The force exerted by a field, \(F=-\cfrac{\partial\psi}{\partial\,x}\) applies for a unit positive test inertia (eg. a unit positive charge). This force is directed towards decreasing energy density \(\psi\). For a negative particle this force is reversed, \(F=+\cfrac{\partial\psi}{\partial\,x}\).
In the case of two positive particles, the directions of decreasing \(\psi\) point away from the two particles and the particle is pull apart, ie they repel.
In the case of two negative particles, the directions of decreasing \(\psi\) push the particles together, but because they are negative particles, the forces are reversed and the particles are repelling each other.
In the case of one positive particle with another negative particle, the positive particle experiences decreasing \(\psi\) in the direction towards the negative particle and is driven by the force towards the negative particle. The negative particle experience decreasing \(\psi\) in the direction away from the positive particle, but since it is negative, the force it experiences is reversed and is driven into the direction of the positive particle. The particles attract each other.
This is consistent with the behavior of charged particles that we are familiar with.
Why does then Newton's apple fall to Earth? Shouldn't like particles repel each other?
Apple is made from earth, on Earth. Apple is like Earth. Apple has the same gravitational potential as Earth. It is moved to higher gravitational potential as it grows on branches of the apple tree. This gained potential is dynamic and is converted to kinetic energy when its support is removed, as the particle seeks to lower its potential energy. How does this happen in the context of energy density? We know that this particle will not experience any force when its energy density is equal to the external energy density of the field surrounding it. At this position, the particle remains at rest. Without any other external force performing work on the particle (ie. no gain in total energy), the particle by itself will move in the direction as to lower the force exerted by the gravitational field around it. (The opposite direction would mean the particle gain energy by itself.) That is to say, the particle will move towards zero force, where its energy density is the same as its surrounding, ie. towards Earth, where it grew from. In this direction, the particle lowers its dynamic potential energy and gain kinetic energy. The apple falls towards Earth. Its energy density remains unchanged. Here, we differentiate between dynamic potential energy or positional potential energy that changes as a particle is displaced in a field from the intrinsic potential energy or energy density of the particle.
This situation is not the same as two like charges repelling each other. It is like a fragment of a charge, with the same energy density as the charge, being displaced from the rest of the charge and falling back towards the charge when it is released.
We have seen before, a particle interacting within the energy density field of another particle of the same type, from which we obtained interesting photon emissions phenomenon.
The notion of like repels and opposite attracts still applies, but not at the quantum level. It applies when the force density exerted by the particle is a constant beyond the upward slope of the force density curve. (cf. post "Boundary Between Wave And Particle Interaction" dated 22 Dec 2014) On Earth, given the size of Earth, the apple and Earth interact as waves.
This suggests that beyond a certain distance into space, apple and Earth do repel each other. This might explain the mysterious extra push deep space probes experience at greater distance from Earth.
