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Is it possible mathematically for photons, which behave as individual Gaussian wave packets, to combine in such a way that the approximate result is a plane wave at one particular frequency (i.e., the classical plane wave solution to Maxwell's wave equations)?

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If you have the capability to create an infinite number of Gaussian wave packets of arbitrary size, amplitude and phase then you can mathematically create a plane wave. In the context of physics, a true plane wave would require an infinite amount of energy. – Joe Jun 10 '13 at 21:51
As others have pointed out, a photon doesn't have to be a wave packet. What's more interesting is the purely classical fact that you can build an arbitrary electromagnetic wave out of such packets, so if you can establish that one such packet is a solution to Maxwell's equations, then so is any other traveling wave with any shape. There is a treatment in this style in Purcell, Electricity and Magnetism, ch. 7. – Ben Crowell Jun 14 '13 at 19:04
up vote 3 down vote accepted

You said:

photons, which behave as individual Gaussian wave packets

This isn't strictly true. Photons aren't restricted to being represented by a gaussian wave packet, or really any type of wave packet allowed by Maxwell's equations. A plane wave of one particular frequency is a perfectly valid photon (except that it would occupy an infinite amount of space). Maxwell's equations are the equivalent of the Schrödinger equation for light, and the electric and magnetic fields are the equivalent of the wavefunction. Any solution to Maxwell's equations is a valid photon wavefunction.

Short Answer:

You don't need a superposition of gaussian wave packets, the classical plane wave solution is fine on it's own. It is, however, possible to express the classical plane wave solution as a superposition of suitable wave packets.

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However, if you knew that a photon had to exist somewhere between some two points in space, would that require the photon to be a wave packet, and would you then need to derive the classical plane wave solution from a superposition of wave packets? – abhishek Jun 10 '13 at 22:43
It wouldn't require the photon to be a wave packet, just that the photon wavefunction would have to be zero outside of that region. However, in that case the classical infinite plane wave solution isn't the solution of Maxwell's equations. – Dan Jun 10 '13 at 23:14

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