What happens in a universe with only two electrons?

Question

What happens in a universe with only two electrons? Do they stay as waves or do they collapse into particles?

Answer 1

The question doesn't have an answer because electrons aren't waves and they aren't particles. This is a common source of confusion, and has led to the endless debates about wave particle duality. Quantum systems are described by a wavefunction that can behave as a wave in some circumstances and behave like a particle in others.However it is vital to understand that the description of an electron as a wave or as a particle is an approximation that works in some circumstance but not in others.

To describe two electrons alone in the universe simply requires us to calculate the wavefunction that describes them. If we assume energies are low enough not to require a relativistic treatment we calculate the wavefunction by solving the Schrodinger equation.

The trouble is that the Schrodinger equation is a partial differential equation, and like all (most?) partial differential equations it doesn't have a unique solution. Instead there are an infinite family of functions that are solutions to the Schrodinger equation, and to find the solution that describes our system we have to use the initial conditions i.e. the state of the system at time zero.

You specify your system is two electrons isolated from anything else, but what is the state of the system at time zero? Suppose we start with both electrons having a well defined momentum, in which case the uncertainty principle tells us they don't have a well defined position. In this case the wavefunction will be wave-like, by which I mean describing the electrons as waves is a good description.

Then suppose we start with both elections having a well defined position, in which case the uncertainty principle tells us they don't have a well defined momentum. This initial state is particle-like, by which I mean describing the electrons as particles is a good description. However the uncertainty in momentum means the particle spreads out with time and becomes more and more wave-like as time goes on.

The point of all this waffle is that we can't answer your question because you've left out something very important i.e. the initial conditions.