3D unitary transformation

newshurik

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

PhilDSP

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!

Bill_K

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.

Jazzdude

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