1. [A, B] = 0 If A and B commute there exist eigenfunctions that are simultaneously eigenfunctions of both operators A and B and one can determine simultaneously the values of the quantities represented by the two operators but if they do not commute one cannot determine the values of the quantities simultaneously

2. [A, B] = 0 [A, B] = k B A I A ’ › = A ’ I A ’ › A { B I A ’ › } = (A ’ + k ) { B I A ’ › }

3. [q, p] = i [H, q] = ω (-ip) [H, p] = ω (-iq)

4. F + = q + i p F − = q − i p [ H, F + ] = − ω F + [ H, F − ] = + ω F − H = − ½ ω ( F + F − − 1 ) H = + ½ ω ( F − F + + 1 )

5. H { F + I E n › } = ( E n − ω ){ F + I E n › } H { F − I E n › } = ( E n − ω ){ F − I E n › }

6. F + I E ↓ › = 0 H I E ↓ › = ½ ω I E ↓ › E (v) = ω (v + ½ )

8. F + = q + i p [H, q] = ω (-ip) F ± = q ± i p F − = q − i p [ H,F ± ] = q ± i p