Spin: why su(2) for an electron

[plmokn2]

Sorry if this is a silly question (I'm doing a fairly basic QM course but am doing a bit of extra reading for interest etc.), but why for an spin hbar/2 particle is the spin operators (the Pauli matricies) the generator of su(2)? I understand why generally generators act as conserved observables, and according to wikipedia su(2) is isomporphic to so(3) (which is plausible to me, even though the details are maybe beyond me, since both have the same number of independent variables and the generators commute in the same ways) but this doesn't explain why su(2) is used?

Any help appreciated.
Thanks

 

[Jhero]

Thank you for your question! It is not a silly question at all, and it's great that you are taking the initiative to do extra reading and expand your knowledge in quantum mechanics.

To answer your question, we must first understand what spin is in the context of quantum mechanics. Spin is an intrinsic property of particles, much like mass or charge. It is a quantum mechanical property that describes the angular momentum of a particle, even when it is at rest. In classical mechanics, angular momentum is a vector quantity, but in quantum mechanics, it is described by operators.

Now, let's talk about the generators of a group. A group is a mathematical object that describes a set of transformations that leave a system unchanged. In the case of spin, the group we are interested in is SU(2), which stands for special unitary group of order 2. This group is important because it describes the transformations that leave the spin of a particle unchanged.

The Pauli matrices, which are the spin operators, are the generators of SU(2). This means that they can generate all the transformations in the group. In other words, any rotation of a spin hbar/2 particle can be described by a combination of the Pauli matrices.

Now, you are correct in saying that SU(2) is isomorphic to SO(3), the special orthogonal group of order 3. This means that the two groups have the same algebraic structure and can be mapped onto each other. However, SU(2) is a more general group than SO(3) and includes all possible rotations in three-dimensional space, while SO(3) only includes proper rotations.

In summary, the reason why SU(2) is used in quantum mechanics is because it is the group that describes the transformations that leave the spin of a particle unchanged. The Pauli matrices, as the generators of SU(2), are the operators that describe these transformations. I hope this helps clarify your understanding. Keep up the good work in your studies!