Building on the Wigner function formalism—which represents quantum states as quasi-probability distributions in phase space—we now explore how this framework can also describe quantum observables, the measurable quantities represented by Hermitian operators.

The Wigner function represents a quantum state as a quasi-probability distribution in phase space, expressed in terms of position and momentum. It provides insight into quantum behavior and nonclassical effects such as interference. This overview highlights its key representations and properties.

The hydrogen atom is a cornerstone of quantum mechanics, traditionally described in three dimensions. Interestingly, its two-dimensional counterpart—where the electron’s motion is confined to a plane—also admits an analytical solution. This overview summarizes that solution.