Quantum numbers are a set of numerical values that describe the energy, angular momentum, and other properties of an atom or subatomic particle. SU(2) is a mathematical group that represents the symmetries of the quantum states of a system, including rotations of particles in space. Quantum numbers and SU(2) are related because the quantum numbers are used to label the quantum states of particles, and these states are transformed by the symmetries of SU(2).
In the SU(2) group, there are a total of three quantum numbers: spin, magnetic quantum number, and principal quantum number. These quantum numbers describe different aspects of the quantum state of a particle, such as its intrinsic angular momentum, its orientation in space, and its energy level.
The Pauli exclusion principle states that no two identical particles can occupy the same quantum state. Quantum numbers and SU(2) play a key role in determining the quantum states of particles, and therefore, they also play a role in enforcing the Pauli exclusion principle. Specifically, particles with different quantum numbers can occupy the same quantum state, but particles with the same quantum numbers cannot.
Quantum numbers and SU(2) can be used to describe and predict the behavior of particles within the framework of quantum mechanics. They help us understand how particles interact with each other and how they behave in different environments. However, the predictions made using these concepts are probabilistic in nature, as particles at the quantum scale do not follow deterministic laws.
SU(2) transformations are rotations in a 3-dimensional space, and they do not change the values of the quantum numbers themselves. However, they can change the quantum states of particles, and therefore, the quantum numbers associated with those states may change. This is because SU(2) transformations affect the orientation and energy of particles, which are described by the quantum numbers.