Pauli Matrices

Author

Eric W. Weisstein
September 14, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/LinearAlgebra/PauliMatrices.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PauliMatrices.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Usage

?Pauli*

Notation

PauliMatrix[k]//TraditionalForm

TraditionalForm/@PauliMatrix/@Range[0,3]//TraditionalForm

Definition

σ[0]={{1,0},{0,1}};
σ[1]={{0,1},{1,0}};
σ[2]={{0,-I},{I,0}};
σ[3]={{1,0},{0,-1}};

Identities

p=Tuples[Range[3],2]

((Subscript[σ, i])^2)=I

Union[σ[#].σ[#]&/@Range[0,3]]

FunctionExpand[PauliMatrix[k].PauliMatrix[k],0≤k≤3&&k∈Integers]

FullSimplify[PauliMatrix[k].PauliMatrix[k],0≤k≤3&&k∈Integers]

(Subscript[σ, i])(Subscript[σ, j])+(Subscript[σ, j])(Subscript[σ, i])=2(Subscript[δ, i,j])I

((σ[#1].σ[#2]+σ[#2].σ[#1]-2IdentityMatrix[2]KroneckerDelta[#1,#2])&@@@p)

(Subscript[σ, i])(Subscript[σ, j])=I (Subscript[δ, i,j])+(Subscript[ϵ, i,j,k])i (Subscript[σ, k])

((σ[#1].σ[#2]- I Plus@@Table[Signature[Join[{##},{k}]] σ[k],{k,3}]-  IdentityMatrix[2] KroneckerDelta[#1,#2])&@@@p)