The title is pretty straightforward. I was wondering if you can have light whose wavelength is not a rational number, but an irrational one. It seems to me there is nothing preventing this from happening, but I am not sure since I've never been exposed to such an instance! Is it possible for light to have a wavelength of, say 100*pi nm?
[Physics] Can light have an irrational wavelength
mathematicsvisible-lightwavelength
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It seems like your are not satisfied by answers involving axioms. I think that you instead want to know the motivation behind the axiom beyond just saying that it works. I am not sure if my answer is the original motivation, but I think it can be viewed as a good motivation for the validity of $p= \frac{h}{\lambda}$. While other answers do a great job at going into the theory, I will tackle the question using more of an experimental motivation.
We will first start with the double slit experiment. This experiment is usually first introduced as evidence of the wave-like nature of light, where light emanating from one slit interferes with light emanating from the other (of course a different interpretation is found if we send single photons through the slits and the same interference pattern arises, but I digress). However, this experiment also works with electrons. You get an interference pattern consistent with treating the electrons as waves with wavelength $$\lambda=\frac hp$$
You get maxima in intensity such that $$\sin\theta_n=\frac{n\lambda}{d}$$
Where $\theta$ is the angle formed by the central maximum, the slit, and the maximum in question, $d$ is the slit separation, and $n$ is an integer.
This would then be a way to experimentally motivate/verify this relationship between momentum and wavelength for matter, but what about photons? The double slit experiment does not give us a way to validate $p=\frac h\lambda$ (that I know of. Maybe you could determine the radiation pressure on the detector?). Let's look at a different experiment.
We know that the energy of a photon from special relativity is $$E=pc$$
So, if our momentum relation is true, it must be that $$E=\frac{hc}{\lambda}=hf$$ which is something that can be verified experimentally to be true. The photoelectric effect is one such experiment we could do, where shining light onto a material causes electrons or other charge carriers to become emitted from that material. The higher the frequency of the light, the more energetic the electrons coming from the material are, and the maximum kinetic energy of an electron can be shown to follow $K_{max}=h(f-f_0)$ where $f$ is the frequency of the light and $f_0$ is the material-dependent threshold frequency (i.e. we need $f>f_0$).
I know that my answer does not get to a fundamental explanation of this relation in question, but I hope it shows why one would want it to be a fundamental idea that holds true when formulating QM. If you want a more fundamental explanation, then I will edit or remove this answer due to some pretty good fundamental answers already here.
I think you have cosmic rays and electromagnetic radiation a little mixed-up.
We all know that without Earths magnetic field, electromagnetic radiation from the sun would cook us within minutes.
No - the Earth's magnetic field protects us from cosmic rays. High energy charged sub-atomic particles, mostly from the sun. The Earth's atmosphere does protect us from Ultra-Violet radiation (i.e. light) which would kill us.
Since visible light is the same thing as cosmic rays, except that its a different wavelength,
No, cosmic rays are charged sub-atomic particles (protons, electrons etc). Visible light, and UV, x-rays, gamma-rays, infrared, are all electromagnetic radiation of different wavelengths
I was wondering if it were possible to use magnetic fields (they would have to be pretty strong) to essentially "block" light the same way it blocks cosmic rays?
Not directly. But magnetic fields do affect how light passes through certain materials. You can use this effect to make very fast shutters by passing light through a crystal and changing the magnetic field.
Best Answer
I agree with you, there is nothing preventing this from happening, not to mention that if it is rational for a certain unit, it could very well be irrational fro another unit (example new unit = $\pi$ meters). And as the choice of unit is arbitrary...
Addendum: for unit dependent quantities, one can chose units that make a given measure rational or not. But there are other quantities where we have no choice as for example $\pi$ or the proton-to-electron mass ratio (those are dimensionless constants).