What is wave particle duality? I am sort of confused about this. Wave particle duality says that sub atomic particles are waves.  There is something more though. What is the actual meaning of wave particle duality?
 A: I answered a question related to this a few days ago, so I suppose I'll try to summarize it here.
Wave particle duality doesn't really say that waves are particles. It says that "particles" aren't really particles, nor are they really waves, they're just little objects that have some properties of waves and some properties of particles, and there are certain situations where one is more visible than the other. I've heard it said (in a very rough sense) that subatomic objects travel like waves, and interact like particles. Again, this is a huge simplification, but there's an important intuition, which is that these objects are always a little like waves and a little like particles. We can describe their position by a function that tells you the probability that the object will be at a particle point in space at a particular time; this function takes the mathematical form of a wave, so we call it a wavefunction, and this is the sense in which particles are like waves. When these objects interact, however, we tend to see them more as particles, like little classical marbles.
The double-slit experiment is a good example of this. Once more, I emphasize that this is a very big simplification, but just for the purposes of giving you a bit of context, we can imagine that as the electron travels through the slits, its wavelike character is more obvious, and so there are noticeable behaviors we normally attribute to classical waves, like interference. When it collides with the backboard, however, its particle-like character is more obvious, and so we see a single point where the electron collided with the wall. But at all times, the electron had both wave and particle characteristics, and that's the essence of wave-particle duality.
A: It's a little easier to think about photons for starters, because we are more used to them.  But it turns out that all the other particles in nature work this way ...
Start with a classical picture of a propagating electric field: a classical wave.  Consider it in the context of a double-slit interference experiment.  The electric field exists everywhere, at least everywhere we have any interest: after the source, between the source and the slits, within the slits, between the slits and the screen, at the screen. We can increase the intensity of the field as much or as little as we like.  We have no trouble understanding this.  
Add quantum mechanics:  We no longer can change the intensity of the field in small amounts. We can only change it in quantum steps.  When we raise the intensity by one quantum, we say that "we have created a photon".   Lower the intensity by one quantum, and "we have destroyed one photon".  Remember that the field exists throughout space.  The "photon" has no specific location.  Each time we create or destroy a photon we add or subtract a quantum of energy from the field.  Not only that, we add or subtract momentum from the field.  The field carries energy and momentum.
Suppose there is an atom in the field with energy levels whose energies differ by the energy of one quantum of the field.  There's a chance (a probability) that the field will interact with the atom, in which case a photon will be destroyed, and the atom will take the energy and be raised to the higher energy state.  It will also take the momentum from the field, and get a little kick in the direction of the field's momentum.
From the point of view of the atom, it looks like it's been hit by a particle.  It has more energy, and has picked up some momentum. (Glossing over some conservation issues.)  The interaction has occurred at the location of that atom.  But the field excitation, the photon, lives everywhere in a pattern described by the classical wave.  The field/wave obeys all the laws of wave physics, but the interactions happens at a particular location, and is indistinguishable from a particle collision.  Wave and particle. 
At the photographic film at the screen of the interference experiment, the field interacts with the molecules of the film at individual locations.  But where those interactions might occur are determined by the law of interference.
It turns out that all elementary particles can be described by fields.  So there are wave equations that tell us what the wavefunction is ... where interactions might occur.  These fields are a little more complicated because of other conservation laws that apply in the subatomic domain: you can't destroy an electron as you can a photon, but you can redirect an electron, which is like destroying one electron and creating another that moves in a different direction.  Or neutron can be destroyed while simultaneously creating and electron, proton, and neutrino.  And so on.
A: It was found through the law of dynamics we calculated and observed the particle to follow on a macroscopic scale failed miserably when applied to microscopic objects. The particle showed some properties that were essentially found to be similar of waves, in the sense then when a lot of particles are observed they give properties similar to that what a wave distribution would have given. As a result we lost out on the determinacy of a particle's state and it was replaced by calculating probabilities whether a particular state would show up on actual measurement. Now the function that gives this probability in each state is a function that follows the same differential equation in some sense that the actual wave equation follows, however things like wavelengths etc. are momentums,energy etc. in this function.
So in the end it is all just about the mathematical function which is used to say that the particle has dual nature. There is actually no wave there whose energy you can measure or which you can interact with, whole energy of the particle is located at one place only.
Kind of like F=ma is replaced by Schrodinger's equation, now the Schrodinger's equation is similar to the wave equation of the wave you see on a string, hence you can physically assume that the particle is travelling like a wave in a very crude sense, kind of like if some other thing's dynamics follows F=ma, you can consider it to be like a particle, however mass will not mass in this case here, it will be some other property completely. However on a heuristic level, it will help you solve and think about problems.
