Undergrad level quantum mechanics is enough to have a basic idea of what is going on. The electron position is calculated as a time- and space- dependent probability density that can also be time-independent if certain conditions are met. This probability density is proportional to the magnitude squared of the wavefunction. The wavefunction is a solution to a partial differential equation like Schrodinger's equation given a specified Hamiltonian and boundary conditions. In this experiment, the potential energy portion of the Hamiltonian (total energy) is controlled via externally applied electric fields. Designing the quantum dot to control the spatial pattern of these electric fields is the secret sauce of the experiment that is not so easily understood at an undergrad level, but as a first pass you can consider canonical problems like the quantum harmonic oscillator. It's been a while, so others should feel free to correct me if I said anything inaccurate.