How Are Isospin Values Determined?

[jeebs]

Hi,
I'm having a little bit of trouble with isospin. What I have gathered so far is that say, for the proton and neutron, apart from their charge they are very similar, so they are considered to be different states of the same particle called the nucleon.

So, we say the nucleon has an isospin of I=1/2, and the proton has third component I₃=+1/2, and the neutron has third component I₃=-1/2, right? And we say the proton and neutron form the nucleon multiplet, right?

Also, there are other multiplets like the pions. the pion multiplet (which has I=1) consists of \pi^+,\pi^0 and \pi^-, which have I₃=+1, 0 and -1 respectively.

What I don't get is where these numbers actually come from. How have the values for I assigned to each multiplet been worked out?

And am I right in thinking that the I₃ values are arbitrary, for example, there was no reason to give protons an I₃ value of +1/2 instead of -1/2, its just something that was decided on?

Thanks.

 

[Redbelly98]

The isospin values are chosen to give the correct number of states.

Eg., "proton" and "neutron" are 2 states. We know that a spin=1/2 particle also has two states, +1/2 and -1/2. So we use I=1/2 for the isospin.

Similarly for the pion system and a spin=1 particle, which both have 3 possible states.

And am I right in thinking that the I₃ values are arbitrary, for example, there was no reason to give protons an I₃ value of +1/2 instead of -1/2, its just something that was decided on?

I suspect that is the case. It may be the convention to rank particles according to charge when assigning I₃ values.