In the double-slit experiment, the Schrödinger equation and the Born rule can indeed predict the interference pattern that is observed on the screen, not just on average, but for each individual particle as well.
The Schrödinger equation describes the time evolution of a quantum system, including particles such as electrons. It is a mathematical equation that governs the wavefunction of the particle, which contains information about the probabilities of different outcomes when the particle is measured. In the case of the double-slit experiment, the wavefunction describes the behavior of the particle as it passes through the two slits and interferes with itself.
The Born rule, or the Born probability rule, is a fundamental principle in quantum mechanics. It states that the probability of obtaining a particular measurement outcome is given by the squared magnitude of the wavefunction. In the context of the double-slit experiment, the interference pattern observed on the screen arises from the constructive and destructive interference of the wavefunctions associated with the two slits.
Importantly, both the Schrödinger equation and the Born rule apply to individual particles. The wavefunction of a single particle passing through the double slits can exhibit interference effects, leading to the characteristic pattern on the screen over time as more particles are detected. This interference pattern arises due to the probabilistic nature of quantum mechanics, where the wavefunction encodes the probabilities of different measurement outcomes.
It's worth noting that when many particles are sent through the double slits, the observed interference pattern becomes more defined and closely resembles the predicted pattern from the Schrödinger equation and the Born rule. This is known as the statistical or average behavior, where the pattern emerges as the result of the accumulation of many individual particle interactions.