Quantum Analysis of the Hydrogen Atom

A mathematical treatment of the hydrogen atom — the simplest atomic system in quantum mechanics, but also the gateway to all of quantum chemistry. The paper works through Schrödinger's equation from scratch, exploiting the spherical symmetry of the Coulomb potential to separate variables and reduce the problem to two ODEs.

The angular part is solved via spherical harmonics — eigenfunctions of the angular momentum operators. The radial part reduces, after a change of variables, to the associated Laguerre differential equation, whose solutions are the generalized Laguerre polynomials. Together these give the full set of quantized energy levels and electronic orbital shapes — the foundations of atomic structure.

The paper is expository rather than novel research, but it is a rigorous treatment of a classical result and demonstrates working comfort with PDEs, special-function theory, and the operator formalism of quantum mechanics.