Proton, neutron, and pi mesons

Question:
Consider a proton, a neutron, and pi mesons. These are also called hadrons which are constructed with quarks. Particularly, these particles consist of up and down quarks. Confirm the structure of each particle and its charge.

Answer:
The up quark has $+\frac{2}{3}e$ as the electric charge. The down quark has $-\frac{1}{3}e$. Protons and neutrons are specifically called baryons which consist of three quarks. A proton contains uud quarks. The charge will be
\[
\mathrm{charge \ of \ proton} = \frac{2}{3}e+\frac{2}{3}e-\frac{1}{3}e = +e
\]
which is experimentally proven.
Likewise, we can find the charge of a neutron (udd).
\[
\mathrm{charge \ of \ neutron} = \frac{2}{3}e-\frac{1}{3}e-\frac{1}{3}e = 0
\]
which is consistent with the experimental result.
Now, mesons are made of a quark and an anti-quark, and note that an anti-quark gets the opposite charge to its particle. The pion has three states:
\begin{eqnarray*}
\pi^+ &=& u\overline{d} = \frac{2}{3}e+\frac{1}{3}e = +e  \\
\pi^- &=& \overline{u}d = -\frac{2}{3}e-\frac{1}{3}e = -e  \\
\pi^0 &=& \frac{u\overline{u} - d\overline{d}}{\sqrt{2}} = 0
\end{eqnarray*}
Notice that $\pi^-$ is the anti-particle of $\pi^+$.

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