# Solving schrodinger equation for quarkonium

1. Jul 2, 2011

### nasibaba

I've tried to solve Schrodinger equation for Charmonium and Bottomonium and there are some problems with that :

1. As we know the Schrodinger equation is independent from quantum number "ml" so there would be same Energy for different "ml" for a specific n. But what we see in PDG book is some how confusing, according to this book the mass of states with different "ml" are not equal !!!
Why this happens and what does it mean ?

2.How can I do the same with dirac equation?! Is there any specific code ?

thanx

2. Jul 2, 2011

### Bill_K

The different energy levels of quarkonium are due to spin-orbit coupling. The two quark spin 1/2's can couple two ways: S = 0 and S = 1. If for example we take L = 1, the different energy levels will be 1P1 for S = 0, and 3P0, 3P1 and 3P2 for S = 1, J = 1, 2, 3.

3. Jul 2, 2011

### nasibaba

First, thank you bill for answering me

I know what you've said but they are eigenvalues !!!
So there wouldn't be any difference in the mass !

Or I'm wrong :(

Last edited: Jul 2, 2011
4. Jul 3, 2011

### Bill_K

L and S have definite values for each state, since they commute with the Hamiltonian. That just means you can label the states according to the eigenvalues of L and S. However since the Hamiltonian depends on S1·S2 and L·S, each state has a different energy. This is analogous to the line splitting you find in other systems, like atoms and also in positronium.

5. Jul 3, 2011

### nasibaba

oh, I got it :)

really really thank you