Driving force and mean of a particle wave function

Question

I am currently undergoing a course on introduction to quantum mechanics and we took the historical approach. I'm currently at DeBroglie wavelength. He introduces the wave particle duality in matter, from which will arise the wavefunction of a particle for example. And we have seen that this wave can take a lot of forms and shares most properties of other waves that we have seen (e.g. water waves, electromagnetic waves, and so on), but I can't seem to grasp two things:

1. We know that physical waves (water waves or sound) need a physical mean of propagation in order to transfer its energy, but electromagnetic radiation doesn't. It can propagate in vacuum! However, this idea that waves propagate in nothingness does not fit in my mindset, it just can't be! Although, one could argue that, for electric charges, the waves travel due to perturbations in their own field, since it extends itself to infinity and so any changes in the field would be noticeable (for photons I'll guess its a bit different, but that's another question). And so, it kind of has some "mean" to propagate, but I can't see where or "on what" do the wave functions live or travel! Do they simply "exist"? That can't be. What created them? and this leads me to the next question;
2. What is the wavefunction driving force? we can easily create sound, and electromagnetic waves by applying some driving force to their system. But for a particle we assume that it merely has a wave associated to it. But how did it even get there in the first place? Does this wave exist from the beginning of time? We could say and argue intensely "It's a property of the particle!!! and when we "mess" with the particle we change the wave function so this wavefunction is always changing, but where does it come from??

I know that it will always be impossible to answer all the why's but if we have a theory that fits beautifully to our experimental data it's worthwhile to answer as many fundamental questions as possible

Answer 1

The setting of physics is not a medium but the underlying space. A medium is to the space what a carpet is to the floor.

So if you write for instance electromagnetism through its classical field theory formulation you will have that the dynamics is contained in the fields, whose excitations can (in some cases) be seen as particles. These perturbations will propagate through space. If you are skeptical about the physical relevance of a field, you can follow the propagation of energy or of even more concretely observable quantities. Every physical information of this kind propagates as dictated by the dynamics of the fields (they are just a convenient set of quantities).

What drives the field dynamics? The answer to this comes by writing the equations of motion for the field. Depending on the theory (the kind of particles in play and how they interact) you will have different "forces" driving the dynamics of each field. The principle underlying this all is the principle of stationary action (you can get from this Lagrangian formulation to the Hamiltonian through the Legendre transformation and reinterpret it with the Hamiltonian flow). So the Lagrangian (or the Hamiltonian) serves at the goal of establishing what the driving forces for the fields will be.

Note that (combinations of) the fields can be seen as canonical variables in the Hamiltonian formalism; this may help you to trace back to quantum mechanics what I wrote.

Now to your second question. You can find solutions to the field equations which have an excitation from the beginning of times to the end: in all this time, this excitation evolves, moves, bounces, interacts with no problems. The origin and fate of the "particle" is captured in this formalism as the excitation can be created and can ultimately vanish (usually being changed into excitations of other fields, i.e. decay).

When dealing with the wave function you may feel like bypassing or renouncing to this global knowledge of the history of the particle, but this is physically consistent since you know what you are observing at the moment you do the observation. In this sense the field theoretic approach maybe enables you to grasp more easily the information about the complete history of the dynamics, but this can be also known in quantum mechanics by use of correlators.

I hope I answered at least a good part of your question!