How Are Spin, SO(3), and SU(2) Related in Quantum Mechanics?

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SUMMARY

The discussion centers on the relationship between spin, SO(3), and SU(2) in quantum mechanics, specifically addressing the homomorphism between the groups and the role of Pauli matrices. It is established that the Pauli matrices serve as generators for the SU(2) group, which correlates with infinitesimal rotations and conservation laws related to spin. The conversation also questions the conventional presentation of these concepts in literature, suggesting that an isomorphism between the generators of SO(3) and SU(2) may exist. Additionally, the distinction between orbital angular momentum and spin is clarified, emphasizing that both are conserved quantities arising from different aspects of angular momentum in quantum systems.

PREREQUISITES
  • Understanding of group theory, particularly the concepts of homomorphism and isomorphism.
  • Familiarity with quantum mechanics, specifically angular momentum and spin.
  • Knowledge of SU(2) and SO(3) groups and their representations.
  • Proficiency in the use of Pauli matrices and their role in quantum mechanics.
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  • Study the relationship between SU(2) and SO(3) in detail, focusing on their homomorphic properties.
  • Explore the implications of the Pauli matrices as generators of rotations in quantum mechanics.
  • Research the conservation laws associated with spin and orbital angular momentum in quantum systems.
  • Examine literature on the mathematical foundations of angular momentum in quantum mechanics, including texts by Sakurai.
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Quantum physicists, students of quantum mechanics, and researchers interested in the mathematical foundations of spin and angular momentum in quantum systems.

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I’ve got a couple of conceptual questions on spin etc, and any help would be appreciated.

First of all reading books (eg. Sakuri) it seems like authors tend to show there’s a homomorphism between the groups SO(3) and SU(2) using Euler angles etc. I know the Pauli matricies act as generators for the group SU(2), so does this and the homomorphism automatically mean the Pauli matricies can be considered to be generators of infinitesimal rotations and so lead to a conservation law (spin)? Also why do books seem to do it this way around rather than showing an isomorphism between the generators of SO(3) and SU(2) (which I suspect exists)?

One more thing, if the Pauli matricies do act as generators of rotations why do we get both orbital angular momentum and spin, when it seems like there’s only really one unique set of generators, so why do we have the two conserved quantities: spin and orbital angular momentum?

Thanks in advance.
 
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