Particles in quantum mechanics are often seen as wave packets - linear superpositions of plane waves summing to a localized excitation. But wave packets dissipate - for example passing such single photon through a prism, its different plane waves should choose different angles - such single photon would dissipate: its energy would be spread on a growing area ... while we know that in reality its energy remains localized: will be finally adsorbed as a whole by e.g. a single atom. Analogously for different particles like electron - any dependence on momentum while scattering would make such wave packet dissipating (e.g. indivisible elementary charge). How is this problem of dissipating particles solved? Aren't there some additional (nonlinear?) mechanisms needed to hold particles together, make these wave packets maintaining their shapes - being so called solitons?
Have you looked at this: http://en.wikipedia.org/wiki/Wave_packet It has a fairly informative section on wave packet diffusion. Note also that a wave packet is a solution of the Schrodinger eqn.
Yes I know, but there are situations without dissipation, so I have written about prism - which would always cause dissipation of photon as just linear superposition of plane waves. So imagine a single excited atom producing single optical photon, which comes through a prism and finally is absorbed by another single atom - suggesting that energy has traveled localized through a concrete trajectory between them. While if it would be just a wave packet, this energy should be dissipated - especially after the prism. What objectively is such single photon, particle? Doesn't the fact that there is no dissipation mean that there are some additional mechanisms needed to hold this wave packet together (make it soliton) ?
Why should there be a problem? Quantum physics is not classical field theory, and quantum wavefunctions do not have the same role and interpretation as classical field configurations. According to quantum physics, if you measure the position of a particle you will always get a localised detection no matter how spread out the particle's wavefunction was before the measurement. The square of the wavefunction just tells you the probability of the photon being detected in different places. This is quantum mechanics 101, and what any textbook and any person who has studied the subject will tell you. In most quantum physics experiments where individual particles are detected, the wavefunction is much more spread out than the actual particle detections we see. This is deliberate: if you want to see an interference pattern, like in the double-slit experiment, then the wavefunction needs to be as spread out as the interference pattern you're hoping to see. Large-ish wavepackets are normal and routine in quantum physics, regardless of dispersion.
I don't know if you have heard this oft-quoted sentence: "If you think you understand quantum mechanics, you don't understand quantum mechanics." What that means is that it is a subject you will never be able "to get your head round". You have to accept it because it works, not because it makes sense. In this case the observed particle, and observation includes being absorbed by an atom, is different in nature to the unobserved particle. Whether it "really" is different, is one of those questions we will never know the answer to, but describing it mathematically in that way is self-consistent and useful. While thinking about this, I have thought of a question. Maybe someone could give me a quick answer. If it has a complicated answer, tell me and I'll start a thread. What is the force of attraction between two touching protons in Newtons? Ignore the force which repels them.
Even for Mach-Zehnder interferometer we draw two classical trajectories, saying only that we don't know which one is chosen. Here situation is even simpler - no interference. Taking Feynman path integrals, their basic approximation is considering the classical trajectory and small variations around it (van Vleck formula) - in QM energy travels through a bit fuzzed classical trajectories. If you would like energy to really travel through more complex trajectories, remember that every change of momentum (direction) requires something to exchange this momentum. Capitan Kremmen, my point is only that they are not just usual wave packets, but those not dissipating: solitons. And still you can use QM for them - there are quantization methods for solitons, like skyrmions.
Skyrmions. New one on me. Please Register or Log in to view the hidden image! I'll have to look that up Added later: In particle theory, the skyrmion is a hypothetical particle related originally to baryons. It was described by Tony Skyrme and consists of a quantum superposition of baryons and resonance states. Skyrmions as topological objects are also important in solid state physics, especially in the emerging technology of spintronics. A two-dimensional skyrmion, as a topological object, is formed, e.g., from a 3d effective-spin "hedgehog" (in the field of micromagnetics: out of a so-called "Bloch point" singularity of homotopy degree +1) by a stereographic projection, whereby the positive northpole spin is mapped onto a far-off edge circle of a 2d-disk, while the negative southpole spin is mapped onto the center of the disk. http://en.wikipedia.org/wiki/Skyrmion There won't be many people here able to discuss skyrmions with you I think.
