In the double-slit experiment with single electrons, the Schrödinger equation and the Born rule can indeed predict the probability distribution for the electrons and provide insights into the interference pattern on the screen. However, it's important to note that the interference pattern arises from the statistical distribution of many individual electron detections, rather than predicting the exact "picture" of a single electron's path or the pattern it will create on the screen.
The Schrödinger equation describes the wave-like behavior of particles, including electrons, in quantum mechanics. When applied to the double-slit experiment, it can provide a mathematical description of the probability amplitude associated with the electron's position at different locations on the screen.
The Born rule, which is a fundamental principle of quantum mechanics, relates the probability amplitude to the probability of finding the electron at a particular position. It states that the probability of finding the electron at a given point on the screen is proportional to the squared magnitude of the probability amplitude at that point.
By solving the Schrödinger equation for the double-slit experiment, one can determine the probability distribution for the electrons on the screen. This distribution exhibits interference patterns, with regions of higher and lower probability that correspond to bright and dark fringes, respectively.
However, it's important to emphasize that the interference pattern arises from the statistical behavior of a large number of electrons and not from predicting the exact path or behavior of a single electron. When a single electron is detected on the screen, it appears as a point-like particle at a specific position, and its path cannot be determined precisely due to the inherent probabilistic nature of quantum mechanics.
In summary, while the Schrödinger equation and the Born rule can predict the probability distribution and provide insights into the interference pattern that emerges in the double-slit experiment with single electrons, they do not predict the exact "picture" of the interference pattern created by a single electron.