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# The Pauli Matrices

The Pauli Matrices are a set of 2×2 matrices which are indispensible in quantum mechanics. They are as follows:

σ1 = (
 0 1 1 0
)
σ2 = (
 0 -i i 0
)
σ3 = (
 1 0 0 -1
)

They may also be expressed as components of the vector σ as follows:

σ = (σ1, σ2, σ3)

You might also see them referred to as σx, σy and σz. These are exactly the same as σ1, σ2 and σ3, it's simply a different convention for labelling the different components of the vector σ.

## Algebraic Properties

Warning: This section is not written in Plain English!

The Pauli matrices are hermitian and unitary.
They have the following commutation and anti-commutation relations:

 [σi, σj] = 2iεijk {σi, σj} = 2δij

Putting these relations together gives us an equation of what happens when we multiply any two Pauli matrices together:

σi σj  =  iεijkδij

 Q Wait! I don't quite see where you got this equation from! A I took the commutation [  ,  ] relation, the anti-commutation relation {  ,  }, added them together then divided by 2.Why? Because in general for two symbols A and B; commutation means  [A, B] = AB - BA,and the anti-commutaion means: {A, B} = AB + BA.So [A, B] + {A, B} = AB - BA + AB + BA = 2ABDividing this by 2 gives AB.

The Pauli Matrices