# How is an arbitrary operator usually denoted in quantum mechanics?

Which symbols are usually used to denote an arbitrary operator in quantum mechanics, such as O in the following example?

$O \mbox{ is Hermitian} \Leftrightarrow \Im{\left< O \right>} = 0$

It's common to put a hat over anything that's an operator instead of a c-number, so that $\hat A$ is an operator, $A$ is a c-number. Then we can use any letter as either an operator or as a c-number. Your $\hat O$ or $O$ to some extent suggests something that is specifically an observable quantity, just as $\hat H$ suggests a Hamiltonian operator, although one would usually expect such uses to be explicitly stated.
• The hat convention seems useful, thanks. Do you know of a letter I could use instead of $\hat{O}$, that carries no strong connotations? – user1778 Jun 20 '11 at 11:57
• As you see, I used $\hat A$. $\hat B$ and $\hat C$ do not have strong connotations unless you're writing in a particular field, provided they're said to be arbitrary operators, $\hat D$ might imply something to do with differentiation, but not if it's declared to be arbitrary. There's a writing style that makes these things work out OK that you should look for. – Peter Morgan Jun 20 '11 at 12:07
• For me $\hat D$ implies some dipole. As you say, it's field dependent... – Frédéric Grosshans Jun 21 '11 at 14:43
• You can also use a different font or style, such as \mathcal (i.e. $O$ becomes $\mathcal{O}$, provided the amsmath package is called). – Olaf Jun 21 '11 at 16:16
In my experience, which covers basic QM and quantum field theory to some extent, people tend to use $A$ or $O$ for a generic operator. The convention that $O$ indicates an observable is not universal - at least, not universally used, although it's probably at least familiar to most physicists. (Same with the hat convention that Peter mentioned; everyone understands it but not everyone uses it all the time.)
If a paper refers to multiple generic operators, then it's typical (again, in my experience) to denote them using sequential letters starting from $A$. So for example, one might talk about a commutator $[A,B]$ or $[[A,B],C]$ or something like that. Other than that, though, I would consider it unusual to see letters other than $A$ or $O$ used to denote generic operators.