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Hi all,

My motivation is understanding some derivations in Quantum Mechanics, but I think my questions are purely algebraic. I have a general question and then a specific one:

General Question - when writing the commutator of commuting vector and a scalar operators (for instance angular momentum and some Hamiltonian) - [itex][\vec A,H]=0[/itex] - what is meant by this *exactly*? I see two possible answers:

1. [itex][A_i,H]=0[/itex] for [itex]i=1,2,3[/itex]

2. [itex][A_1+A_2+A_3,H]=0[/itex] in which case we could have [itex][A_i,H]\ne0[/itex] for some [itex]i[/itex] .

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 [itex]\vec A[/itex] and [itex]\vec B[/itex] commute with [itex]H[/itex], does [itex]\vec A \cdot \vec B[/itex] 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!

My motivation is understanding some derivations in Quantum Mechanics, but I think my questions are purely algebraic. I have a general question and then a specific one:

General Question - when writing the commutator of commuting vector and a scalar operators (for instance angular momentum and some Hamiltonian) - [itex][\vec A,H]=0[/itex] - what is meant by this *exactly*? I see two possible answers:

1. [itex][A_i,H]=0[/itex] for [itex]i=1,2,3[/itex]

2. [itex][A_1+A_2+A_3,H]=0[/itex] in which case we could have [itex][A_i,H]\ne0[/itex] for some [itex]i[/itex] .

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 [itex]\vec A[/itex] and [itex]\vec B[/itex] commute with [itex]H[/itex], does [itex]\vec A \cdot \vec B[/itex] 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!

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