P=1,0,0,-1 Q=0,i,-i,0 R=0,1,1,0 in 2 by 2 matrices show that (R,P) = RP - PR = 2iQ?

Question

P=1,0,0,-1 Q=0,i,-i,0 R=0,1,1,0 in 2 by 2 matrices show that (R,P) = RP - PR = 2iQ
Note: the first two elements are for the first row in that order. e.g. in P=1,0,0,-1.....1 and 0 is first row, while 0 and -1 is second row in that order.

Answer 1

Ah... commutator relations of Pauli's spin matrices.

RP = [0 1 | 1 0] [1 0 | 0 -1] = [0 -1 | 1 0]
PR = [1 0 | 0 -1] [0 1 | 1 0] = [0 1 | -1 0]

{R,P} = RP - PR = [0 -1 | 1 0] - [0 1 | -1 0] = [0 -2 | 2 0]

2iQ = 2i [0 i | -i 0] = [0 -2 | 2 0]

Clearly, {R,P} = 2iQ.



You also have [P,Q] = PQ - QP = 2iR and [Q,R] = QR - RQ = 2iP.

Answer 2

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Answer 3

use matlab and do it