wave-particle-duality
I know that there is a particle model that describes the absorption of electromagnetic radiation in matter – A photon with energy E can excite ("absorbed") an atom if it has energy gap of the same size E.
What about a wave model for absorption, or is it only a particle phenomena ?
I'd like to add a slightly different take on both Anna V's and Ben Crowell's answers.
Classical optics IS the theory of one photon. There is NO approximation in this statement in free space (classical optics is actually a little bit more than this generally, but I'll get to that below). Maxwell's equations to the lone photon are EXACTLY what the Dirac equation is to the lone electron (indeed the two equation sets can be written in forms where they are the same aside from a mass term in the Dirac equation coupling the left and right circularly polarized fields together, whereas the two polarisations stay uncoupled in the mass-free Maxwell equations). Photons are Bosons, which means you can put as many of them as you like into the right same state: so you can build up classical states which correspond EXACTLY to one photon states. When you do experiments where one photon is transferred at a time, you solve Maxwell's equations for a field inside your experimental set up, normalise the solution field so that the total electromagnetic field energy is unity, and then the energy density $\frac{1}{2}\epsilon_0 |\mathbf{E}|^2 + \frac{1}{2}\mu_0 |\mathbf{H}|^2$ becomes the probability density that your one photon will be photodetected at the point in question.
The full quantised theory of light works like this: each monochromatic free space mode (plane wave) is replaced by a quantum mechanical harmonic oscillator (which you have likely dealt with). The motivation for this is that free space classical plane waves oscillate sinusoidally with time and so a classical free space wave is assumed to corresponds to a coherent quantum state of the corresponding quantum harmonic oscillator. You may recall that energy may be given to/withdrawn from a quantum harmonic oscillator only in discrete packets. It is these discrete packets which are the "particles". In this bigger picture, a one photon state is a quantum superposition of next-to-ground-state (one photon) states of the infinite collection of plane wave harmonic oscillators that are "THE FIELD"; the spatial Fourier transform of the superposition coefficients propagates precisely following Maxwell's equations. Otherwise put in the Heisenberg picture: in a one photon state, the electric and magnetic field observables evolve with time precisely following Maxwell's equations. Note that in this description, it's very hard to say where the "particles" actually are: this is how I like to think of it: "the Electromagnetic Field communicates with the outside world (the other electron, quark, ... fields making up the universe) in discrete data packets, and these packets are what we call photons". Also note that, in the light of Anna V's comments below: one doesn't have to stick with plane waves: one can choose any complete orthonormal set of fields and assign quantum harmonic oscillators to each of these and the description is wholly equivalent. So one chooses whatever the basis states make the analysis of their particular problem easiest.
Going back to our one photon state evolving following Maxwell's equations: when we put dielectrics and other matter into the description, we no longer purely have photons. If we represent the "atoms" of the matter by two level quantum systems, when light propagates in matter it is not really just light: it is a quantum superposition of free photons and excited matter states. The description of lossy materials in this picture is a little more complicated but it can be done: lossy materials are continuums of quantum oscillators that the photon has an extremely low probability of remission from once it's absorbed there). So here then is how I like to think of classical optics:
Classical Optics = The Theory of One Photon + Optical Materials Science
Let's go back to the statement about putting many Bosons into the same state and thus building a classical light field that is mathematically the same as a one photon state. Mostly that's all there is to macroscopic optics: and this is what I believe Dirac meant when he said famously that "each photon interferes only with itself". In macroscopic states built by simply copying Bosons, it should be pretty clear that whether you calculate their propagations wholly separately as lone photons and then sum up their probability densities to get the field intensity, or if you simply classically calculate the field intensity, you'll get the right same result. Most macroscopic light fields behave like this and it is actually very hard to find deviations from this behavior. Aside from the experiment where we turn the light level right down low so that interference patterns are built up "click click click" one photon at time, the photon is extremely hard to observe experimentally as a quantum and not as a classical light field. Mathematically what all this means is that macroscopic states behave as though they are products of one photon states (special Glauber "coherent" states - actually discovered by Schrödinger): and the last twist in the quantum optics tale is the phenomenon of entanglement. This is what we see in the seldom and very hard to set up situations where the "product" behaviour no longer holds: but you may care to see the Wikipedia page on quantum entanglement or ask another question to find out about that one!
