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Excuse any ignorance, but something was on my mind today and my professor didn't give me a very clear answer...

Obviously the Fourier Transform is used pretty constantly in QM. What about the wavelet transform?

My (perhaps flawed) understanding is that the uncertainty principle arises interestingly in the fourier transform (for example, you can get either good time or frequency resolution), and that the wavelet transform was a sort of alleviation of that, providing both time and frequency resolutions.

Equations from the wiki show that the fourier transform is a function of 1 variable - $\xi$ being frequency:

$$f(\xi)=\int^\infty_{-\infty}f(x)e^{-2i\pi\xi x}\,\mathrm{d}x$$

and the wavelet transform being a function of two variables, (what seems to be both time and frequency):

$$X(a,b)=\frac{1}{\sqrt{a}}\int^\infty_{-\infty}\overline{\Psi\left(\frac{t-b}{a}\right)}x(t)\,\mathrm{d}t$$

Does the fact that wavelets are localized in both time and frequency make it hard to use for QM? Or is it utilized at all?

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  • $\begingroup$ I witnessed the early work by Grossman and Morlet around 1980. It was originally used to analyse seismic signals. But soon it was applied to acoustic signals as well. Filtering noise from old grammophone recordings has become a useful application. At the time quantum mechanics was not a subject of interest. $\endgroup$ – Urgje Nov 6 '15 at 11:30
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    $\begingroup$ This question (v2) seems like a list question. $\endgroup$ – Qmechanic Nov 6 '15 at 13:14
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It is perfectly possible to use wavelets to analyse quantum mechanical situations. The wavelets are localised in both time and frequency but they are themselves subject to the uncertainty principle - if you want a better time resolution, you need to pay for it with a coarser frequency resolution. The uncertainty principle is a universal wave phenomenon and wavelets, being waves, cannot escape it.

That said, just because you can use a particular tool does not mean that it will be particularly useful. Wavelets are useful for analysing experiments which are both time and energy resolved, and that is generally quite challenging, so most bits of QM get along perfectly fine without them. Even time-dependent spectroscopies like 2D spectroscopy and is generalisations work perfectly fine with just the standard Fourier transform.

On the spatial side, for example, you do want to use wavefunctions which are (approximately) localised in both position and momentum. Here the unambiguous tool of choice are the coherent states of the harmonic oscillator, for a variety of reasons. Compared to them, wavelets are clunky and not really supported by any physical intuition.

In my neck of the woods we do care, very much, about resolving the time at which different energies occur. (More specifically, if you drive atoms and molecules with a strong last field they will emit high-order harmonics of the laser, in a process that's driven by a collision between an ionised electron and its parent ion. Different harmonics are emitted at different times, which means you can observe very fast time dependent phenomena by simply looking at the spectrum, but you need to be able to invert the time-energy mapping - example, example.)

However, even here people tend to use Gabor window spectra (multiply by a gaussian of fixed width and then Fourier transform, then scan over the window's position) (example). Wavelets could probably work well enough here (and some people do use them, see e.g. this or this examples) but the community isn't that much into them, I think mostly because of cultural factors.

Having said that, you should keep in mind that QM is a huge field with a long history, and no one can really survey all of that literature. You shouldn't be too quick to dismiss the concept, and certainly some googling is in order. Google scholar shows about 19,500 results for 'quantum mechanics wavelet'; many of those are just incidental mentions of QM on wavelet papers but there are lots of relevant hits there.

On the other hand, the wavelet transforms have not really reached the fundamental importance placed on the Fourier transform. This is partly a matter of unicity (there's a single Fourier transform, but there isn't a single all-round convenient wavelet transform because the choice of mother wavelet is a case-by-case thing), partly a matter of intuition (Fourier transforms are intimately tied with changes of representation to a conjugate variable, while wavelet transforms are harder to re-cast in terms of observables) and partly a matter of ease of access (so few introductory courses go to the lengths required to talk about them), among other factors.

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  • $\begingroup$ Here is an example where a wavelet basis is used to derive a kinetic "Boltzmann"-type equation (where one wishes to recover a semi-classical concept of phase space cells). $\endgroup$ – Mark Mitchison Nov 6 '15 at 18:10
  • $\begingroup$ @Mark Ugh, if ever a preprint called for downloading the source and re-latexing it into a readable source, that's the one. $\endgroup$ – Emilio Pisanty Nov 6 '15 at 18:15
  • $\begingroup$ Oh wow, I didn't even open it. The PRA version is rather lovely, but behind a paywall, obviously. $\endgroup$ – Mark Mitchison Nov 6 '15 at 19:04

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