There are also simpler solitons, like fluxons in superconductors or for example here is a nice animation of annihilation of a 1D soliton-antisoliton pair: http://en.wikipedia.org/wiki/Topological_defect#Images They have rest energy (mass) due to Higgs-like potential - we see that this energy is released while the annihilation. Running this animation backward, you get pair creation. Are there other ways for solving the problem (of dissipating wave packets) than some additional mechanisms holding them together (making them solitons) ?
This stuff about solitons is fascinating, but I think it's a mistake to draw too much of a connection between the bound wave packets of solitons and the particle nature of photons/electrons. When a quantum wave function is allowed to evolve freely, absent any confining force, it dissipates. This is true for electrons, photons, or whatever other fundamental particles you want to look at. When we take a measurement, the wavefunction collapses, and the entire particle appears wherever we measured it to be. As far as I know, no one has come up with a satisfying explanation for why this occurs, but it certainly does happen. The fact that the particle is localized upon measurement like this does not, however, imply that its wavefunction was well-localized before the measurement. As arfa brane pointed out, many interference experiments rely on the fact that the wavefunction can dissipate before we collapse it with a measurement. Long story short: A "particle" is a particle because it is localized upon measurement, even if its wave function dissipates between measurements. Captain Kremmen: I don't think your question about protons is answerable, because according to the standard model, protons are point particles. For two point particles to be "touching" they would have to be directly on top of each other, which would lead to singularities (not in the sense of black holes, just fractions with zero denominator). If you wanted to determine the forces between two protons that are very close to each other such that their wavefunctions overlap, you would use quantum chromodynamics.
Sure, but what about energy distribution of a single photon, or charge distribution of a single electron - does it also dissipate? Why do you think it is a mistake? For example let us look at a simple model of electric charge as topological charge - there is also conservation law for it, Hopf theorem corresponds to Gauss law. So imagine there is a vector field with Higgs-like potential, for example V(v)=(1-|v|^2)^2, making that energetically preferred are unitary vectors (in vacuum). However, there can appear topological singularities, like hedgehog V(v)=v/|v| of topological charge 1. There is a problem in the center of such singularity, as no direction can be chosen there - but the field can get out of Higgs potential minimum, to v=0 in the center of such topological soliton - avoiding singularity and giving it rest energy/mass (released while annihilation like in animation for 1D solitons). Here is such pair of opposite unitary charges in 2D (lines are along these unitary vectors): Please Register or Log in to view the hidden image! If these charges get closer to each other, we see that stress (and so energy) of the field decreases - meaning that they attract. Analogously the same charges would repel. In 3D if would define electric field from curvature of the vector field, such that hedgehog have electric field like elementary charge, and analogously magnetic field - taking standard EM Lagrangian for such defined field, we get standard Maxwell equations for dynamics of such topological charges: http://iopscience.iop.org/1742-6596/361/1/012022/ Why do you think that such view on particles - as nondissipating wave packets: solitons, is generally a mistake?
I have to admit that I don't have much experience in topology, so it's likely that I'm missing something. But my basic argument is that treating electrons as non-dissipating wave packets is a mistake because electrons do dissipate. Take an electron, drop it in a vacuum, and leave it alone for a while; when you come back to check on it, its wavefunction will be all over the place. Of course, you can then measure the electron to collapse it back down to something more point-like, but that doesn't change the fact that the wavefunction exhibited dissipation. By contrast, my understanding of solitons is that they are configurations in which the wavefunction itself is non-dissipative, and the "particle" travels along in a well-localized bundle even without measurement. If I'm mischaracterizing solitons, I'd love to hear a quick description of solitons in as non-technical terms as possible. But if I'm right about the nature of solitons, it's incorrect to treat electrons as them because their wavefunctions evolve in different ways.
The dissipation you are talking about is our subjective lost of information - you have it also classically: if you know that something is exactly in given place in given moment, this Dirac delta of its probability accordingly to your knowledge will became wider and wider Gaussian. But do you really think that elementary charge can objectively dissipate? Such soliton indeed travels along some trajectory, but everything is happening in a wave propagating field - we have wave-particle duality: with the corpuscle, there are conjugated waves it creates. Like when you make Madelung transform of Schrodinger equation: substitute psi = R exp(iS). Now for density (R^2) you get standard continuity equation, while for action (S) you get classical Hamilton-Jacobi equation, but modified by h-order correction from this wave nature - the corpuscle travels through nearly classical trajectory, but modified by waves "piloting" it. Exactly like in double slit interference for Couder's walking droplets: corpuscle travels single trajectory, while waves it creates travel all trajectories - influencing behavior of the corpuscle. ... and generally there are also quantization methods for solitons.
