In the double-slit experiment, the behavior of particles, such as electrons, is described by quantum mechanics, which introduces wave-particle duality. According to this concept, particles like electrons can exhibit both particle-like and wave-like properties depending on how they are observed or measured.
When an electron is fired towards the double-slit setup, it behaves as a wave, and its wavefunction spreads out and interferes with itself as it passes through the slits. The wavelength of the electron's wavefunction determines the interference pattern that will be observed on the screen behind the slits.
The Compton wavelength of an electron, which you mentioned, is a property associated with its particle-like behavior and is related to its momentum. It represents the scale at which the wave-like behavior of the electron becomes significant.
In the context of the double-slit experiment, the Compton wavelength of the electron is much smaller than the typical distance between the slits and the screen. As a result, the wavefunction of the electron is spread out over a large distance and can simultaneously pass through both slits. This allows for interference to occur, leading to the characteristic interference pattern observed on the screen.
It's important to note that the wave-particle duality and the behavior of subatomic particles at the quantum level can be complex and counterintuitive. The double-slit experiment is just one example that highlights the wave-like behavior of particles. Explaining the full intricacies of quantum mechanics and wave-particle duality would require a more in-depth discussion of the subject.