Is there a recognised standard for typesetting quantum mechanical operators?

Question

Firstly, I wasn't sure exactly where to put this. It's a typesetting query but the scope is greater than $\TeX$; however it's specific also to physics and even more specific to this site.

I've recently been reading a style guide for scientific publications (based on ISO 31-11), however there was no mention of quantum mechanical operators. I've seen them written a few ways and was wondering if there was a decision handed down from "up above" that any particular way is best.

• $H$ -- I see this most commonly but I suspect it's mostly due to (mild) laziness to not distinguish it from a variable.
• $\hat{H}$ -- This is nicer to me because it makes the distinction between operator and variable. From what I understand of the ISO the italic means it's subject to change, which is true of the form of an operator, but not really its meaning? So I'm not totally sure if that's appropriate here.
• $\mathrm{H}$ -- Roman lettering is used for functions e.g. $\sin{x}$, $\mathrm{erf}(x)$, and even the differential operator (as in $\frac{\mathrm{d}}{\mathrm{d}x}$) so this seems to me like the most suitable category to put operators in.
• $\hat{\mathrm{H}}$ -- Probably the least ambiguous but may also be redundant.

Which would be the best to use? Am I being too pedantic?

Answer 1

My taste, never overload your notation unless its necessary.

Many people in quantum information try to avoid "hats" or further ornaments for operators that are just linear maps. Simple capital letters are fine to write Hamiltonians, channels, unitaries and measurements (italics are not really important, but its a de-facto standard). When people write many-body hamiltonians in terms of smaller k-body interactions, it is common that they use low-case letters for the latter (example, the Hubbard model). Also, mind that in finite dimensional systems linear maps are in one-to-one correspondence to matrices.

On the other hand, thinking of linear maps as matrices forces you to choose a basis. It might be more clear in some contexts to use a symbol-with-hat to denote an operator without mentioning the basis and the same symbol without hat for a matrix representation. However, I find that this practice can make your notation more complicated without earning much, since typically there is a natural default basis in every problem.

Answer 2

As you see, there are different notations for quantum mechanical. Typically, even within a journal there is no one typesetting (style guides usually don't touch this topic).

Besides the ones you mentioned, sometimes people use:

• bold font (e.g. ${\mathbf H}$),
• small font for operators acting on subsystems.

Try looking at common notations used by your field. If there is no consensus, just use the one that you like the most.

However, what is important is if you use it consistently and if it is clear to the reader. There aren't many more irritating things than reading a paper when even types of symbols are not clearly stated and one needs to guess.

Answer 3

The last ISO-IUPAP-IUPAC overall rule is that symbols representing scientific quantities are italic, but symbols representing units, or labels, are Roman. Mathematical operators such as $\frac{\mathrm{d}}{\mathrm{d}x}$ are Roman as well.

Quantum mechanical operators are not labels neither mere mathematical operators, therefore they would go in italic shape. Of courser, you can find this rule violated by authors who do not follow standards and in the older literature.

$\widehat H$ --the \widehat command looks better-- is the ordinary representation of the Hamiltonian operator, but $H$ is also used when there is no room for confusion (e.g. as in $H \Psi = E \Psi$). Since each hermitian operator is associated to a matrix, the matrix notation $\boldsymbol H$ is also used --italic is recommended for matrices of physical quantities--.

The advantage of the hat notation is that allows for a direct extension to superoperators: $\widehat{\widehat L} \widehat \rho = \lambda \widehat \rho$, but it is too intrusive for more complex equations.

Answer 4

ISO is a modernist, slightly unhappy, attempt to come up with a standard. The best authors use italics for all variables and for all single-letter functions or constants even if based on a word.
Example: erf for error function is not italicised, but $l$ for logarithm (go back to Euler's original papers, you'll see they used that abbreviation) is italicised, but $\log$ is not italicised. Therefore $e$ and $i$ are also italicised. Now,
$dx$ is italicised because $d$ is not an operator, it is part of the two-letter notation. It isn't as if $dx\over dt$ was the quotient of the result of applying the operator d to x by the result of applying the operator d to t....it is just a portmanteau notation. (My failure to italicise the «x» and the «t» in the previous sentence is to demonstrate the real origin of this convention: the sentence as I typeset it is quite hard to read. Italicising single letter math symbols makes it easier to read.)

In differential geometry, $d$ is exterior differentiation, i.e., it is a map between two spaces. I.e., it is a function. Single letters for functions have always been italicised even if derived from a word, and that is why Spivak and all normal authors do so. Hence, if $x$ is regarded as a function on a space (instead of as a variable, as above), then $dx$ is a differential form obtained by applying the function $f$ to its input, the function $x$.

In brief, the answer to your question is use ordinary (i.e, italic) $H$ for the Hamiltonian, as did Dirac and Wigner and nearly everyone, there is no need to make it bold (a very old fashioned method to indicate that a variable was not simply a numerical variable) or to put a hat on it. Putting a hat on it can be pedagogically useful for textbooks for undergraduates, but it is like putting an arrow on top of x to indicate it is a vector....grown-ups don't do that anymore.