Historically wave and particle has been perceived as totally different phenomenons (before 20th century). Now is it widely accepted and there are experimental results to show that in fact both matter and light have a dual nature. Let's take light for example. Depending on the experiment it could behave as a wave(as in interference and diffraction) and it could be taken as a particle (as in explanation of photoelectric effect). If it has properties of both wave and particle at the same time, shouldn't we be able to explain the experiments with both wave and particle nature and not selecting either wave or particle? Why particle or wave, why not particle and wave? Please help me understand it and this question has been bugging me for quite some time now. I understand the standard explanation of both but I am having trouble putting them together. If there is anything I am missing please provide some link or reference to read.
Light and matter are neither particles nor waves. We use the particle and wave analogies to allow us to apply some level of intuition to the effects. The interference effects in light (and matter interference experiments) follow similar equations to waves in water so when we want to talk about the interference, we call light and matter a wave. When we want to talk about effects such as the absorption of a discrete quanta of light in the photoelectric effect or the billiards like collisions of two atoms in a gas, we talk about light and matter as particles.
Reiterating: light and matter are neither particles nor waves... they are something else, something different than a sum of the two catch phrases. That difference generates amazing effects that defy our macro-world based intuition.
As for calculations, quantum field theory (QFT) treats both matter and light in a way that handles both the wave and particle like effects in one theory. One thing to note: light and matter have significantly different properties in QFT. The 'particle' properties of light are not the same as the 'particle' properties of matter.
In many cases the particle interpretation is perfectly right.
It's known as the path integral formulation. Basically, what you do is consider that when a particle travels from point $A$ to point $B$, it goes through every possible trajectory (including back and forth in time) all at the same time! In fact, every paths have the same probability, they just differ in phase.
An interesting example would be the treatment of the double slit experiment. If you sum the contribution of all possible paths, you get the classical result using waves.
This way, you can forget about the wave nature of particles (in your question, photons). I think that there are other possible explanations which don't need waves, like Heisenberg's picture.
Because it's a matter of our perception
To quote Jason A:
Light and matter are neither particles nor waves... they are something else, something different than a sum of the two catch phrases. That difference generates amazing effects that defy our macro-world based intuition.
I think math is the only way to exactly describe it. The point is, we don't have that thing in our language. We have a name for a phenomenon of transmitting energy in a medium: wave. We have a name for a separated inelastic entity: particle. We just don't have a name for a thing that behaves in appearance like both, so we have to invent one: wave - particle duality.
Similarly, that's also the difficult of translators. From the Wikipedia page of the German word Geist:
Depending on context it can be translated as the English words mind, spirit, or ghost, covering the semantic field of these three English nouns.
No single English word can translate geist wholly, correctly and satisfactorily. No single English word can translate the mathematical nature of light. It's the matter of us, not the matter of nature.