Dear students,
on friday I noticed a few question marks on your faces, so I suspect my lecture was not sufficiently clear, especially regarding the division of the matrix elements in total isospin values. I gave for granted the consequences of commuting operators on the vector spaces, but since that may be unclear, I prepare a little integration. You can find it in the lecture notes for Module 3
https://drive.google.com/open?id=1xvkPtvCdWObHfQaCOf7D1TZSuIQRG5xm&usp=drive_fsI also added a plot of the pi proton cross section that I did not have time to show. We will discuss it later in the course, but it may be useful to take a look. The value of the 1/2 and 3/2 matrix elements change with energy following the dynamics of the strong interactions Hamiltonian. If for some reason (like the Delta resonance) there is only the I=3/2 isospin state selected, then we can numerically estimated the cross section ratios for the pi+ and pi- channels.
https://drive.google.com/open?id=1xvHFXSq1bh8CgN_CSRh5fmbd5gNgRz4u&usp=drive_fs
Ciao,
FF