Wave Particle Duality and its relation to the Wave Function

[torquerotates]

Ok so from my understanding, the wave particle duality of matter is simply the fact that matter sometimes behaves as a wave and sometimes behaves as a particle. Ok, but I wonder if this has anything to do with the wavefunction. The wave function gives us the probability spread of matter at any location. That means that once we know where that matter is, the wavefunction collapses and probability is one. Now the part that I don't get is if we don't make the measurement, the matter is a wave and does this wave have the exact same spread as the wavefunction? That is, does the probability of finding that matter(or particle) at give point correspond to the actual amplitude of that matter manifested as a physical wave?

 

[The_Duck]

Let's take an electron to be concrete. You are making a distinction between the electron and the electron's wave function that I don't think really exists. In the mathematics of quantum mechanics, the wave function of an electron tells you everything there is to know about the electron. The electron *is* the wave function, in some sense. You have some idea, I think, of "matter manifested as a physical wave" as distinct from the "wave function." There is just the wave function.

When we say that some electron is "behaving as a wave" what we mean is that we are leaving the wave function alone for a while and not measuring anything so that the wave function does not collapse. Left alone, the wave function behaves somewhat similar to familiar water waves or sound waves, which is of course why it is called the "wave" function.

When we say that the electron is "behaving like a particle" what we mean is that we are measuring the electron's position. When we do such a measurement we find that the electron is in one spot and not spread out at all, which fits our idea of what a "particle" should be like.

 

[arkajad]

The wave function gives us the probability spread of matter at any location.

No. The wave function gives us the probability of a detection at a given point (be it in configuration or a momentum space or elsewhere). It is better to think of it as a state of a field. Quantum mechanics may be misleading. Quantum fields are technically more difficult but less misleading in a philosophical sense. The book "Introduction to realistic quantum physics" by Guliano Preparata, which some will certainly find somewhat controversial, nevertheless elucidates pretty well some of these points.

He writes in the Introduction:

And it is in the typical notion of "wave-particle complementarity" of the Copenhagen "vulgata" that any hope of a realistic, objective interpretation of QM fades away, leaving in its place a well defined set of rules to "compute" the statistical distributions of the outcomes of given observations (or observables) on a statistical ensemble of identical physical systems.
...
Let's begin with the idea that quantum behaviour has to do with the microscopic world only. The well-known collective phenomena of Ferromagnetism, Superconductivity and Superfluidity, whose classical impossibility is easy to demonstrate, bear witness to the fact that there exist macroscopic pieces of matter whose behaviour cannot be described by classical physics. If not classical, what is then the relevant physics? Most contemporary physicists have no doubt that QFT must be involved in the still mysterious workings of these fascinating phenomena, but, beyond a few phenomenological attempts, such as the Landau-Ginsburg approach, no real headway has ever been made into this kind of physics. And, I contend, the fallacious philosophy of QM is largely responsible for this unfortunate state of affairs. Conversely, couldn't the situation improve, indeed change drastically if we were to realize the centrality of QFT and find that QM is but some kind of approximation of QFT in a well defined, limiting physical situation?"