The Quark Model — Question 5

Question 5

A normalized meson color singlet is a coherent superposition

Use orthonormal color states $|r\bar r\rangle,|g\bar g\rangle,|b\bar b\rangle$. A color singlet is $|1\rangle=A(|r\bar r\rangle+|g\bar g\rangle+|b\bar b\rangle)$ with real positive A.

1. Normalize the state and calculate the weight of each basis component.

2. Find its overlap with $(|r\bar r\rangle-|g\bar g\rangle)/\sqrt2$ and explain why the singlet is not a single fixed-color pair.

Question 5 - Solution

A normalized meson color singlet is a coherent superposition

(a) Method and calculation

$$1=\langle1|1\rangle=3A^2\Rightarrow\boxed{A=1/\sqrt3}.$$$$\boxed{|\langle r\bar r|1\rangle|^2=|\langle g\bar g|1\rangle|^2=|\langle b\bar b|1\rangle|^2=1/3}.$$

(b) Interpretation and further calculation

$$\boxed{\langle8|1\rangle=(1-1)/\sqrt6=0}.$$$$|r\bar r\rangle=\frac1{\sqrt3}|1\rangle+\text{nonsinglet components}.$$

Physical check

Color labels are internal quantum numbers, not visible colors. The basis-component weights are a formal decomposition, not freely isolated colored-particle probabilities.