Episode 1 of Quantum on the Back of an Envelope. In this first episode of Quantum on the Back of an Envelope , we explore an unexpected phenomenon in quantum mechanics: how a perfectly smooth wave function, evolving under the Schrödinger equation, can spontaneously develop a singularity over time.

For a freely evolving particle, the position probability density approaches the form of the momentum probability density in the long-time limit. This correspondence underlies techniques such as measuring a cloud’s momentum distribution after free expansion. This overview outlines the origin of this equivalence.

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.