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Assume that a charged particle accelerates/vibrates at a certain frequency and as a result produces e.g. red light. If every frequency has just a single wavelength associated with it, then the charge can never be a source of white light. So how do we get white light?

What is the equation that relates the vibration frequency of a charge with the wavelength emitted?

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White light is an ideal. You could argue that we never actually get truly perfectly white light. However, many processes are capable of outputting a very smooth spectrum because there's enough different energy transitions possible that we cannot distinguish any quantization.

An excellent example of this appears in the conduction bands for many metallic compounds. In these bands, energy of electrons is quantized, just as it always is. However, there are so many bands that they start to blur together (both by actual overlap and because scientific instruments just can't detect the boundary).

As for the equations, it's not actually just the vibration that matters, but the changes in states. If you can determine the difference in energy between the two states, you know the energy of a photon that would be emitted during that state change.

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The vibration frequency is also the emitted EM wave frequency, so the relation you are looking for is simply $\lambda = \frac{c}{\nu}$. If you force an electron to oscillate with a sine wave, it will emit a sinusoidal wave with a single frequency, but if you choose a more complicated function, the emitted spectrum will also be more complicated. In principle you just have to apply an inverse Fourier transform to the spectrum you want to obtain and force an electron to oscillate with that function, if I'm not missing some obscure nonlinear effects.

That said, the electron emits photons when it vibrates and even it it emits a full white spectrum, it will really emit photons of different frequencies at different times.

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The question that you are asking seems to be making an assumption that already enforces its conclusion. If a charge oscillates as a fixed frequency, then yes, it can only produce that single frequency. Perhaps, what you really want to know is whether there could be a way for a charge to move/accelerate so that it could be radiating white light. Well, imagine the charge makes random motions. Such motions could potentially produce any frequency. So in that case a single charge could in principle produce white light.

The relationship between frequency and wavelength is independent of the source that produced it, but it does depend on the medium through which it propagates. In vacuum light propagates at the speed of light $c$. Then the relationship between frequency $\nu$ and wavelength $\lambda$ is simply: $$ c=\nu\lambda . $$

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  • $\begingroup$ Independent of source? I thought it depends on frequency of vibration of source $\endgroup$ – user282856 Dec 16 '16 at 18:15
  • $\begingroup$ Yes, the source determines the frequency, but that's it. The source does not determine the wavelength. The medium does. $\endgroup$ – flippiefanus Dec 16 '16 at 18:18
  • $\begingroup$ why? Because speed of light is defined by the medium? $\endgroup$ – user282856 Dec 18 '16 at 17:18
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    $\begingroup$ Yes, the medium determines the speed of light. More generally, the medium determines the dispersion relation. This may be more complicated than the simple equation that relates the frequency and wavelength via the speed. $\endgroup$ – flippiefanus Dec 19 '16 at 4:47

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