Does the wavefunction propagate as a Gaussian wave or is it just virtual (a mathematical model)?

Question

I am not asking about wavefunction collapse. I do understand that QM is one of the most experimentally proven theories, but there are different interpretations. What I am asking about is whether the wavefunction travels like a real particle or a virtual particle (just a mathematical model).

I have read these questions:

"Reality" of EM waves vs. wavefunction of individual photons - why not treat the wave function as equally "Real"?

Is there a direct physical interpretation for the complex wavefunction?

Is the wavefunction a real physical wave or only a mathematical abstraction?

where Bob bee says:

Yes, it is physical enough. It is real enough. Eigenstates or projections or some other description of the state of the particle, such as a wave function, are equivalent for your purposes. And the fact is that they have an amplitude (sqrt of probability) and a phase. Both are real. So, whichever words, the property (and we call it all those like prob. amplitude, projection into eigenstates, wave function, etc) is real, is physical. Not just a math concept. As you said, otherwise they would not interfere.

On the nature of the collapse of the wave function

where John Rennie says:

The wave function is not an actual wave - like an electromagnetic wave. It is a collection of numbers that summarizes our knowledge about the physical system and that can be used to make predictions. Any attempt to "overinterpret" the wave function and "visualize" it as a real wave that objectively exists etc. is fundamentally flawed.

Do photons oscillate or not?

where ACuriousMind says:

The wavefunction that models a freely travelling particle is usually a Gaußian wavepacket. This moves, but it does not "oscillate".

If the wavefunction really moves, like a Gaussian wavepacket, then it could be a wave, and if it does have amplitude and phase, it could be a wave, and if it interferes (causes interference), then it could mean that it is a wave too.

Yet, wavefunctions has no physical significance at singularities, at the initial singularity, because the wavefunction is a probability density (its square modulus), not a probability, so the wavefunction started to gain physical significance later, the wavefunction itself might be a human creation, like virtual particles, just a mathematical model.

So basically the wavefunction could either be like a real particle (traveling as a wave) or a virtual particle (just a mathematical model).

Question:

1. Does the wavefunction propagate as a Gaussian wave, or is it just a mathematical model, a virtual wavefunction?

Answer 1

The wave function does not “travel” like photon. For one thing the wave function describing two particles in 3d lives in 6-dimensional space, whereas a photon would always travel in 3d space. For another the wavefunction can be complex so one should really concentrate on the time-dependence of $\vert \psi(x,t)\vert^2$ rather than $\psi(x,t)$ itself.

Moreover, a Gaussian wavepacket is just a convenient example because of the simple properties of Gaussian distributions. Since the Gaussian is non-negative, there is nothing oscillating here. In addition, there is nothing to prevent the wavepacket to have any particular other shape, have multiple local maxima (v.g a sum of two separated Gaussians): it basically depends on the initial conditions, i.e. on the initial shape of the wave packet. Moreover, one can craft wavepacket which do not deform as a function of time: the best example would be a particle described by a coherent state in a harmonic well.

Answer 2

The way I understood this question, I think it is something which humanity has no way to investigate. There is no way to guarantee that a description of what is there is the "actual way it is" and not just a mathematical model. There could always be something more fundamental underlying what is known, and it's not even knowable if math is the language nature would use to describe how it works in the end. You can rule out theories from being the "ultimately correct theory" if they contradict experiment, but you can't prove that they are the final answer.

In the case of Quantum Mechanics / Quantum Field Theory, we certainly don't have the final answer, because of various known problems in the theory, so it seems like there's a good chance that nature won't describe reality using a wave function in the end; the concept of wave function in position space has big issues in QFT anyway.