In a perfectly elastic collision, is the kinetic energy transferred instantaneously? Given that we're assuming a perfectly elastic collision implies a perfectly inelastic elemental particle, there should be no mechanical delays, but wouldn't that imply energy transfer could surpass c through an appropriately dense medium? I can't help but wonder if/how Planck Constant might come into play - or does that only apply to motion/propagation through a vacuum, as viewed through the lens of the wave/particle duality interpretation?
If the medium is incompressible what would be the limitation on the speed? Say, two neutrinos colliding, for example.
I don't think this can be right. For macroscopic matter, there is no such thing as incompressibility. The colliding objects will progressively deform and rebound elastically, a process that takes a finite time and thus involves an impulse (F x t) equal to the change of momentum of the colliding objects. It is not instantaneous. For colliding atoms, the electron probability clouds will deform, due to Coulomb repulsion as the atoms approach, and their energy will rise, as the mutual k.e. is converted to electronic energy, with the converse occurring once the direction of mutual motion has changed and the atoms separate. But you raise an interesting question about collisions between fundamental particles in unbound states. These are modelled as wave packets, the "size" of which is inversely related to the uncertainty in their momentum. But since the wavefunction is related to probability I suppose that for a pair of individual "particles" the collision either happens or it does not, with a probability given by the value of a wavefunction for the interaction, i.e. the combined state represented by the converging particles. But I'm speaking a bit ex ano here and you should probably wait for a more authoritative contribution.
Theoretically, as compressibility tends toward zero, shouldn't the speed of sound tend toward infinity? If so, what's the limiting factor?
You have the right idea here. One thing to keep in mind is that even when we're talking about fundamental particles, their collisions can still involve long-range interactions. Two free electrons, for instance, will repel each other via the long-distance electromagnetic field, and as their wavepackets approach each other, we expect them to deform and change course even before they overlap significantly. In other cases, like a photon colliding with an electron, the interaction is indeed pointlike, and the collision dynamics are in some sense probabilistic. But one common trap that you shouldn't fall into is the assumption that because a collision is "probabilistic", it either happens in full with probability p or doesn't happen at all with probability 1-p. This might lead one to assume that the outgoing wavepackets split into two parts: a scattered part that diverges wildly from its original path as though two hard particles had bounced off each other, and an unscattered part that doesn't diverge at all. This would be true for classical probability distributions, but because quantum waves can interfere with themselves, we'll get something that looks a lot more like continuous deformation. To see why this is, it helps to think of the wavefunctions not as probability distributions, but as objects in their own right that are evolving with time. The time evolution of a wavefunction is determined by the Hamiltonian operator, which is basically just a fancy way of saying its energy. Modify the energy, and you can change the way in which the wave propagates. For an electron and a photon the Hamiltonian will contain an interaction term that is "pointlike" in the sense that it can be written as a delta function: if the electron and the photon share the same position, they get extra energy, otherwise nothing happens. If the two wavefunctions overlap partially, the portions that overlap will get more energy from this delta function than the parts that don't, and so their time evolution will be different from what we would expect of a free particle. The more strongly the two wavefunctions overlap, the more pronounced this change will be. If we integrate the effects of this modified Hamiltonian over the full duration of the collision, we'll find that the free-particle wavepackets have transformed into "scattered" wavepackets which can look like anything from slightly squished free-particle packets to spherical waves radiating from the collision site, depending on the extent of overlap and the size of the interaction term in the Hamiltonian. Actually solving for these output modes is the domain of scattering theory, which is one of the more challenging topics I've encountered in grad school. Compressibility is kind of a macroscopic property that breaks down in relativistic scenarios. When we talk about the compressibility of a substance, we're generally working in the "quasi-static" approximation, where the interactions between particles can just be written as a potential function of their relative positions. The textbook example of this is when we say that the energy of two charged particles increases like 1/r as they approach each other; even though our system is changing, it is "quasi-static" in the sense that we assume the particles' electric fields have plenty of time to adjust as they move, so we can use the 1/r law which is a result from electrostatics. This approximation no longer holds as the speed of sound approaches the speed of light, because the time it takes for the electric field to change as the particles move can no longer be written off as instantaneous. Instead, we now have to consider how the particles emit radiation as they accelerate, and how they absorb the radiation from other particles to create an effective interaction. The resulting dynamics will be pretty complicated, and probably quite different from everyday compression waves, but we can say one thing for certain: because the particles can only "talk to" one another by passing photons back and forth, no interaction between them can go faster than the speed of light.
The limiting factor is the thing that creates the compressibility in the first place: the chemical bonds between the atoms and forces holding the electrons in their shells.
Many thanks for this contribution, Fednis. Your description of the Hamiltonian for an interacting system is just what I was had in mind when speaking about a wavefunction for the combined state. What you say about the various possible shapes of scattered waveforms is also interesting. We never covered the QM of scattering processes in quantum chemistry when I was at university, so this is exactly the bit I felt I could not speak about.Please Register or Log in to view the hidden image!
