# Equal commutators

Hi.

Cohen-Tannoudji has this section in his quantum mechanics book where he derives a bunch of relations which are true for operators having the commutation relation $[Q,P]=i\hbar$. Is there any special significance to this value of a commutator? Would things be much different if it had the value 1 ?

Also, if we have two sets of operators with the same commutator, i.e. $[x,p]=[Q,P]=i\hbar$, what does this tell us about the relations between the operators, if anything?

nicksauce
Homework Helper
Take the hermitian conjugate of each side. If the operators are Hermitian, then you get
[P,Q] = -ih = -[Q,P], which is exactly what you expect based on the definition of a commutator. If you had [Q,P] = 1, then this process would lead to a contradiction if your operators are Hermitian.

Ah, so that's why it's important that it's imaginary! Great. What about sets of operators with the same commutator? I know that if we have two Hamiltonians of the same form, i.e.
$$H=a^{\dagger} a+\frac{1}{2}$$
$$H=b^{\dagger} b+\frac{1}{2}$$

and $[a^{\dagger},a]=[b^{\dagger},b]$ then the Hamiltonians will have the same eigenvalues. Is there more we can say? I've heard that all of quantum mechanics can be based on commutators...

nicksauce