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

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.

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.