3D unitary transformationby newshurik Tags: complex function, gellmann matrices, group theory, su3 

#1
Aug1711, 09:35 AM

P: 5

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
Aug1811, 03:20 PM

P: 563

Hi Sasha,
You might want to look up information on the SchrödingerBloch equation or Schrödinger equation associated with a Bloch sphere rotation. Here's a paper you might find useful: QuantumtoClassical Correspondence and HubbardStratonovich Dynamical Systems, a LieAlgebraic Approach  Victor Galitski http://arxiv.org/PS_cache/arxiv/pdf/...012.2873v2.pdf Also http://wwwbcf.usc.edu/~tbrun/Course/lecture05.pdf Don't miss the picture on page 12! 



#3
Aug2011, 06:29 PM

Sci Advisor
Thanks
P: 3,864

Does your wavefunction have a definite angular momentum, i.e. does it contain a spherical harmonic Y_{lm}? If so, you rotate it using a rotation matrix D^{l}_{mm'}. See a good book on Angular Momentum such as Edmonds.




#4
Aug2111, 04:14 PM

P: 178

3D unitary transformation
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 


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