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I SO(3) rotation of eigenvectors

  1. Nov 22, 2016 #1
    Consider the eigenvectors ##(0, 1)## and ##(1, 0)## for the quantum system described the magnetic field ##\vec{B} = (0,0,B)##.

    Say I now rotate the magnetic field as ##\vec{B} = (B\sin\theta\cos\phi,B\sin\theta\sin\phi,B\cos\theta)##.

    Then the eigenvectors are supposed to change as ##(\cos(\theta/2),e^{i\phi}\sin(\theta/2))## and ##(-\sin(\theta/2),e^{i\phi}\cos(\theta/2))##.

    How can I prove that under SO(3) rotation, this is how the eigenvectors change?

    P.S. : I've written the eigenvectors as row vectors, but they should be column vectors.
     
  2. jcsd
  3. Nov 22, 2016 #2

    hilbert2

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  4. Nov 22, 2016 #3
    I was wondering if you could explain why a general rotation matrix is given by:

    ##\begin{pmatrix}
    -\sin(\theta/2) & \cos(\theta/2) \\
    e^{i\phi}\cos(\theta/2) & e^{i\phi}\sin(\theta/2)\\
    \end{pmatrix}##?
     
  5. Nov 22, 2016 #4

    hilbert2

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    Use the angles ##\theta## and ##\phi## to parametrize the 2x2 spin matrix ##S_u## (along axis u) and solve its eigenvectors. Both the 3D spatial rotation and the 2D state space rotation have two parameters in them.
     
  6. Nov 22, 2016 #5
    Some steps would be helpful.
     
  7. Nov 22, 2016 #6

    vanhees71

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    Note that the full rotation group is a three-parameter group. Most common are two ways to parametrize it. One is to use a vector ##\vec{\phi}## in a sphere of radius ##\pi## (with diametral points on the boundary identified). Then the spin-1/2 representation is given by
    $$\hat{D}(\vec{\phi})=\exp(-\mathrm{i} \vec{\phi} \cdot \hat{\vec{\sigma}}/2),$$
    where ##\hat{\vec{\sigma}}## are the Pauli matrices.

    The other way are Euler angles. In quantum theory one usually uses the variant, where you rotate around the 3- and the 2-axis:
    $$\hat{R}(\alpha,\beta,\gamma)=\hat{D}_3(\alpha) \hat{D}_2(\beta) \hat{D}_3(\gamma), \quad \alpha,\gamma \in [0,2 \pi[, \quad \beta \in [0,\pi[.$$
    The matrices ##\hat{D}_j## (##j \in \{1,2,3 \}##) are given by
    $$\hat{D}_j(\phi)=\exp(-\mathrm{i} \sigma_j \phi/2).$$
     
  8. Nov 22, 2016 #7

    hilbert2

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    You will find the matrices representing ##S_{x}, S_{y}## and ##S_z## from the site I linked to. Now create the spin operator ##S_u =
    \sin\theta\cos\phi S_x + \sin\theta\sin\phi S_y + \cos\theta S_z## and calculate its eigenvalues and eigenvectors with the usual method of characteristic polynomial, linear pair of equations...
     
    Last edited: Nov 22, 2016
  9. Nov 22, 2016 #8
    In my case, I have a two-sphere. So, I was wondering which Pauli matrices correspond to ##\phi## and ##\theta##.

    Also, what does it mean for the diametral points on the boundary to be identified?
     
  10. Nov 22, 2016 #9

    vanhees71

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    I don't know, what you want to achieve with your rotation. If you want a point on the two-sphere the meaning of the usual angles (polar and azimuthal) is to rotate the point on the northpole first by ##\theta## (##\theta \in (0,\pi)##) around the ##x## axis and then the result by ##\phi## (##\phi \in [0,2 \pi)##) around the (new) ##z## axis, i.e.,
    $$\hat{D}(\theta,\phi)=\hat{D}_3(\phi) \hat{D}_1(\theta).$$
     
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