Pauli Spin Matrices

The Pauli spin matrices , , and represent the intrinsic angular momentum components of spin- particles in quantum mechanics. Their matrix products are given by , where I is the 2×2 identity matrix, O is the 2×2 zero matrix and is the Levi-Civita permutation symbol. These products lead to the commutation and anticommutation relations and . The Pauli matrices transform as a 3-dimensional pseudovector (axial vector) related to the angular-momentum operators for spin- by . These, in turn, obey the canonical commutation relations . The three Pauli spin matrices are generators for the Lie group SU(2).
In this Demonstration, you can display the products, commutators, or anticommutators of any two Pauli matrices. It is instructive to explore the combinations that represent spin-ladder operators.

SNAPSHOTS

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DETAILS

Snapshots 1, 2: you can derive the commutation relations for the ladder operators and
Snapshot 3: the Pauli matrices mutually anticommute
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