Chapter7_10

# Chapter7_10 - PHY3063 R D Field The Pauli Matrices Consider...

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PHY3063 R. D. Field Department of Physics Chapter7_10.doc University of Florida The Pauli Matrices Consider the three Pauli matrices = 0 1 1 0 x σ = 0 0 i i y σ = 1 0 0 1 z σ These matricides have σ i = σ i det( σ i ) = -1 Tr( σ i ) = 0 [ σ i , σ j ] = 2i ε ijk σ k { σ i , σ j } = 2 δ ij . They are hermitian, traceless, and they anti-commute ( σ x 2 = σ y 2 = σ z 2 = 1). Define + = ± = ± 0 1 1 0 0 1 1 0 ) ( 2 1 2 1 2 1 y x i σ σ σ hence = + 0 0 1 0 σ = 0 1 0 0 σ also = + 0 0 0 1 ) ( 2 1 z I σ = 1 0 0 0 ) ( 2 1 z I σ where = 1 0 0 1 I . Arbitrary 2×2 Matrix: Any 2×2 matrix can be written as σ σ σ σ r r + = + + + = + + + =
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