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My text book says-
When a monochromatic light is incident on a surface separating two media, the refracted and reflected light both have the same frequency as the incident frequency.
Can anyone explain why? I think we must look into the behavior of the atoms of the oscillator to understand the above statement.
In refraction and reflection the incoming electromagnetic wave causes the electron density of the refracting material to oscillate. This happens because at any point in space the wave produces an oscillating electric field (and magnetic field, though that isn't relevant here) so any material that has a non-zero polarisability will respond by developing an oscillating dipole. This oscillating dipole then emits EM radiation, as any oscillating dipole will do. However the emitted wave will have a phase shift relative to the incoming wave, and this causes the velocity of the EM wave in the solid to be different from the speed in the vacuum. Hence the refractive index is different from 1 and we get refraction and reflection. A search of this site will find several questions that go into this process in more detail.
The point of all this is that the oscillations of the electron density in the material are at the same frequency of the incoming wave because they are driven by it. Therefore the frequency of the reradiated light is also the same frequency as the incoming wave. The process cannot change the frequency of the light.
The energy of a photon doesn't change when moving from one medium to another as pointed out by Andrew in a comment.
Considering that $E = h\nu$, $\nu$ being the frequency of the photon and $h$ Planck's constant, we see that the frequency has to stay the same when going from one medium to another. Since the frequency is the same, then the wavelength of the refracted photon will change in the process.
It's actually important to recall that this is true only by definition, because it's part and parcel of what refraction means. But refraction is not the only thing that can happen when light passes between two mediums, even though it is probably by far the most common thing that happens in this situation.
There are solutions of Maxwell's equations where light passes between two mediums and the interaction of electromagnetic field and medium is linear, and, experimentally, we observe these interactions often and in keeping with Maxwellian theory. In such a case, the physics is as described by John Rennie's Answer.
But intense beams incident on a frequency doubling or other nonlinear material, where the physics is no longer linear, do not conserve the frequency. Hence my somewhat pedantic point about being true by definition. One can't prove that frequency is conserved in general, as shown by the nonlinear counterexamples.
There are very good intuitive answers already, I would like to provide a mathematical version.
The propagation of electromagnetic fields is governed by the Maxwell equations. A self-consistent solution of these equations has to fulfil certain conditions at the interface of a material.
These conditions have to be fulfilled at all times.
Say the conditions match at a certain time. You know that wave solutions oscillate with $e^{i\omega t}$. So if $\omega$ is different in the two media, then you clearly can't fulfill the interface conditions at subsequent times. Therefore the frequencies have to be equal.
Bottom line:
Consistency with Maxwell's equations requires that the frequencies in the media are the same.
Light reflects and refracts due to the interaction of incident light with the atoms of the medium. These atoms always take up the frequency of the incident light which forces them to vibrate and emit light of same frequency. Hence, the frequency remains the same.
