Spin Matrix for Arbitrary Spin: A Closer Look

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SUMMARY

The spin matrix S_i, defined as h-bar divided by 2π times the i-th Pauli spin matrix, is specifically applicable to spin-1/2 systems. For constructing spin matrices for arbitrary spins, one must utilize unitary irreducible representations of the rotation group. For instance, to create spin matrices for spin-1, a 3-dimensional irreducible representation is necessary, which involves three Hermitian generators S_i corresponding to the spin values.

PREREQUISITES
  • Understanding of Pauli spin matrices
  • Familiarity with quantum mechanics concepts
  • Knowledge of unitary irreducible representations
  • Basic grasp of the rotation group in physics
NEXT STEPS
  • Study the derivation of Pauli matrices in quantum mechanics
  • Learn about unitary irreducible representations of the rotation group
  • Explore the construction of spin matrices for higher spins
  • Investigate Hermitian operators in quantum mechanics
USEFUL FOR

Physicists, quantum mechanics students, and researchers interested in the mathematical formulation of spin and its representations in quantum theory.

captain
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i am having trouble understanding if the spin matrix S_i (the i th compenent of the spin matrix which is h-bar divided by 2pi times i th pauli spin matrix) is only for spin 1/2 and if it is then what would be the spin matrix for an arbitrary spin?
 
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If you know how the pauli matrices are derived, I am sure you can figure out how to construct spin matrices for higher spin.
 
captain said:
i am having trouble understanding if the spin matrix S_i (the i th compenent of the spin matrix which is h-bar divided by 2pi times i th pauli spin matrix) is only for spin 1/2 and if it is then what would be the spin matrix for an arbitrary spin?

A hint: particles of different spin are closely associated with unitary irreducible representations of the rotation group. There are representations of integer dimensions: 1,2,3,4,... which correspond to spin values 0, 1/2, 1, 3/2,... So, in order to build spin matrices for spin 1 you can build a 3-dimensional irreducible unitary representation of the rotation group. Then 3 Hermitean generators S_i of this representation will be the spin matrices you are interested in.

Eugene.
 

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