The Wigner-Eckart Theorem
is a formula that tells us about all "simple constraints" that group theory - the mathematical incarnation of the wisdom about symmetries, especially in the $SO(3)\approx SU(2)$ case (rotations in a three-dimensional space) - implies about matrix elements of tensor operators - those that transform as some representation of the same symmetry.
The dependence on the indices labeling basis vectors of the representations - $m_1, m_2, m_3$ in the $SU(2)$ case - is totally determined. Instead of thinking that some matrix elements depend on many variables, physicists may realize that the symmetry guarantees that the matrix elements only depend on a few labels labeling the whole "multiplets" of the states and operators rather than on all the labels identifying the individual components. It's always critically important to know how much freedom or how much uncertainty there is about some observables - for example, experimenters don't want to repeat their experiment $(2J+1)^3$ times without a good reason - and we would get a totally wrong idea without this theorem.
To see that the theorem is used all the time, check e.g. these 5700 papers
many of which are highly cited ones. The topics of the papers include optics, nanotubes, X-rays, spectroscopy, condensed matter physics, mathematical physics involving integrable systems, quantum chemistry, nuclear physics, and virtually all other branches of physics that depend on quantum mechanics. In many other cases, the theorem is being used without mentioning its name, and its generalizations are being used all the time in advanced theoretical physics, e.g. in the contexts with groups that are much more complicated than $SO(3)\approx SU(2)$.
It's interesting to note that in some sense, quantum mechanics allows the symmetry to impose as many constraints as classical physics. In classical physics, we could start with an initial state labeled by numbers $I_i$, apply some operations depending on parameters $O_j$, and we would obtain a final state described by parameters $F_k$. Classical physics would tell us Yes/No - whether we can get from $I_i$ via $O_j$ to $F_k$: that's the counterpart of the quantum probability amplitude.
The rotational $SO(3)$ symmetry - which has 3 parameters (3 independent rotations - or the latitude; longitude of the axis, and the angle) would only tell us that if we rotate all objects $I_i,O_j,F_k$ by the same rotation, we obtain a valid proposition again (Yes goes to Yes, No goes to No). So the dependence on 3 parameters - corresponding to the 3 rotations - is eliminated. In quantum mechanics, we also eliminate the dependence on 3 parameters - in this case $m_1,m_2,m_3$, the projections $j_z$ for the two state vectors and for the operator sandwiched in between them. In some proper counting, this is true for any $d$-dimensional group of symmetries, I think.
This post imported from StackExchange Physics at 2014-04-01 05:46 (UCT), posted by SE-user Luboš Motl