Equal Commutators: What Do They Tell Us?

[daudaudaudau]

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]

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.

 

[daudaudaudau]

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]

I'm not too sure what we can say about sets of operators with the same commutator... hopefully someone else can help you out.