The double-slit experiment demonstrates that light and matter can display characteristics of both classically defined waves and particles. What more is that this phenomenon displays the probabilistic nature of quantum mechanical design. Below is a short video on the subject to catch you up to speed.
As you saw on the video, the act of intelligent observation affects the functioning of the particles as they travel. If we observe to see which slit the particle travels through, effectively breaks down the super-positioning of the particle in the wave function, and causes it to act as a particle would travel. The key here is that observational energies have affected matter.
So, scientists are left to ponder the implications here. There are a few ways in which mainstream science has looked at the mechanisms involved and the implications. Below is a list of the mainstream interpretations for the experiment, from the scientific standpoint.
The Copenhagen Interpretation
The Copenhagen interpretation, put forth by some of the pioneers in the field of quantum mechanics, asserts that it is undesirable to posit anything that goes beyond the mathematical formulae and the kinds of physical apparatus and reactions that enable us to gain some knowledge of what goes on at the atomic scale. One of the mathematical constructs that enables experimenters to predict very accurately certain experimental results is sometimes called a probability wave. In its mathematical form it is analogous to the description of a physical wave, but its "crests" and "troughs" indicate levels of probability for the occurrence of certain phenomena (e.g., a spark of light at a certain point on a detector screen) that can be observed in the macro world of ordinary human experience.
The probability "wave" can be said to "pass through space" because the probability values that one can compute from its mathematical representation are dependent on time. One cannot speak of the location of any particle such as a photon between the time it is emitted and the time it is detected simply because in order to say that something is located somewhere at a certain time one has to detect it. The requirement for the eventual appearance of an interference pattern is that particles be emitted, and that there be a screen with at least two distinct paths for the particle to take from the emitter to the detection screen. Experiments observe nothing whatsoever between the time of emission of the particle and its arrival at the detection screen. If a ray tracing is next made as if a light wave (as understood in classical physics) is wide enough to take both paths, then that ray tracing will accurately predict the appearance of maxima and minima on the detector screen when many particles pass through the apparatus and gradually "paint" the expected interference pattern.
Path-integral Formulation
The Copenhagen interpretation is similar to the path integral formulation of quantum mechanics provided by Feynman. The path integral formulation replaces the classical notion of a single, unique trajectory for a system, with a sum over all possible trajectories. The trajectories are added together by using functional integration.
Each path is considered equally likely, and thus contributes the same amount. However, the phase of this contribution at any given point along the path is determined by the action along the path.
All these contributions are then added together, and the magnitude of the final result is squared, to get the probability distribution for the position of a particle.
As is always the case when calculating probability, the results must then be normalized by imposing.
To summarize, the probability distribution of the outcome is the normalized square of the norm of the superposition, over all paths from the point of origin to the final point, of waves propagating proportionally to the action along each path. The differences in the cumulative action along the different paths (and thus the relative phases of the contributions) produces the interference pattern observed by the double-slit experiment. Feynman stressed that his formulation is merely a mathematical description, not an attempt to describe a real process that we can measure.
Relational Interpretation
According to the relational interpretation of quantum mechanics, first proposed by Carlo Rovelli, observations such as those in the double-slit experiment result specifically from the interaction between the observer (measuring device) and the object being observed (physically interacted with), not any absolute property possessed by the object. In the case of an electron, if it is initially "observed" at a particular slit, then the observer–particle (photon–electron) interaction includes information about the electron's position. This partially constrains the particle's eventual location at the screen. If it is "observed" (measured with a photon) not at a particular slit but rather at the screen, then there is no "which path" information as part of the interaction, so the electron's "observed" position on the screen is determined strictly by its probability function. This makes the resulting pattern on the screen the same as if each individual electron had passed through both slits. It has also been suggested that space and distance themselves are relational, and that an electron can appear to be in "two places at once"—for example, at both slits—because its spatial relations to particular points on the screen remain identical from both slit locations.
Many Worlds Interpretation
The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction and denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternate histories and futures are real, each representing an actual "world" (or "universe"). In layman's terms, the hypothesis states there is a very large—perhaps infinite—number of universes, and everything that could possibly have happened in our past, but did not, has occurred in the past of some other universe or universes. The theory is also referred to as MWI, the relative state formulation, the Everett interpretation, the theory of the universal wavefunction, many-universes interpretation, multi-history or just many-worlds.
De Broglie's Double Solution Theory
In de Broglie's double solution theory there are two waves. There is the wave-function which is statistical, non-physical and is used to determine the probabilistic results of experiments. There is also a physical wave in the "sub-quantum medium" which guides the particle.
In a double-slit experiment the particle always travels through a single slit and the associated wave in the "subquantic medium" passes through both. As the wave exits the slits it creates wave interference which alters the direction the particle travels as it exits a single slit. Over time the particles form an interference pattern. Strongly detecting the particle exiting a single slit destroys the cohesion between the particle and its associated wave, the particle continues on the trajectory it was traveling and does not form an interference pattern.
This is not de Broglie–Bohm theory (also known as Pilot wave and Bohmian mechanics). De Broglie realized the pilot-wave existed in fictitious configuration space and went back to his original double solution theory.
My Proposed Theory
This reality is a simulation being run on a quantum computing system that realizes every possible outcome. It utilizes an advanced Artificial Intelligence that is reactive to observational energies which affect the matter that surrounds the observer. It takes direct observation to enable manifestation of matter. This simulation is more likely designed around the human species, and resembles an ancestral simulation designed to study ourselves. Since all possibilities are observed due to the quantum nature of the simulation, the makers of the simulation are free to learn and advance their understandings of the universe through allowing multiple versions of reality to be realized. We live in a stream of just one of those infinite possibilities that exist. In another, where the double-slit is studied, particle placement would vary. The probabilistic nature of quantum energies gives me the evidence that we are in fact living in a reality that is dictated by choice. We experience one choice, but every choice is observed by the makers of the sim.