Les commutateurs obtenus dans l'équation (4.26) sont les suivants :

`color(blue) ("["sigma_x,sigma_y"]"=2"i"sigma_z)`

`color(#6ace3b) ("["sigma_y,sigma_z"]"=2"i"sigma_x)`

`color(chocolate) ("["sigma_z,sigma_x"]"=2"i"sigma_y)`

en partant des matrices de Pauli :

`sigma_x = [(0, 1),(1,0)], sigma_y = [(0, -"i"),("i",0)], sigma_z = [(1, 0),(0,-1)] `

On obtient donc :

`color(blue) ("["sigma_x,sigma_y"]")=sigma_x sigma_y-sigma_y sigma_x `

`= [(0, 1),(1,0)][(0, -"i"),("i",0)] - [(0, -"i"),("i",0)][(0, 1),(1,0)] `

`= [("i",0),(0, -"i")] - [(-"i", 0),(0, "i")] = [(2"i", 0),(0,-2"i")]`

`=2"i" [(1, 0),(0,-1)]`

`color(blue) (=2"i"sigma_z)`

`color(#6ace3b)("["sigma_y,sigma_z"]")=sigma_y sigma_z - sigma_z sigma_y `

`= [(0, -"i"),("i",0)][(1, 0),(0,-1)] - [(1, 0),(0,-1)][(0, -"i"),("i",0)] `

`= [(0, "i"),("i", 0)] - [(0, -"i"),(-"i",0 )] = [(0, 2"i"),(2"i", 0)]`

`=2"i" [(0, 1),(1, 0)]`

`color(#6ace3b) (=2"i"sigma_x)`

`color(chocolate)("["sigma_z,sigma_x"]")=sigma_z sigma_x - sigma_x sigma_z `

`= [(1, 0),(0,-1)][(0, 1),(1,0)] - [(0, 1),(1,0)][(1, 0),(0,-1)] `

`= [(0, 1),(-1, 0)] - [(0, -1),(1,0 )] = [(0, 2),(-2, 0)]`

`=2"i" [(0, -"i"),("i", 0)]`

`color(chocolate) (=2"i"sigma_y)`

les trois résultats que l'on voulait démontrer.