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.
Take the photon, it is periodic in nature, at times reflecting like a wave, sometimes impacting a particle as a particle. You dont speak of both wave-like and particle-like at the same time since the photon is alternating between a wave-like and a particle-like nature.
I may not answer the question but I might be able to give you the reason for your dilemma. Here's what we know. Light undergoes interference and diffraction and interference. How do we know for sure? Because we did the math and came up with a mathematical formula and tried to see if the experiment agreed with the theory. We figured out the equation of a wave by observing another wave in the macroscopic world (water waves or a wave on a string maybe) and we tried applying it to the interference pattern of light. And it seemed to work fine. Which means it has a wave nature. Now we try to imagine the wave nature of light as thought it were a water ripple or a wave on a string.
Next we observe that the photoelectric effect, where photons knock our electrons from the surface of the metal. Einstein gave a theory and all theories in physics need to have a mathematical expression since math is the only absolute language. We don't need to be able to visualize the meaning of the expression but so long as it conforms to the rule of mathematics, it's sufficient (you can't imagine the 4th dimension, doesn't mean it isn't there). Coming back, Einstein's theory treated light as a particle and the explanation held up. Hence we know that it has a particle characteristic.
Nobody has told how it switches between particle and wave characteristics, if it even switches or if there's something much different happening to give us the result that we are getting.
Then quantum mechanics was born. The thing about quantum mechanics is that it is purely mathematical. It will have a set of initial conditions, operated on by mathematical logic and a computed result. This result is obtained in such a way that we try to get the phenomenon as observable by us through experimentation. The idea that light or electrons should only have either wave or particle nature is purely classical. In quantum mechanics, there is a chance for every thing... every thing!
From the observations we know the behavior and characteristics. We have mathematical expressions (it's called mathematical models) to explain the phenomena (they work). But unfortunately we can't imagine the phenomena... it's just beyond us. That we need to accept. You solve the equation, look at the properties of the equation and infer the consequences.
There is no real 3D model to light or matter right now. You should learn the math to understand it.
But to understand the experiments, you can check out the minute physics videos in youtube regarding dual nature and others. Might give you some idea.