A page from the "Quantum Behavior" chapter in Feynman's book "Six Easy Pieces":

Water Wave Experiment

He uses the following notation:

$h_1e^{i\omega t}$

But the "h" has a caret (^) on top. What does it denote in this context?

Side note: I used a picture instead of a MathJax-formatted quotation because the caret (^) is only used in the book format of his lectures, not in the online version.

  • 1
    $\begingroup$ \hat{h} gives you $\hat{h}$. Usually $\hat{f}$ denotes the Fourier transform of $f$. No idea whether that makes physical sense there. $\endgroup$ Dec 6 '20 at 15:03
  • $\begingroup$ Thanks, Daniel. I guess I should post this question in the Physics SE then? $\endgroup$ Dec 6 '20 at 15:04
  • $\begingroup$ I don't know. There are people here to whom Physics makes more sense than it does to me, so maybe one of them knows. $\endgroup$ Dec 6 '20 at 15:07
  • $\begingroup$ The title asks about what the caret means on this page, but the text of the question asks "What is it used for as a mathematical symbol, in general?". Those are very different questions in that it's probably almost never used in math in a way compatible with its use on this page. Which are you interested in for this question? $\endgroup$
    – Mark S.
    Dec 6 '20 at 18:36
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    $\begingroup$ Hey @MarkS., well, I thought that it's a generic symbol. I edited my question to make it clear that I wish to understand the role of the caret in the book. $\endgroup$ Dec 6 '20 at 19:44

Feynman isn't really using something standard here, although I would be unsurprised if it's not unique, or not even truly original. The issue here is that, while the waves he's considering depend in general on both time and space, they are all of the same-$\omega$ form $f(x)e^{i\omega t}$ (or if they're not, separable basis elements of the relevant space of functions are).

So it makes sense to write all waves as $e^{i\omega t}$ times something time-independent, although it may still be complex. In fact, even if there's no overall space-dependence, this still makes sense. For example,$$he^{i(\omega t-\phi)}=\hat{h}e^{i\omega t},\,\hat{h}:=he^{-i\phi}$$is how this notation encodes the relative phase of multiple waves, even if $\phi$ is a constant for each wave. Doing all this lets Feynman reduce a lot of differential equations to characteristic polynomial equations.


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