What does does [H1,H2] = 0 mean if each H is a hamiltonian of a quantum system

What does does [H1,H2] = 0 mean if each H is a hamiltonian of a quantum system

I'm trying to get through a research paper on theoretical quantum biology and I just want to make sure I'm interpreting this relation nearly correctly. Is this equivalent to saying that the two Hamiltonians are orthogonal? Or that the eigenstates of the Hamiltonian are orthogonal?

Here is an excerpt from the paper for context. I don't have enough time to learn all of the math right now as this is partially for a class presentation for the undergrad physics seminar course i'm in, but I want to get the gist of the theory.

Here's a link to the paper for anyone interested. http://arxiv.org/abs/1408.5798

• It means that the Hamiltonians commute (or, in the case of the paper, don't commute). Two Hamiltonians that don't commute cannot, in particular, have the same eigenstates. If $A, B$ are two operators then $[A, B]$ denotes their commutator $AB - BA$. – Qiaochu Yuan May 2 '15 at 23:54
• @QiaochuYuan Sorry, was busy typing as you answered. Should I delete my answer? – user12802 May 3 '15 at 0:04
• @QiaochuYuan So does writing [A,B] = 0 say that the two Hamiltonians do not commute, which says that the order in which you apply them to a state matters, whereas if [A,B] =/ 0 the Hamiltonians do commute and the order does not matter? – lthermin May 3 '15 at 0:25
• No, it is precisely the opposite. $[A,B] = AB - BA$ so $[A,B] = 0$ if and only if $AB = BA$. – JHance May 3 '15 at 0:26
• @JHance Awesome thanks for clearing that up. – lthermin May 3 '15 at 0:37