The Pauli spin matrices σ 1 , σ 2 and σ 3 are defined by σ 1 = [ 0 1 1 0 ] , σ 2 = [ 0 − i i 0 ] , and σ 3 = [ 1 0 0 − 1 ] . Verify that they satisfy σ 1 σ 2 = i σ 3 , σ 2 σ 3 = i σ 1 , σ 3 σ 1 = i σ 2 . If A and B are n × n matrices, we define their commutator , denoted [ A , B ] , by [ A , B ] = A B − B A Thus, [ A , B ] = 0 if and only if A and B commute. That is, A B = B A . Problems 19-22 required the commutator.
The Pauli spin matrices σ 1 , σ 2 and σ 3 are defined by σ 1 = [ 0 1 1 0 ] , σ 2 = [ 0 − i i 0 ] , and σ 3 = [ 1 0 0 − 1 ] . Verify that they satisfy σ 1 σ 2 = i σ 3 , σ 2 σ 3 = i σ 1 , σ 3 σ 1 = i σ 2 . If A and B are n × n matrices, we define their commutator , denoted [ A , B ] , by [ A , B ] = A B − B A Thus, [ A , B ] = 0 if and only if A and B commute. That is, A B = B A . Problems 19-22 required the commutator.
Solution Summary: The author explains the Pauli spin matrices for the given condition, i.e.
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