Quantum Mechanical Interpretation of Rutherford Experiment Ernest Rutherford performed the gold foil experiment; alpha particles were fired at a gold foil and the alpha particles were scattered. This result disproved Thomson's plum pudding model of atoms.
This got me wondering, how does QM fit into this picture? How do we use the wave nature of alpha particles to explain what is going on in this experiment? When does the wavefunction collapse? Did the wavefunction spread out across the entire space prior to measurement? If so, doesn't it mean that the particles were not "fired" but rather just under the influence of the potentials of the gold atoms?
P.S. My understanding of the physical interpretation of QM is all over the place so my question might not even be valid in the first place. If so, sorry for that.
 A: There are two parts in the question. First, given potential, how to find the scattering amplitudes as a function of scattering angle. Second, how to obtain the scattering potential in the first place.
The answer to the first question is twofold again. Normally, yes you need QM to compute scattering amplitudes correctly. However Rutherford did not do that and just did the calculation using classical scattering theory for a Coulombic potential. But there he was helped by one of those rare lucky coinsidences that helped researchers when they needed to leap the gap. Namely, the scattering amplitudes for Coulombic field coinside for classical and quantum theories. This is a feature of Coulombic potential that makes it special in this sense.
Second, the scattering potential. This is where QM was necessary. Classical theory could not (and still cannot) accommodate for a point-like concentration of positive charge in the center of an atom without compromising its long-term stability. Rutherford reached this conclusion with his classical scattering calculation, which he was lucky is still valid in QM domain, and the rest is history.
A: The Rutherford experiment

Rutherford scattering is the elastic scattering of charged particles by the Coulomb interaction. It is a physical phenomenon explained by Ernest Rutherford in 19111 that led to the development of the planetary Rutherford model of the atom and eventually the Bohr model. Rutherford scattering was first referred to as Coulomb scattering because it relies only upon the static electric (Coulomb) potential, and the minimum distance between particles is set entirely by this potential. The classical Rutherford scattering process of alpha particles against gold nuclei is an example of "elastic scattering" because neither the alpha particles nor the gold nuclei are internally excited

This can be mathematically  model with classical mechanics , no need of quantum mechanics.
The experimental data that showed the need for a new theory for the atomic dimensions were the photoelectric effect, the black body radiation and the spectra of atoms. These lead to the semiclassical Bohr model and finally to quantum mechanics for describing with accuracy interactions at the atomic level.
Quantum mechanics uses the solutions of the differential wave equations with extra postulates that pick up those solutions that describe the measurements possible in those dimensions.
A main postulate is that the solutions for a given system are not solutions for individual particles but for a statistical accumulation, predicting the probability of measurement . I.e. there should be an accumulation of same boundary condition events to see how the quantum theory fits the data.

Expectation Value Postulate
For a physical system described by a wavefunction $Ψ $, the expectation value of any physical observable q can be expressed in terms of the corresponding operator Q as follows:


It is a probabilistic theory.

How do we use the wave nature of alpha particles to explain what is going on in this experiment?

The so called "wave nature" is an envelope that can roughly describe this probabilistic expectation, the wave functions are sinusoidal solutions of the wave equation, but is not needed to model the experiment.

When does the wavefunction collapse?

Wave function collapse is a confusing way of stating that a measurement or an interaction picks up one of the probable states that the wavefunction models. After interaction/measurement a differen wave function appears due to the different boundary conditions that result with an interaction.

Did the wavefunction spread out across the entire space prior to measurement?

The wavefunction is defined over the space of the problem by construction of the theory. The measurement picks up one probable point, ( like throwing 1 and 6 in a two die throw)

If so, doesn't it mean that the particles were not "fired" but rather just under the influence of the potentials of the gold atoms?

Certainly the wavefunction is the proper quantum mechanical mathematical solution for the available Coulomb potentials of the experiment :"alpha particle scattering off gold nucleus".
A: Yes the wavefunction spreads out across the entire space prior to measurement.
But don't forget the wavefunction is itself a mathematical tool.
Consider the following example from ordinary probability. Suppose you have a piece of paper and two envelopes. The paper is put into one of the envelopes but you don't know which. So the probability distribution describing your knowledge is 50:50 between the envelopes. If the envelopes are located on a table, separated by 10 cm, then the probability distribution as function of position has two bumps separated by 10 cm. Now send one of the envelopes to Australia. Now the probability distribution has two bumps separated by many thousands of miles. Now open the envelope near you. At this stage the probability distribution suddenly collapses to either a bump near you (if the paper was in your envelope) or a bump in Australia.
The above facts concerning classical probability distributions do not capture all that one can say about quantum wavefunctions. One difference is that the distribution I described was about your state of knowledge rather than the physical situation itself, whereas the wavefunction is about the physical situation itself (or, if you prefer, the maximum amount of knowledge that anyone could have). Nonetheless you should not think of wavefunctions as just like waves; they are subtle tools and what they do is give the distribution of quantum amplitude. Quantum amplitude is much debated in interpretations of quantum theory. It is the thing whose modulus-squared give a probability for some event or scenario. In the Rutherford scattering experiment, there is a quantum amplitude for "alpha particle left source and arrived at detector located at $x$" for each $x$. The quantum amplitude, then, is a complex number $\psi$ which depends on $x$, so we write it as $\psi(x)$ and thus you have a wavefunction. And in the end it is complicated permanent things such as an electrical discharge involving billions of atoms (e.g. in a particle detector), or dots on a photographic film made from billions of chemical reactions which are observed. It is the probabilities of those complicated permanent events (called "measurements") which the mathematical formalism of quantum theory ultimately describes.
