## 3D unitary transformation

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
 PhysOrg.com physics news on PhysOrg.com >> Study provides better understanding of water's freezing behavior at nanoscale>> Soft matter offers new ways to study how ordered materials arrange themselves>> Making quantum encryption practical
 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/...012.2873v2.pdf Also http://www-bcf.usc.edu/~tbrun/Course/lecture05.pdf Don't miss the picture on page 12!
 Blog Entries: 1 Recognitions: 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.

## 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

 Tags complex function, gell-mann matrices, group theory, su3