# Can matter waves of bound systems experience dispersion? If so, how would it manifest itself experimentally?

Electromagnetic waves experience dispersion and these result in a "chirp" in frequency after traversing some distance. This chirp can be heard when waves from a lightning strike in one magnetic hemisphere of the Earth propagate along magnetic Earth's magnetic field lines through charged particles trapped in them and are received in the other magnetic hemisphere, and are used to estimate distances to Fast Radio Bursts (Just how fast is a Fast Radio Burst thought to be?) since dispersion by the interstellar medium is well-characterized.

Question: Are there conditions under which matter waves of bound systems like molecules experience dispersion? If so, how would it manifest itself experimentally?

I don't know if this is a "for example" or a separate question, but it helps reflect my current lack of understanding of the problem. Atoms and molecules have been demonstrated to exhibit diffraction and even Bragg scattering from electromagnetic standing waves1 when there is a dipole moment. Since diffraction is dispersive, do different "frequency components of the molecule's wave" (whatever that means) get spread out in a sort of a spectrum?

1recent news "Bragg Diffraction of Large Organic Molecules"

Can matter waves of bound systems experience dispersion? If so, how would it manifest itself experimentally

Electromagnetic waves have variations in energy as a function of (x,y,z,t)

The quantum mechanical waves that describe particles and molecules are not energy waves. Each particle/molecule is described by a wavefunction $$Ψ$$ , a complex function which is a solution of the quantum mechanical equation, BUT is not measureable. What is measured in experiments is $$Ψ^*Ψ$$ which , according to the postulates of quantum mechanics, is a probability to find the particle/molecule at (x,y,z,t). See the double slit experiment one electron at a time

Each electron is a dot seemingly random, the wave nature appears with the accumulation of events which is a probability distribution.

So any type of interference is possible ,but it will manifest itself in the probability distributions , in no sense in matter waves that can be timed, as far as I can see.

As far as "chirps" concerned: we are in the QM regime when talking of bound states. They are quantum mechanically bound, and described mathematically by QM wavefunctions. This means that any single measurement can show no chirps. Any chirps will be seen in the probability distributions.

• Okay so in the electromagnetic examples (whistlers and dispersion measurements of radioastronomical signals) there are zillions of photons in each pulse and the radio reception and frequency vs time analysis does the equivalent of building up a histogram of them, whereas in an experiment based on counting individual molecules no matter what happens in a dispersive media we're still counting individuals and each one collapses into a single count upon measurement.
– uhoh
Jul 26, 2020 at 0:24
• So actually I don't understand drawing the distinction between electromagnetic waves and matter waves as one being an "energy wave" and the other being complex and quantum mechanical. Maybe it's more that the electromagnetic examples I gave are collections of a very large number of photons -- we usually don't count radio photons experimentally due to their individual very low energies -- while we often do count individual electrons, or alpha particles or in this case molecules.
– uhoh
Jul 26, 2020 at 0:27
• As always there are different ways to look at the same problem, I guess for me it's better to think of the radio waves as a zillion individual photons, with each completely quantum mechanical in nature than it is a single "radio wave packet" treated classically. The QM treatment should always give the right answer, it's just that it's harder sometimes. Thank you very much for your answer!
– uhoh
Jul 26, 2020 at 0:32
• Single photns behave the same as the electrons google.gr/… . The classical em and its mathematcs emerges from a superposition of the photon wavefunctions, not a simple addition. and yes it is very complicated motls.blogspot.com/2011/11/… Jul 26, 2020 at 3:44
• Would the answer to "Can matter waves of bound systems experience dispersion?" be "No"? or just "Not in this case?" If it's the latter, then I can accept this answer and then ask a new, more general question, something along the lines of the concept of matter wave dispersion making sense in any context ever, or if there are any proposed methods for its detection. I'll have to think about exactly how to ask that though.
– uhoh
Aug 6, 2020 at 1:49

As stated in the comment by @ChiralAnomaly, all wave packets corresponding to particles with nonzero mass should experience dispersion, because velocity is (classically) proportional to momentum and all wave packets have a spread of momentum due to the uncertainty principle.

An experiment to prove this dispersion would be relatively easy: for example, diffraction of a particle beam by a crystal will produce first- and second- order beams with angular spread that depends on the momentum spread of the original wave packets of the particles.

A single detected particle does not exhibit dispersion, because (if its velocity is detected) it can only have one velocity. Dispersion and wave phenomena in general are features of the probability distribution, which can only be measured in the form of a large number of detections of particles associated with identical wave packets.

In the case of a bound particle such as an electron in a hydrogen atom, it's not obvious what "dispersion" would mean. However if the velocity of the electron could be measured at exactly the same point for a large number of identical atoms, the magnitude of the velocity would always be the same because the energy spectrum is discrete. The direction of the velocity would vary randomly. So, if "dispersion" means having a spread of magnitude of velocity, there is no dispersion on a typical bound system. If there are some bound systems without discrete energy spectra, those systems would exhibit "dispersion" in that sense.

• Please see the edit to my answer. Aug 6, 2020 at 23:52
• OK, please see the further edit, which directly addresses bound particles. Aug 7, 2020 at 1:58
• Thanks for the edit! While I chose a large organic molecule specifically for its size and complexity, you've chosen a single hydrogen atom for your example; the smallest possible system without going sub-atomic. Intuitively we like to think of the electron as well-bound to the proton and spherical, which minimizes any consideration of the differential effects of the dispersive medium or field on it relative to the proton, but I still see this as avoiding the question.
– uhoh
Aug 7, 2020 at 2:03
• Can you make a similar argument for a large organic molecule as I've discussed in the question, where there is far less coupling between the wave functions of the individual constituents within the system?
– uhoh
Aug 7, 2020 at 2:03
• I suspect that a large molecule can look a lot like a system with energy "bands" rather than a discrete energy spectrum - but I don't know. Probably if it is isolated it has a discrete spectrum but with very closely spaced lines that look continuous. Aug 7, 2020 at 2:05