The two-dimensional hydrogen atom hides an unexpected simplicity. By applying a clever coordinate transformation, its bound-state problem can be mapped exactly onto that of a two-dimensional harmonic oscillator. This elegant correspondence offers a fresh and intuitive perspective on one of the rare exactly solvable systems in quantum mechanics.

The harmonic oscillator is a foundational model in physics, valued for its solvability and broad relevance. When the oscillator’s frequency varies in time—as in ion traps with dynamic fields—it can still be treated analytically in certain cases. This overview presents an exactly solvable example illustrating such behavior.

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 conti