PHYS598MAP Spring 2005 Blog

1. AMO physicists are not consistent when it comes to units. That's OK, as long as you have a conversion table and some idea of what's going on in terms of the physics.

2. Angular momentum in quantum mechanics is all about commutation relations and rotations. Any operator that satisfies the right commutation relations will have eigenkets that rotate like vectors.

3. You can rotate quantum mechanical states using rotation operators. You just need the right matrix elements to write the rotated state in terms of the unrotated basis states. You can find the non-trivial matrix elements of the rotation operator on the Clebsch-Gordan handout.

4. Addition of two angular momenta J1 and J2 produces new states that are eigenkets of J², J1², J2² and J_z. These states |j,m> can be written in terms of the |j1,m1> and |j2,m2> states using Clebsch-Gordan coefficients. The quantum number j takes values that differ by integers between j1+j2 and |j1-j2|. It's always true that m=m1+m2. This means that |j,m> is only made up of states |j1,m1> and |j2,m2> where m=m1+m2. I didn't emphasize this point or the notion of "stretched states" enough in class. We'll briefly go over those ideas on Tuesday. 3-j symbols are Clebsch-Gordan coefficients in disguise.

5. Three angular momenta can be added in different ways. The different representations that you end up with can be connected by 6-j symbols, which we will see in other contexts.

   
   

I flubbed the right explanation of what's happening is real space for the problem that we did at the end of class (the decay problem). We'll go over the right picture on Tuesday, as well as talk about the meaning of parity in that problem.

On Tuesday we'll continue with spherical tensors and the Wigner-Eckart theorem. I think that we'll also be able to get to Hydrogen.

I'll send another email when pre-flight set 3 are ready. Lecture notes will be on the website shortly.

Cheers,

Brian