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I read that interference in seen in waves with similar frequency, so if interference is possible for cosmic rays then wouldn't we need another cosmic ray to see this effect? If interference is possible then what are the necessary conditions for it? Suppose we make another wave by constructive interference of many waves to finally make a single wave of similar frequency as cosmic rays', then can't we observe constructive or destructive interference between our big wave and cosmic rays?

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Cosmic rays are usually particles coming down to the surface of the sea and are various,muons, protons, electrons, from various sources and with a spectrum of energies.

cosmic

Cosmic flux versus particle energy

The "ray" was assigned as a generic term, at the same time when nuclear radiation was called alpha beta and gamma rays. We know now they are helium nuclei, electrons, and only gammas are electromagnetic radiation.

I read that cosmic rays in seen in waves with similar frequency

These "waves" describing the particles counted in the figure, are not energy waves, as light that shows interference patterns. They are probability waves. Please study this answer describing how, even for photons, it is a probability distribution that shows the interference. More so for particles.

This means that to show an interference of cosmic particles with each other one should have more than one particle in a region in space compatible with the heisenberg uncertainty, so that the two or more wavefunctions could interfere and give a probability interference pattern.

Looking at the flux above, even for the low energy particles ( large wavelengths of possible probability interference) there are only about 1000 arriving for a meter square, which is 10000 cm square, per second. Considering that quantum effects need nanometers and nanoseconds to be detectable, there is not enough flux coming down from cosmic rays to display interference between particles arriving.

edit after edit of question

Suppose we make another wave by constructive interference of many waves

The beauty of the probability waves, in quantum-mechanical-particle behavior, is that they do not have to arrive together to show interference patterns. This is an experiment one photon at a time:

sinlgphot

Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames

each dot the footprint of a photon. There exist similar experiments for single electrons at a time.

to finally make a single wave of similar frequency as cosmic rays',

So? (please note that these are particles)

then can't we observe constructive or destructive interference between our big wave and cosmic rays?

No. There would still be the problem of trying to locate a needle in a haystack ( a cosmic particle in a specific region that detectors-and-created-beam can be localized at dimensions where quantum mechanical effects can be measured)

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