The Pauli exclusion principle is a fundamental principle in quantum mechanics that states that no two identical fermions can occupy the same quantum state simultaneously. Fermions are particles that have half-integer spin, such as electrons, protons, and quarks.
In the case of a proton, which consists of three quarks (two up quarks and one down quark), the Pauli exclusion principle plays a crucial role in determining the overall properties of the system. Quarks are also fermions and are subject to the exclusion principle.
The Pauli exclusion principle applies to all the intrinsic properties of fermions, including their spin and their quantum numbers. While the up quarks in a proton have the same charge and the down quark has a different charge, they still have other quantum numbers, such as spin, that need to be considered.
Each quark in a proton must occupy a different quantum state, which includes not only their charge but also their spin. The Pauli exclusion principle dictates that no two identical fermions can occupy the same quantum state simultaneously, and this applies to both the charge and the spin states.
Therefore, even though the up quarks in a proton have the same charge, their spins must be different in order to satisfy the Pauli exclusion principle. Similarly, the down quark, which has a different charge and is a different particle, must also occupy a distinct quantum state with a different spin.
Overall, the combination of different charges and different spin states for the quarks in a proton allows them to satisfy the Pauli exclusion principle and form a stable, composite particle.