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If you're traveling relative to the radiation source (or vice versa, depending on how you look at it) then yes, electromagnetic radiation will be blue shifted if you're approaching the source and redshifted if you're moving away from the source.

That kind of shift is (I believe) the first evidence we had of the expansion of the universe.

If you're asking about other kind of radiation, like alpha and beta particles, I have if the relative velocity has any impact on how they're perceived. I mean an electron is an electron, regardless of velocity, as far as I know.

Yes, the redshift observed from stars in distant galaxies indicate that they are moving away from us. That the universe is expanding.

Yes, radiation in relative motion towards or from you is "pitch shifted". The concept of redshift comes from this, if you do the calculations you have that if a light source moving away from you is "redshifted", i.e. the wavelengths are shifted towards the red (they get longer) with respect to a still source. The opposite is called "blueshift", since when the source moves towards you it's shifted towards the blue (shorter wavelengths). In experimental physics the blueshift/redshift is widely used to measure astronomical distances (you know, that redshift thingy that astrophysicists use to indicate cosmological distances, that's expansion-induced redshift), and also it's used to evaluate relative speeds. As an example, if you look at the spectrum of stars in Andromeda, since you more or less basically know already the theoretical (still) values of the spectrums, what you get is that differences from the theoretical value are exactly what you'd expect in a rotating disk, with one side coming towards you (blueshift) and one going away from you (redshift). With appropriate measurements of this you can therefore know how fast it's rotating and how quick is coming towards or further from you. Note that this shift is always there when there's relative movement, therefore the cutoff velocity for seeing it depends only on instrument sensibility, don't expect hopping on a very fast plane and starting to see the world in front of you bluer and the one behind you redder 😅

For the case of alpha and beta radiation (gamma and neutron too actually), yes, it would be red/blue-shifted. You'd see that in energy shifts since E~hc/λ. You'd still be able to discern the type of radiation via the usual means (charge, penetration depth, interaction,...) but its energy spectrum would be shifted

I get that everything undergoes a Lorenz transformation, but I don't quite get what frequency shifting means in terms of a fermion.

With a photon if you know the frequency it's supposed to have, you can measure the frequency it has and see the delta.

What's the comparable measure of an electron?

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The wavelength of an electron can be found with de Broglie's formula:

l=h/p

where h is Planck's constant and p is the momentum of the electron. This is the wavelength that gets blue shifted.

If we pretend fermions are classical particles and the rules of quantum mechanics don't apply, then we would see that the particle would have a different kinetic energy and impuls, since those quantities depend on the velocity and hence the reference frame. So if I shoot you with a bullet and you move away from me at 90% the bullets speed, it wouldn't smack you as hard. In other words: because of the movement, the impulse and kinetic energy of the particle was reduced. Totally classical concept that is in complete agreement with the quantum world. I guess that answers your question.

Now we can relate these findings to the non-classical finding that fermions undergo interference, which we can measure e.g. in a slit-experiment. We see this effect in electron microscopes near the edges of objects, for example, where very non-classical diffraction happens. Or when we shoot electrons through pulverized crystals and observe the diffraction pattern. So, we can work out the frequency. From either of those.

But if fermions have a frequency and an energy, how cool would it be if these two seemingly unrelaged properties were actually related? Turns out we can test this with the above described accelerator experiments and the two are totally the same physical property and perfectly related, just measured through different effects. Nice!

Now we figure, fermions should abide to Doppler's effect, yes? Because both sound waves and EM waves do it and we just found out that electrons behave very wavy? Now I didn't learn about a particular experiment that tested this, but since QM theory has not yet collapsed I assume someone tested it and it all agrees nicely.

In the end we started with classical particles and got a nice test for particle-wave-duality.

No one is pretending fermions are classical, I'm remaining in the domain of classical quantum mechanics, there is no relativity, there is no spin and therefore no fermion or boson statistics. The doppler effect is a direct consequence of the wave nature of the functions that describe both light and particles (as probability waves). Plus energy and wave frequency have always been related. What quantum mechanics adds is that particles have discrete energy levels. Looking at the photoelectric effect in particular you can prove that energy is directly proportional to the frequency of light. I don't understand what wasn't clear on my explanation, in case reply to me and ask please so that I can better explain

Oh well, I thought you were OP and asking a question. Never mind then.

I guess we could argue about what "classical physics" means, I would say QM and GR don't belong in there and SR is mayby a little gray-area-ish, because "classical" EM (the one Maxwell describes) has special relativity tatooed on its forehead. But that's not an interesting discussion imho. It's like asking whether a calzone is a pizza or not.

I think we try to explain different things. I made a suggestion how we could discover the wave nature of electrons, which is what I think OPs question was about. I may have misunderstood what you were trying to do. It seems you start with "obviously everything is a particle and a wave" which seems a few steps ahead of the question.

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Depends a bit on what you mean by that word "radiation". Radiation from radioactive elements is mostly electromagnetic (EM) energy ("light" even if it might be outside visible range) so redshift or blueshift will happen (doppler effect as it applies to light/EM radiation, which moves as a waveform of fixed frequency).

If it is a particle emission (alpha or beta decay, say), then there is no waveform involved so there will be no change in perceived wavelength at impact with the particle (no wave at all). The impact will be less (or more if a head-on collision) forceful, is all.

There is always also some EM emission whether or not there is particle emission, and the EM emission will do the wavelength shift if the receiver is moving fast enough to cover a significant proportion of a wavelength during the time interval of the individual wave. If the receiver moves a third of a wavelength distance in the time it takes for a second wave to arrive, then the perceived wavelength will be reduced or increased by as much as 1/3 (maximum change if wave and receiver are moving in the same or opposite direction but less if the two are moving obliquely).

Redshift with light from stars and galaxies is how we know that the universe is expanding; the further away the source, the more it redshifts, and further away means more time since the light began the trip. How we know how far away the source is, well, that is a different problem.

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