Analysis of the Isospin of meson and baryon bounded states (particle physics)

[jonjacson]

Homework Statement

Part A)

Establish which of the following combinations of particles can exist in a state of I=1 :

a) [tex]\pi^0\pi^0[/tex]
b) [tex]\pi^+\pi^-[/tex]
c) [tex]\pi^+\pi^+[/tex]
d) [tex]\Sigma^0\pi^0[/tex]
e) [tex]\Lambda^0\pi^0[/tex]

Part B) of the problem is:

In what states of isospin may exist the following systems?

f) [tex]\pi^+\pi^-\pi^0[/tex]

g) [tex]\pi^0\pi^0\pi^0[/tex]

Homework Equations

Conservation of the Isospin.

Addition rules of angular momenta (the same rules of the Isospin)

The Attempt at a Solution

Well I have searched which are the values of the Isospin of the particles, and this is what i have found:

- I=1 for the pions

- I=1 for the sigma barion

- I=0 for the lambda baryon

Using the addition rules of the angular momenta in every case give me this result:

J = j1+j2 , j1+j2-1, j1+j2-2 ... /j1-j2/ ; in the case of j1 > j2

a) I=1, I=1, so the total Isospin of the coupled state could be I=2,1,0 , and this three states of total Isospin are including 1 of the statement, so this combination can exist.

b) c) d) are the same case of the a).

e) Now we have I=0 for the lambda baryon and I=1 for the pion so:

Total I= 1, 0 and the state is posible again because I=1 is a posible value. It is very strange that the first statement of the problem talks about "wich of the following..." in singular, but ¿does it mean that there is only one answer?.

Surprisingly i have found that all the states are possible, and this is worrying me.

For the part B) we have:

f)

[tex]\pi^+\pi^-\pi^0[/tex] , all of them have I=1 , I have used the results of part A) and then i have added the Isospin of the third particle.

So for the [tex]\pi^+\pi^-[/tex] , I have I= 2, 1 ,0 , let's analyze the three cases:

I=2, the pi0 isospin is I=1 so : I total= 3,2,1

I=1, and I=1 so I total=2,1,0

I=0, and I=1 so I total = 1, 0

But the Isospin must be conserved, so the only possibility is I total = 3.

¿is this right?g)

If i am not wrong is the same case as f) .

But this looks very strange, i don't understand why there are a lot of parts similar, I am almost convinced that i am doing something wrong.

¿have I applied correctly the theory of the angular momentum? ¿Should I calculate the eigenstates, eigenvalues... with clebsh gordan coefficients?.

I have to send some problems (one of them is the one of this thread) to my teacher and it will be the 15 % of the value of the exam, so it is very important for me, if I am rigth ¿Could somebody confirm it?.

Thanks.

 

[jonjacson]

Now i understand the mistake in part A) I had forgotten conservation of the spin in a)b)c)d) , these combinations can only have I=2 due to conservation, and the correct answer of the problem is part e) .

It's much more logical.

¿What do you think?.

 

[jonjacson]

Hello Is there anybody here?

 

[jonjacson]

There aren't answers in this thread ¿Can anybody explain to me why? . I think that I have followed the rules of the forum, perhaps I am doing something wrong ¿Could you tell me what it is?.

Thanks in advance, I am very sorry for this.

 

[paranormal]

Dear student,

Thank you for your thorough analysis of the problem. It seems like you have correctly applied the theory of angular momentum and the conservation of isospin. Your approach of using the addition rules of angular momentum to determine the possible values of total isospin is correct.

For part A, you have correctly determined that all combinations of particles can exist in a state of I=1. This is because the pions have an isospin of I=1 and the sigma baryon has an isospin of I=1, so any combination of these particles will result in a total isospin of I=1.

For part B, your analysis is also correct. The possible values of total isospin for the system \pi^+\pi^-\pi^0 are I=3,2,1,0, and the only possible value that conserves isospin is I=3. Similarly, for the system \pi^0\pi^0\pi^0, the possible values of total isospin are I=3,2,1,0, and the only possible value that conserves isospin is I=3.

Overall, your approach and analysis seem correct. However, it is always a good idea to double-check your calculations and seek feedback from your teacher to ensure that you have fully understood the concepts and applied them correctly. I wish you all the best for your exam.

Best regards,
[Your name]