Solve Hamiltonian Operator Question: What is [\hat{H}, i\hbar]?

[Jenkz]

Homework Statement

I'm completing a question regarding position and momentum operators, however I'm stuck on one term.

What does [$$\hat{H}$$, i$$\hbar$$] equal? Or what does it mean?

Thanks.

 

[fzero]

If you have $$i \hbar$$ in an operator equation, it usually means $$i \hbar \hat{I}$$, where $$\hat{I}$$ is the identity operator.

 

[Jenkz]

Ok maybe putting it into context might help.

The question needs us to prove that [H,xp] =[H,px]

Where xp =px + [x,p]

 

[fzero]

When we write

$$[\hat{x},\hat{p}] = i \hbar ,$$

this is an operator equation, so we should really have an operator on the RHS. This operator is the identity operator and it's usually left out because it's clear from the context. To be precise, we'd write

$$[\hat{x},\hat{p}] = i \hbar \hat{I}.$$

 

[Jenkz]

It doesn't really seem to make a difference in my case if I have the identity operator. Anyhow, I've realized what I need to do, so thanks anyways :)