Do Wave Functions Vanish at Infinity?

Episode 5 of Quantum on the Back of an Envelope.

In this episode of Quantum on the Back of an Envelope, In this episode of Quantum on the Back of an Envelope, we ask whether a quantum mechanical wave function really has to vanish at infinity — an assumption often made in introductory textbooks and courses. To explore this, we examine a specific example (taken from "Principles of Advanced Mathematical Physics" by Robert D. Richtmyer) of a wave function that is perfectly continuous, differentiable, and normalizable, yet not only refuses to vanish, but actually becomes unbounded at infinity.