3D unitary transformation

  1. Hello,

    I have a 3D complex wave function and I want to apply a unitary transformation to rotate it with respect to arbitrary axis.

    Anybody have any ideas how I can do that?

    Sasha
     
  2. jcsd
  3. Hi Sasha,

    You might want to look up information on the Schrödinger-Bloch equation or Schrödinger equation associated with a Bloch sphere rotation. Here's a paper you might find useful:

    Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical
    Systems, a Lie-Algebraic Approach - Victor Galitski

    http://arxiv.org/PS_cache/arxiv/pdf/1012/1012.2873v2.pdf

    Also
    http://www-bcf.usc.edu/~tbrun/Course/lecture05.pdf

    Don't miss the picture on page 12!
     
    Last edited: Aug 18, 2011
  4. Bill_K

    Bill_K 4,160
    Science Advisor

    Does your wavefunction have a definite angular momentum, i.e. does it contain a spherical harmonic Ylm? If so, you rotate it using a rotation matrix Dlmm'. See a good book on Angular Momentum such as Edmonds.
     
  5. Spatial rotations are generated by the angular momentum operators. So the general answer to your request is picking a representation of the angular momentum operators (Lx,Ly,Lz) and evaluating the operator exponential

    exp(i*n.L) = exp(i*(nx*Lx+ny*Ly+nz*Lz))

    for a vector (nx,ny,nz) that specifies the axis of rotation and the rotation angle by its magnitude.

    In an angular momentum eigenbasis that is aligned with n that unitary operator is diagonal. So you might find expanding in that basis to be simpler than evaluating the most general operator exponential.

    Cheers,

    Jazz
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook