# 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?

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

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## 2 Answers

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.

At the end of the day, however, it's a matter of art to get a paper to look recognizably like the other papers in a field. The use of a particular symbol for a particular purpose can come almost to define a particular area of Physics, and people in that field may stop reading a paper for flagrantly breaking such a rule unless a good enough reason is given for doing something else. To have those nuances down pat requires that you read enough papers in the field you want to write in carefully enough to notice what is used consistently for what.

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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.

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