Louis de Broglie said that electrons are waves. But how they repel each other?
Answer
Electrons aren't waves or particles. Instead they are excitations in a quantum field called the electron field. The interactions between electrons are described by the equations of motion of the electron field.
However de Broglie was correct in the sense that if we consider an isolated electron this can be approximately described as a state of the electron field called a Fock state, and this is basically a plane wave. So for an isolated free electron it is a very good description to consider it as a wave. There is more on this in my answer to What is a subatomic particle? if you're interested in pursuing it.
However if we have two electrons then the state of the field is not just two Fock states superimposed. If it were then the electrons wouldn't interact so there would be no repulsion between them. In fact we can't solve the equations for the field state directly so we use a technique called perturbation theory to approximate what happens. If we consider two electrons heading towards each other, interacting and then heading away from each other. The calculation is done by summing up a series of calculations represented by Feynmann diagrams. You've probably seen Feynman diagrams like this:
This shows the two electrons interacting by exchanging a virtual photon, but be careful about interpreting this literally as it's just a representation of an integral called a propagator. No virtual photon actually exists. What we are doing with these diagrams is calculating how the two Fock states interact and in the classical limit this gives us the force between the two electrons.
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