My motivation is understanding some derivations in Quantum Mechanics, but since my question is purely Algebraic, I think it is suited for these forums. I have a general question and then a specific one:
General Question - when writing the commutator of a commuting vector and a scalar - $[\vec A,H]=0$ - what exactly is meant by this? I see two possible answers:
- $[A_i,H]=0$ for $i=1,2,3$
- $[A_1+A_2+A_3,H]=0$ in which case we could have $[A_i,H]\ne0$ for some $i$ .
It seems to me that in the QM context almost always what is meant is the first option but I'm not certain...
Specific Question - if $\vec A$ and $\vec B$ commute with $H$, does $\vec A \cdot \vec B$ also necessarily commute? If the answer to the question above is #1, then obviously it does. If the answer is #2 then I guess not?
Would greatly appreciate the clarifications. Thanks!
In QM one often considers a Hilbert space $X$ and studies operators $A$ on $X$. So, if your vector $A$ is actually an element of $L(X)$, the vector space of let's say bounded operators on $H$, then you have to interpret your scalar $H$ as an operator. I guess this means that you just think of $H$ as $H I$, where $I$ is the identity operator on $X$.