Pauli Spin Matrices

The Pauli spin matrices , and are central to the representation of spin-particles in quantum mechanics. Their matrix products are given by =where I is the 2⨯2 identity matrix and , the Levi-Civita permutation symbol. These products lead to the commutation and anticommutation relations σiσljσi= ⅈ ϵijkσk and σiσljσi=2δij{I}, respectively. The Pauli matrices transform as a 3-dimensional pseuodovector (axial vector) related to the angular-momentum operators for spin- by . These, in turn, obey the canonical commutation relations .
The three Pauli spin matrices, along with the unit matrix I, 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 , which represent spin-ladder operators.

SNAPSHOTS

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DETAILS

Snapshots 1, 2: You can derive the commutation relations for the ladder operators and .
Snapshots 3: The Pauli matrices mutually anticommute.
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