Pauli Spin Matrices

 

1/3

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 σ_iσ_l-σ_jσ_i= ⅈ ϵ_ijkσ_k and σ_iσ_l+σ_jσ_i=2δ_ij{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.

• Contributed by: S. M. Blinder

SNAPSHOTS

DETAILS

Snapshots 1, 2: You can derive the commutation relations for the ladder operators and .

Snapshots 3: The Pauli matrices mutually anticommute.

RELATED LINKS

Pauli Matrices (Wolfram MathWorld)

PERMANENT CITATION

 S. M. Blinder

"Pauli Spin Matrices"
 http://demonstrations.wolfram.com/PauliSpinMatrices/
 Wolfram Demonstrations Project
 Published: March 7, 2011

 

https://demonstrations.wolfram.com/PauliSpinMatrices/