The dissipation of a wavepacket is fundamentally different from classical propagation of uncertainty. Incomplete classical information gives you a probability distribution, and probabilities just sum. On the other hand, coherent quantum dissipation (maybe "diffusion" would be a more correct term here) gives you a distribution of probability amplitudes, which can interfere in non-trivial ways. To look at it a different way, classical dissipation comes about when we have imperfect knowledge of the initial conditions. By contrast, even if our initial knowledge of the wavefunction saturates the Heisenberg Uncertainty Principle, the Schrodinger equation predicts that allowing it to evolve freely will lead to diffusion. I agree that it's bizarre to think of electrons as dissipating, but it's really a natural extension of accepting the wave/partical duality. Sounds like you're talking about the Pilot Wave interpretation of quantum mechanics. I've never understood the Pilot Wave interpretation very well, so I'm glad to be discussing it with someone who seems to know what they're talking about. Please Register or Log in to view the hidden image! I'd love to hear more, but for now I have two points to make. First, if only for the sake of other readers, it's worth noting that the Pilot Wave interpretation is a minority viewpoint; it's perfectly consistent with experiments, but it is fundamentally non-local, so many physicists shy away from it. Second, as far as I understand it, the Pilot Wave interpretation treats particles as actual 100% particles that are merely guided by associated quantum waves. If we work under this assumption where the electron is a "particle hiding in a wave" rather than "both a wave and a particle", I don't see any reason to assume the electron would dissipate. Its pilot wave would dissipate, sure, but not the electron itself.
Indeed the fact that classically we sum probabilities, while in QM we sum amplitudes seems disturbing. However, for my last PhD thesis (theoretical physics, here are slides) I was studying Maximal Entropy Random Walk (MERW). It is about the question how to choose stochastic propagator (transition probabilities) - it turns out that the standard way usually only approximates the fundamental principle required for thermodynamical models: the maximal uncertainty principle. MERW is choosing the only stochastic model which actually maximizes entropy production. Locally it looks like standard random walk (GRW), but after longer time it leads to stationary probability exactly like for the quantum ground state - with strong (Anderson's) localization properties ( http://prl.aps.org/abstract/PRL/v102/i16/e160602 ). For example here is a comparison of their behavior of regular lattice with removed some vertices (denoted by squares): Please Register or Log in to view the hidden image! A long story short, these corrected stochastic models which are no longer in disagreement with quantum predictions, is assuming uniform or Boltzmann distribution among possible trajectories - it is very similar to euclidean path integrals. And they have also these squares relating amplitudes and probabilities: Please Register or Log in to view the hidden image! Intuitively, while taking distribution among possible paths, to get accidentally given value in given moment you need to simultaneously get it from ensemble of past and future half-paths. Probability for each of them is the amplitude, so to get it randomly twice, you have to square the amplitude. So generally I don't think there is a large difference between quantum and classical probability, but we just need to properly understand it. Anyway, I didn't want to get into discussion about probability, so I was focusing on dissipation of more objective entities, like energy or charge - don't you see it disturbing that energy of a single photon dissipates after going through a prism ... and then is gathered back into a single atom this photon excites? I was talking just about substituting psi = R exp(iS) to the Schrodinger equation: you get continuity equation for density (R^2) and Hamiton-Jacobi for action (S) modified by h-order corrections from the wave nature ... you don't need to "believe in dBB interpretation" for that - pure Schrodinger's equation, who originally based it on classical Hamilton-Jacobi equation, says that. And if you want to get intuitions about such view on the wave-corpuscle duality, I think the best are these Couder's experiments with droplets conjugated with waves they create - for example: - interference in double-slit experiment (particle goes a single way, but "pilot" waves it creates go through all paths - leading to interference): http://prl.aps.org/abstract/PRL/v97/i15/e154101 , - tunneling (the field depends on the whole history, making that getting through a barrier is practically random): http://prl.aps.org/abstract/PRL/v102/i24/e240401 , - orbit quantization (to find a resonance with the field, the clock has to perform an integer number of periods while enclosing an orbit): http://www.pnas.org/content/107/41/17515 , - "classical Zeeman effect" (Coriolis force instead of Lorentz): http://prl.aps.org/abstract/PRL/v108/i26/e264503. here is a nice video about them: http://www.youtube.com/watch?v=nmC0ygr08tE
