The Hydrogen Atom — Basics
Why Hydrogen?
Hydrogen is the simplest atom, consisting of just one electron bound to one proton. Despite its simplicity, it provides the paradigm for quantum mechanics in atoms:
- It is one of the very few systems where the Schrödinger equation can be solved exactly.
- Its spectrum was central to the historical development of quantum theory.
- Many techniques used for hydrogen generalize to more complex atoms and molecules.
Scope of this Section
In this chapter, we study hydrogen in the nonrelativistic Schrödinger picture.
Our goals are:
- Solve the time-independent Schrödinger equation for the Coulomb potential.
- Separate variables in spherical coordinates and identify radial and angular parts.
See Orbital Angular Momentum for the derivation of the spherical harmonics and the angular eigenproblem.
- Derive the hydrogenic eigenenergies and their degeneracies.
See The Hydrogen Atom for the radial equation and energy quantization.
- Explore the structure of the spatial wavefunctions (orbitals) through graphical representations.
See Visualizing Hydrogen Orbitals — placeholder / forthcoming.
- Connect angular momentum quantum numbers \((\ell,m)\) with the orbital shapes and nodal structure.
This material provides the foundation for later refinements (spin, relativistic corrections, Lamb shift, hyperfine structure) treated in the Hydrogen (Details) chapter.
A Final Remark
Hydrogen is not just a “toy problem.” It remains a testing ground for quantum mechanics and a reference point for experiments across physics, from spectroscopy to astrophysics. By mastering the hydrogen atom at this basic level, we gain insight into both the methods of quantum mechanics and the rich structure of atomic physics.