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since electromagnetic radiation possess the property of both wave and particle(photon). and both theory are applicable but how we have to find out that which theory is suitable or applicable in particular explanation. for example in laser traping of atom we use photon concept rather than wave why?

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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 You may recall that energy may be given to/withdrawn from a quantum harmonic oscillator only in discrete packets.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".

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since electromagnetic radiation possess the property of both wave and particle(photon).

There is some confusion here: electromagnetic radiation is a classical physics concept, and yes it does display wave behavior classically.

The photon is an elementary particle, which depending on the experiment displays a wave property or a particle property.The same is true for all elementary particles. The classical electromagnetic wave is composed of zillions of photons which conspire to build up the frequency and behavior of the classical wave.

and both theory are applicable

the classical wave is applicable to for classical optics,

but how we have to find out that which theory is suitable or applicable in particular explanation. for example in laser traping of atom we use photon concept rather than wave why?

In laser trapping the transitions are quantum mechanical and the photon is used because it is a quantum mechanical entity suitable for describing the microcosm. The wave/particle dual nature concept for an elementary particle defines its indeterminacy when trying to localize it.

A classical wave is a collective emergent phenomenon from an ensemble of photons and is suitable for macroscopic observations. When atomic transitions are studied the classical wave is not suitable.

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  • $\begingroup$ why classical wave theory is not applicable for the atomic transition. $\endgroup$ – Rahul kumar walia Aug 17 '13 at 15:00
  • $\begingroup$ Because an atomic transition is an individual photon interacting with an individual atom. The classical wave does not know about photons. The photons though do build up an emergent classical wave , as shown in the link I supplied. $\endgroup$ – anna v Aug 17 '13 at 15:08
  • $\begingroup$ From Ben's link I see that optical trapping is of dimensions suitable to use the classical electromagnetic form, since the particles are in dimensions of microns. My answer is suitable to the creation of the laser light, but that is an other story, and for atomic trapping as in prl.aps.org/files/RevModPhys.70.721.pdf . Once atomic dimensions are reached quantum mechanical entities have to be considered,the photon in this case. $\endgroup$ – anna v Aug 17 '13 at 16:27
  • $\begingroup$ zillions of photons? i thought certain experiments showed that individual photons could exhibit wavelike properties? $\endgroup$ – Michael Aug 19 '13 at 3:52
  • $\begingroup$ @Michael The wave like properties of all elementary particles are held in the probability function which is the square of the wavefunction of the particle. The photon is an "elementary particle". In the case of the photon the synergy between the quantum level and the classical maxwell equation level that gives the amplitude of the classical wave in the varying electric and magnetic fields is such that the same frequency appears in both cases. An individual photon is defined just by the spin and energy and mass=0. The link to Motl's blog derives how the synergy works mathematically. $\endgroup$ – anna v Aug 19 '13 at 4:24
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both theory are applicable but how we have to find out that which theory is suitable or applicable in particular explanation.

There is no particle theory of light. There is a wave theory and a wave-particle theory. For example, the equation $E=h\nu$ can't be an element of a particle theory of light, since the left-hand side refers to a particle (the amount of energy per particle), and the right-hand side refers to a wave (the frequency of the wave).

Sometimes people will claim that light acts like a particle in some experiments and a wave in others. This is wrong for the reasons given above, and also because it implies that there are no experiments in which it acts like both. For example, you can observe double-slit diffraction with individual photons.

The wave-particle theory is valid in all cases. So the question becomes this: under what circumstances is it valid to save ourselves work by using the pure wave theory as an approximation? One way of answering this is that the pure wave theory applies when the density of photons is high enough to allow us to talk about measuring the value of a classical field at a certain point in space. A quantitative criterion for this is the concentration defined by the average number of photons found in a volume $\lambda^3$, where $\lambda$ is the wavelength. If this concentration is large, then the classical theory applies.

or example in laser traping of atom we use photon concept rather than wave why?

I could be wrong about this example, since it's not my specialty, but I don't think it's true that laser trapping has to be discussed in terms of photons. The laser beam has a high concentration according to the definition above, and can be treated classically. This description is totally classical:

Proper explanation of optical trapping behavior depends upon the size of the trapped particle relative to the wavelength of light used to trap it. In cases where the dimensions of the particle are much greater than the wavelength, a simple ray optics treatment is sufficient. If the wavelength of light far exceeds the particle dimensions, the particles can be treated as electric dipoles in an electric field. For optical trapping of dielectric objects of dimensions within an order of magnitude of the trapping beam wavelength, the only accurate models involve the treatment of either time dependent or time harmonic Maxwell equations using appropriate boundary conditions.

(The preceding paragraph in the Wikipedia article does talk about photons, but that doesn't mean that the use of the photon theory is mandatory.)

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