The time-dependent Schrödinger equation is usually expressed in position space, with the wave function depending on position and time. This overview presents an alternative formulation in momentum space, where the wave function depends on momentum and time.

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.