3D unitary transformation


by newshurik
Tags: complex function, gell-mann matrices, group theory, su3
newshurik
newshurik is offline
#1
Aug17-11, 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
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PhilDSP
PhilDSP is offline
#2
Aug18-11, 03:20 PM
P: 563
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
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#3
Aug20-11, 06:29 PM
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Thanks
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P: 3,864
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.

Jazzdude
Jazzdude is offline
#4
Aug21-11, 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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