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
Hi Sasha, You might want to look up information on the SchrÃ¶dinger-Bloch equation or SchrÃ¶dinger 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!
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.
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