Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also the case that these matrices (for all positive $j$ multiple of $1/2$) are a representation of the rotation group $SU(2)$, which is a double cover of $SO(3)$.

Now, I can see this for the specific case $j=1$ (3x3 matrices representing rotations in 3-dimensional Euclidean space), but what does it mean for other values of $j$? Say for $j=3/2$, does that represent the subgroup of 3-dimensional rotations given some parameter, in a 4-dimensional ambient geometry?

Thanks.

share|improve this question

1 Answer 1

Spherical harmonics form a basis of solutions of Laplace's equation, and (three-dimmensional in this case) Laplacian is invariant under rotations, so if f(x,y,z) is a solution to this equation, then g(x,y,z) = f( R(x,y,z : a,b,c) from SO(3)) So, if you have a soution (or a certain quantum-mechanical wave function of some spherically symmetric system) in the basis of spherical harmonics, you can calculate, how its changes in rotated system (using Wigner D-matrix, which is exactly the representation of SO(3) in the basis of spherical harmonics). This is for SO(3) and integer j values (related to the angular momentum of the system).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.