PHYS598MAP Spring 2005 Blog

1. The spherical basis.

2. Irreducible tensors. These are operators that transform like spherical harmonics under rotations. They have a rank k, which is like l, and components q, which are like m. We're mostly going to see vector operators in this class, which are rank 1 tensors. We'll also see rank 0 tensors, or scalars, wich are kind of boring.

3. The Wigner-Eckart theorem (WET). I regret bringing this up before we're going to use it for something. Regardless, it is extremely important in many areas of physics. It tells you that the matrix element (between states with angular momentum quantum numbers j and j') of the q component of a rank k vector is proportional to:

   
   

   • the Clebsch-Gordan coefficient for adding j and k to get j'

     
     

   • the reduced matrix element, which is independent of m, m', and q

     
     

   We talked about selection rules based on the Clebsch-Gordan coefficient in the WET for vector operators. There was some confusion (caused by me) over the so-called "parity rule." This is indeed, as Shizhong stated, really a rule that comes from direct evaulation of the matrix element of a vector operator with respect to real-space angular momentum states. I will clear this up at the beginning of next class.

4. The projection theorem. Ellen — what I wrote down in class was correct. This tells you that matrix elements of any vector operator (for states of well-defined angular momentum) are proportional to the matrix elements of the angular momentum operator. We will see how this is useful in a homework problem on the Zeeman effect in Hydrogen.

Lecture notes on angular momentum and the WET are on the website now. I have cleaned up my notation, and standardized to Sakurai's notation (which is just as confusing as anyone else's notation). There are some good tricks in the lecture notes for evaluating matrix elements of tensor products.

Learn to ignore notation! It's a red herring that will prevent you from learning physics. In real life, no one uses the same notation for anything. We'll see in class that people in my field can not even agree to within a factor of 2 on the definition of the Rabi rate for transitions in a two-level system. Also, once your brain starts to decay as you age past 30, you also won't be able to remember the details behind subjects like irreducible tensors. Unless you're Julian Schwinger, you'll just remember the important ideas. Then you look up details in a book when you want to calculate something.

See you and Hydrogen tomorrow!

Brian DeMarco