The Hydrogen Atom: a Review on the Birth of Modern Quantum Mechanics

The purpose of this work is to retrace the steps that were made by scientists of XX century, like Bohr, Schrodinger, Heisenberg, Pauli, Dirac, for the formulation of what today represents the modern quantum mechanics and that, within two decades, put in question the classical physics. In this context, the study of the electronic structure of hydrogen atom has been the main starting point for the formulation of the theory and, till now, remains the only real case for which the quantum equation of motion can be solved exactly. The results obtained by each theory will be discussed critically, highlighting limits and potentials that allowed the further development of the quantum theory.

💡 Research Summary

The paper offers a comprehensive historical and technical review of how the hydrogen atom served as the crucible for the birth of modern quantum mechanics during the first two decades of the twentieth century. It begins by outlining the failure of classical physics to explain the discrete spectral lines of hydrogen, setting the stage for a series of revolutionary ideas. Niels Bohr’s 1913 model introduced quantized angular momentum (L = nħ) and produced the celebrated energy formula Eₙ = −13.6 eV/n², successfully reproducing the Balmer series but leaving the nature of electron radiation and multi‑electron systems unexplained.

The narrative then moves to Erwin Schrödinger’s wave mechanics (1926), where the time‑independent Schrödinger equation is separated in spherical coordinates, yielding exact analytic solutions for hydrogen: radial functions expressed through associated Laguerre polynomials and angular parts given by spherical harmonics. The quantum numbers (n, ℓ, m) emerge naturally, and the probability density |ψ|² provides the modern “electron cloud” picture.

Heisenberg’s matrix mechanics (1925) is presented as a complementary, non‑commutative formulation where observable quantities are operators obeying