Answer to Question #146337 in Field Theory for Afraid

Question #146337
Write properties of pauli matrices.
1
Expert's answer
2020-11-27T07:19:36-0500

First let's write what are Pauli matrices :

"\\sigma_1 = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}, \\sigma_2=\\begin{pmatrix} 0 & -i \\\\ i & 0 \\end{pmatrix}, \\sigma_3= \\begin{pmatrix} 1 & 0 \\\\0 & -1 \\end{pmatrix}"

The main properties are :

  • All three are Hermitian and unitary matrices.
  • All of them have eigenvalues 1 and -1
  • By "[A,B]" I note the commutator of "A" and "B" :

"[\\sigma_1,\\sigma_2]=\\sigma_3; [\\sigma_2,\\sigma_3]=\\sigma_1; [\\sigma_3,\\sigma_1]=\\sigma_2"

These are the main properties of Pauli matrices, but there is, obviously, a lot more of them. You can also check Wikipedia : Pauli matrices - Wikipedia

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