Yes, the mathematical model of quantum mechanics (QM) can indeed yield the probability for a single electron in a double-slit experiment to hit a particular point on the screen, given the initial conditions of the experiment.
In the double-slit experiment, a beam of particles, such as electrons, is fired toward a barrier with two narrow slits. Behind the barrier, there is a screen where the particles are detected. When particles pass through the slits, they can interfere with each other, resulting in an interference pattern on the screen.
The probability distribution for the electron to hit a specific point on the screen is determined by the wavefunction of the electron. The wavefunction describes the quantum state of the electron and contains information about its position, momentum, and other observable properties.
Initially, the wavefunction of the electron is prepared to represent a localized particle at the source. As the electron travels toward the double slits, its wavefunction evolves according to the Schrödinger equation, which governs the time evolution of quantum systems.
When the electron passes through the double slits, its wavefunction splits into two components, each passing through one of the slits. These components can interfere with each other, leading to an interference pattern on the screen.
To calculate the probability for the electron to hit a particular point on the screen, the mathematical formalism of QM involves taking the absolute square of the electron's wavefunction. The resulting probability density represents the likelihood of detecting the electron at different positions on the screen.
By applying QM calculations to the double-slit experiment, one can obtain the probability distribution for the electron's position on the screen, which would show the characteristic interference pattern associated with wave-like behavior. This probability distribution can be compared to experimental measurements to validate the predictions of quantum mechanics.