Coulombs law and the Dual nature of subatomic particles Coulombs law states that,
'The force between two particles acts along the line joining the centers of the charges'
The Quantum Model says that subatomic particles are waves.
How can there be a 'center' for a wave?
Lets say we have 2 electrons separated by a distance 'r' so in accordance to Coulomb's Law the force acting on both will be in opposite directions and along the line joining their centers and will be (1/4pi ε) * e^2/r^2* Here it assumes electron to be a particle, right?
So I'm having problems understanding how this is true with the Quantum Model in mind?!
 A: 
The Quantum Model says that subatomic particles are waves.

This is not correct. What the quantum model posits axiomatically  "the probability of finding the particle at (x,y,z,t)  is described by the complex conjugate squared of the wavefunction that is the solution of the quantum mechanical equation for the problem at hand".
For two electrons, the coulomb potential is entered in the quantum mechanical equation and the $Ψ$ solution contains  the effect of the coulomb potential, and  the probability distribution of electron- electron scattering will be predicted by  the $Ψ^*Ψ$. There is no way to predict a specific electron on a specific electron, it is all probabilistic.
This is how the diagram that gives the calculation looks in quantum mechanics:

At the level of quantum mechanics particles , classical mechanics cannot be used to predict interactions.
A: To truly understand the Nature of a collision between two charged particles you need quantum field theory. The particles travel in all possible ways from two initial points in spacetime to two final points in spacetime (or from two separated points in momentum space to two other points, which is more commonly done, as it is the momentum that is measured in practice) in all possible ways you can imagine. They can do this by not interacting at all, interacting to first order (one exchange of photon), interacting in second-order  (exchanging two photons in all possible ways), interacting by means of three virtual photons, etc. Every Feynman n-th order diagram contains all possible ways for two charged particles to travel from two initial points to two final points with n virtual photon exchanges. There is an infinity of these diagrams all contributing to the process. Some are shown below (in the second picture there are the so-called disconnected diagrams shown, which I'll leave to you to figure out why they are called disconnected). One has to take all diagrams into consideration for calculating the scattering amplitude of the process. Your examples are just a few. But already a first order calculation will do to give a good approximation of the amplitude that the two particles will end up at the two other points (in space or in momentum space). All contributions will be such that the final result will look like two particles that have traveled along a straight line.


The above diagrams are Feynman diagrams of first and second-order respectively. The external lines represent the particle on which measurements can be made (in the momentum representation). These are connected diagrams. There are also disconnected diagrams:

To address your question, you can assign a middle to the wave function. Say as the most probable place for a particle to be.
A: Your statement of Coulomb's law assumes spherical charge distributions and is not general. The general statement of the Coulomb energy of two charge distributions $\rho_1$ and $\rho_2$ is
$$E_C = \int \int d{\bf r}_1 d{\bf r}_2 \frac{ \rho_1({\bf r}_1) \, \rho_2({\bf r}_2) }{ |{\bf r}_1 - {\bf r}_2 |} \, .$$
For two electrons quantum mechanics states the Coulomb energy as
$$J = \int \int d{\bf r}_1 d{\bf r}_2 \frac{ |\psi_1({\bf r}_1)|^2 \, |\psi_2({\bf r}_2)|^2 }{ |{\bf r}_1 - {\bf r}_2 |} \, .$$
in complete agreement with Coulombs law.
Note however that electrons may also experience exchange interaction
$$K = \int \int d{\bf r}_1 d{\bf r}_2 \frac{ \psi^*_1({\bf r}^{\,}_1) \psi^*_2({\bf r}^{\,}_2) \, \psi^{\,}_1({\bf r}^{\,}_2) \psi^{\,}_2({\bf r}^{\,}_1)}{ |{\bf r}_1 - {\bf r}_2 |} \, .$$
due to the antisymmetry many-electron wave functions under exchange of two electrons.
