The Eightfold Way — Question 9

Question 9

An isospin ladder distinguishes the two central octet states

Use dimensionless isospin operators $I_+|I,m\rangle=\sqrt{I(I+1)-m(m+1)}|I,m+1\rangle$. Sigma zero has I=1,m=0; Lambda has I=0,m=0.

1. Apply the raising operator to both states.

2. Find the squared ladder matrix element for Sigma zero, and explain why it is not itself a probability.

Question 9 - Solution

An isospin ladder distinguishes the two central octet states

(a) Method and calculation

$$\boxed{I_+|\Sigma^0\rangle=\sqrt2\,|\Sigma^+\rangle}.$$$$\boxed{I_+|\Lambda\rangle=0}.$$

(b) Interpretation and further calculation

$$\boxed{|\langle\Sigma^+|I_+|\Sigma^0\rangle|^2=2}.$$$$P=\big|\langle f|U(t)|i\rangle\big|^2\le1,\quad I_+\ne U(t).$$

Physical check

A ladder operator is not a normalized time-evolution operator. Interaction strength and duration are needed to calculate transition probabilities.