One may compute the commutation relations between the a and a † † operators and the Hamiltonian: [ H ^ ^ , a ] = [ ℏ ℏ ω ω ( a a † † − − 1 2 ) , a ] = ℏ ℏ ω ω [ a a † † , a ] = ℏ ℏ ω ω ( a [ a † † , a ] + [ a , a ] a † † ) = − − ℏ ℏ ω ω a . [ H ^ ^ , a † † ] = ℏ ℏ ω ω a † † .