Lastly, to think about laser trapping, as Ben says indeed a classical field theory is enough. What you might be thinking of is laser cooling and in this case Anna V's comments apply. Here the atom is withdrawing one quantum at a time from the electromagnetic field and likewise emitting a few quantums at a time. But it's not all "particle-like" - for instance, the Fermi golden rule calculations that give the cross sections for these particle transitions all involve overlap integrals between the atomic dipoles and the wave fields - note that this is a very classical looking calculation wholly analogous to the analysis of the interaction between a short ($\ll \lambda$) dipole antenna and classical electromagnetic field. As in Ben's answer, both the wave and particle aspects of the photon's behaviour are thus showing themselves here. Also - I might be guessing here as laser cooling is not my field - an optical photon's momentum is quite an appreciable chunk of a slow atom's momentum. Optical photons are of the order of 1eV so that their momentum is of the order $10^{-27}\mathrm{kg\,m\,s^{-1}}$ and a proton's mass is of the order of $10^{-27}\mathrm{kg}$, so the transfers are way too chunky to be reduced to continuous momentum / energy flux calculations and still hope for an accurate picture.
Some further reading on the subject of quantum optics can be found in R. Loudon, "The Quantum Theory of Light" and the first chapter of Scully and Zubairy, "Quantum Optics".
Here are some wordy, math-free answers.
a phonon is the minimal amount of energy which can be stored in an lattice vibration in a given mode
Sounds good.
I.e. that when a crystal vibration interacts with matter it does so by the creation/destruction of whole phonons at a time, which may also get absorbed at more or less precise locations, e.g. the energy of a single phonon is absorbed by a localized electron.
A phonon is a periodic motion of the atoms in a solid, so I'd argue that it's always interacting with matter since it is matter in motion.
Localization of phonons is a tricky business. The textbook derivations for phonons result in vibrations (waves) that extend through the whole material. However, they're usually treated as localized.
You can make any function by adding up waves of different wavelengths (the waves form a basis), so you can build up localized phonon "packets" from lattice vibrations of different frequencies. Unlike photons, phonons have non-linear dispersion relations -- meaning that waves of different frequencies travel at different speeds (unlike light where all frequencies travel at the same speed, at least in a vacuum), so the packets will eventually fall apart if left alone. However, they can stick together long enough that they can be thought of as particles. If the frequencies in the packet are of a narrow range, you can think of the packet as having a frequency equal to the average frequency of its constituent waves.
This localization makes sense if electrons are likewise localized. If an electrons scatters with a phonon, and that electron is localized, that means the electron is only really interacting with nearby atoms. So, any lattice vibration the electron creates should be initially localized to that region too.
I should add that a major form of phonon scattering is with other phonons. It turns out that you can't have two-phonon processes (two phonons colliding and create two other phonons); you can only have three-phonon processes and higher (e.g. two phonons merge to create a third). You don't have to think of these processes as being localized in space.
Finally I would like to understand how phonon exchange can effectively establish an attractive force between electrons
The atoms in a lattice are charged, so they can pull on nearby electrons. If several atoms are pulling on one electron, then the atoms are effectively pulling on each other and are brought closer together. If the electron is moving, it can leave a wake of atoms that are closer together (a phonon). Atoms being closer together means more positive change in an area, and that in turn can draw in another electron -- effectively attracting the electrons together.
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Best Answer
In the particle model, as you rightly pointed out, absorption is described in terms of photons — quantized packets of electromagnetic energy. When a photon with energy E encounters an atom, it can be absorbed if the atom has an energy gap of the same size E. This process is quantitatively described by Einstein's theory of the photoelectric effect and is a cornerstone of quantum mechanics.
Now, let's turn to the wave model. In classical electromagnetism, light is treated as a wave, characterized by its electric and magnetic fields. When an electromagnetic wave encounters a material, its oscillating electric field can interact with the charged particles (such as electrons) within the material. This interaction depends on the frequency of the electromagnetic wave and the natural frequencies of the electrons in the material.
If the frequency of the incoming electromagnetic wave matches a natural frequency of oscillation of electrons in the material, resonance occurs. At resonance, electrons absorb energy from the wave efficiently. This absorption leads to a transfer of energy from the electromagnetic wave to the material, resulting in the wave's amplitude decreasing as it passes through the material — a phenomenon we interpret as absorption.
So, in the wave model, absorption is not about discrete energy packets being transferred, but rather about the resonant transfer of energy from the wave to the material at specific frequencies. This model is particularly useful for explaining phenomena like why certain materials are transparent at some wavelengths but opaque at others.
In conclusion, the absorption of electromagnetic radiation in matter can indeed be described using both the particle and wave models. While the particle model (photon absorption) provides a more intuitive explanation for discrete energy exchanges, the wave model (resonant energy transfer) offers insight into the frequency-dependent nature of absorption. Both perspectives are complementary, reflecting the dual nature of light as both a wave and a particle, a fundamental concept in modern physics.