Solving schrodinger equation for quarkonium

[nasibaba]

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

1. As we know the Schrödinger 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

 

[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.

 

[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 :(

 

[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.

 

[nasibaba]

oh, I got it :)

really really thank you