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".
