Why we need $SU(2)$ symmetry? When we use it?

Question

I am trying to learn Quantum mechanics and I am familiar with Pauli matrice but not with group theory. I want to understand SU2 symmetry in common language. When we talk about Pauli matrix x we simply say that it flips the spin. What SU2 symmetry does?What is the advantage of using SU2 symmetry? Can someone please explain this in simple language?

Answer 1

The Pauli matrices turn up in a number of places. They're basically the Lie algebra of SO(3), the 3d rotation group. They're useful in QM to describe spin.

$SU(2)$ is isomorphic to $Spin(3)$ which is the double cover of $SO(2)$. This is helpful in describing 2-component spinors.

You might find it worthwhile working through Shankars book QM as he approaches the subject in a fairly straight-forward and transparent way.