How to find the decay rate of processes: a→ b+c

[Jamiemma1995]

Homework Statement

Show using isospin symmetry that the probabilities of the decays ∆+ → π^+n and ∆+ → π_0p are expected to be in the ratio

Γ(∆+ → π^+n)/ Γ(∆+ → π_0p)=1 /2

I unfortunately missed the classes that covered this last week due to severe health problems , I don't want anyone to solve it for me I want to learn , but my problem is I don't know how to find the decay rate of processes. Could anyone please explain the method, and give the relevant equations and then I can work it out for myself .

Relevant Equations

N.A.

As mentioned above I'm unable to begin because I don't know the relevant equations or method.

 

[samalkhaiat]

You need the following equations for the relevant iso-spin states:
$$|\pi^{+}n\rangle \equiv |1, 1 \rangle |\frac{1}{2} , - \frac{1}{2} \rangle = \sqrt {\frac{1}{3}} \ | \frac{3}{2} , \frac{1}{2} \rangle + \sqrt{\frac{2}{3}} \ | \frac{1}{2} , \frac{1}{2} \rangle ,$$$$|\pi^{0}p \rangle \equiv |1 , 0 \rangle |\frac{1}{2} , + \frac{1}{2} \rangle = \sqrt{\frac{2}{3}} \ |\frac{3}{2} , \frac{1}{2} \rangle - \sqrt{\frac{1}{3}} \ |\frac{1}{2} , \frac{1}{2}\rangle .$$ Now, solve these equations for $|\Delta^{+} \rangle \equiv |\frac{3}{2} , \frac{1}{2} \rangle$ and obtain equation of the form $$|\Delta^{+} \rangle = a \ |\pi^{+}n\rangle + b \ |\pi^{0}p \rangle .$$ Make sure that $|a|^{2} + |b|^{2} = 1$. Now $$\frac{\Gamma ( \Delta^{+} \to \pi^{+} n )}{ \Gamma ( \Delta^{+} \to \pi^{0} p )} = \frac{|a|^{2}}{|b|^{2}} .$$

 

[Orodruin]

Make sure that $|a|^{2} + |b|^{2} = 1$.

Normalising the state is generally a good advice, but not that relevant here since the sought result is a decay rate ratio.

 

[vela]

Could anyone please explain the method, and give the relevant equations and then I can work it out for myself?

Don't you have a textbook? I'm sure it explains how to do this kind of calculation as well as giving examples.

 

[samalkhaiat]

Normalising the state is generally a good advice, but not that relevant here since the sought result is a decay rate ratio.

The advice was about solving the equations correctly. Normalization is not the issue here, $10^{23}|\Delta^{+}\rangle$ and $|\Delta^{+}\rangle$ lead to the same ratio. However, obtaining $a^{2} + b^{2} = 1$ means that his solution is definitely correct.

 

[Orodruin]

The advice was about solving the equations correctly. Normalization is not the issue here, $10^{23}|\Delta^{+}\rangle$ and $|\Delta^{+}\rangle$ lead to the same ratio. However, obtaining $a^{2} + b^{2} = 1$ means that his solution is definitely correct.

It is correct either way if you include the argument with the ratio. In fact, it is a very common way and important tool when solving for ratios to only care about proportionality and not the constant in front.