations
    Spin operator in arbitrary direction:
    n.[itex]\sigma[/itex] = [itex]\hbar[/itex]/2(cos[itex]\phi[/itex]sin[itex]\theta[/itex][itex]\sigma_x[/itex] + sin[itex]\phi[/itex]sin[itex]\theta[/itex][itex]\sigma_y[/itex]+cos[itex]\theta\sigma_z[/itex])

    [itex]\sigma_x[/itex],[itex]\sigma_y[/itex],[itex]\sigma_z/[itex] are the Pauli spin matrices.


    3. The attempt at a solution

    The way I did it was to express the pauli matrices in their matrix form, sum up the expression to get one matrix, then solve the eigenvalue equation
    n.[itex]\sigma[/itex][itex]\Psi[/itex] = [itex]\lambda[/itex][itex]\Psi[/itex].

    This gives me the answer [itex]\pm[/itex][itex]\hbar[/itex]/2

    My question is: Is there a better/quicker way to do this (and problems similar to this) without having to solve the eigenvalue equation directly? I have other similar questions where solving the eigenvalue equation becomes long and tedious.
     
  2. jcsd
  3. Oct 30, 2011 #2

    diazona

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    Homework Helper

    What you've done is the "direct" way to solve these problems. There are tricks you can use in special cases, but I'm not sure if it's possible to make it much easier in general. You could use a coordinate system which is rotated such that the z axis lies along the direction [itex]\hat{n}[/itex], so that the spin operator is just [itex]\sigma_z[/itex]. For this you would have to use the rotation matrix that converts [itex]\Psi[/itex] into the new coordinate system, [itex]\exp[i\theta(\sigma_y\cos\phi + \sigma_x\sin\phi)/2][/itex] (or something like that).
     
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