In particle physics, the quantum version of course, we talk about particles -- including photons-, and not much about waves. that's simply because many of the the measuring techniques stem from old classical-physics based instruments. Toward the bottom of the energy chain, we describe electrons going through crystals in terms of waves -- diffraction and all that. The reality, some say, is that sometimes electrons are waves, sometimes particles.However, it is far more appropriate to say that sometimes electrons BEHAVE like particles, sometimes like waves -- as determined by experiment. We really don't have a clue about the structure of electrons; we know a bit about their behavior. (Is water liquid, solid or gas?)
However, the very structure of Quantum Field Theory, directly involves both wave and particle features. QFT is a big word subject, and is considerably more abstract and mathematically sophisticated than ordinary QM. A key difference is that QFT is about particle transformations -- A->B + C --,
electron->electron + photon.
(Yes, I know about energy conservation; it's not important here and now.)
When you work through the problem you find the following: before radiation there was no photon in the system, then there was a photon. How can we deal with this creation process? What physicists did was to invent the answer. Why not invent an operator -- like position or spin - that changes the number of photons or whatever? Thus we talk about a creation operator and a destruction operator. The Lord giveth and the Lord taketh away.
Phew. The deal is that the standard natural mathematical expression of these transformations involves these creation and descruction operators to create or destroy particles in the system. But they do this in association with the particle's wave-function; both notions are there from the beginning. Ultimately, a creation operator creates a particle and a wave together -- there is a great deal of poetic license here.
Nature is weird, and does not always conform to human notions. Sometimes wave behavior, sometimes particle behavior; crazy making. Who knows from?
Why? -- Lot's of history.
Regards,
Reilly Atkinson