Chapter 5 Further Reading: The Hydrogen Atom

Primary Textbook References

Griffiths, D. J. — Introduction to Quantum Mechanics (3rd ed., 2018)

  • Chapter 4 covers the hydrogen atom in full. Sections 4.1 (angular equation, spherical harmonics), 4.2 (radial equation), and 4.3 (hydrogen spectrum) correspond directly to our Sections 5.3--5.5.
  • Griffiths provides an exceptionally clear treatment of the radial equation, with every step of the asymptotic analysis and series solution written out. If you found our derivation in Section 5.4 too compressed, start here.
  • The discussion of "accidental" degeneracy (Section 4.2.2) is brief but points toward the deeper symmetry.

Sakurai, J. J. & Napolitano, J. — Modern Quantum Mechanics (3rd ed., 2021)

  • Chapter 3 covers angular momentum and the hydrogen atom from the operator perspective. Sakurai develops spherical harmonics algebraically using the ladder operators $\hat{L}_\pm$, which we will encounter in Chapter 12.
  • Section 3.7 treats the hydrogen atom. The approach is more abstract than Griffiths but reveals the algebraic structure beautifully.
  • If you want to see how the same physics looks in Dirac notation (which we will adopt in Chapter 8), this is the place to start.

Shankar, R. — Principles of Quantum Mechanics (2nd ed., 1994)

  • Chapter 13 provides the most thorough treatment of the hydrogen atom at the intermediate level. Shankar devotes 70+ pages to the topic.
  • The discussion of the $SO(4)$ symmetry (Section 13.4) is particularly valuable --- it explains the "accidental" degeneracy at a level accessible to advanced undergraduates.
  • The treatment of radial wavefunctions includes many worked examples and visualizations.

Cohen-Tannoudji, C., Diu, B., & Laloe, F. — Quantum Mechanics (2019 reprint)

  • Complement E$_{\text{VII}}$ gives a meticulous treatment of the hydrogen atom.
  • The two-volume structure allows for extensive mathematical appendices on Legendre polynomials, Laguerre polynomials, and spherical harmonics.
  • Recommended for readers who want complete proofs of orthogonality relations and normalization integrals.

Mathematical References

Arfken, G. B., Weber, H. J., & Harris, F. E. — Mathematical Methods for Physicists (7th ed., 2013)

  • Chapter 15 (Legendre functions) and Chapter 16 (spherical harmonics) provide exhaustive treatments of the special functions encountered in this chapter.
  • The chapter on associated Laguerre polynomials (Chapter 18) includes useful integral formulas.

Boas, M. L. — Mathematical Methods in the Physical Sciences (3rd ed., 2006)

  • Chapter 12 on series solutions of differential equations and Chapter 13 on special functions provide accessible introductions.
  • Less rigorous than Arfken but excellent for building intuition.

Historical Sources

Pais, A. — Niels Bohr's Times: In Physics, Philosophy, and Polity (1991)

  • The definitive biography of Bohr. Chapters 8--9 cover the development of the Bohr model and its reception. Essential reading for understanding the historical context of Section 5.5.

Mehra, J. & Rechenberg, H. — The Historical Development of Quantum Theory (6 volumes, 1982--2001)

  • Volume 5, Part 2 covers Schrodinger's derivation of the hydrogen atom in extraordinary historical detail. For serious history-of-science enthusiasts only.

Schrodinger, E. — "Quantization as an Eigenvalue Problem" (1926)

  • The original paper in Annalen der Physik. Schrodinger solved the hydrogen atom in his very first paper on wave mechanics. Available in English translation in various anthologies.
  • Reading the original is illuminating: Schrodinger's notation is different from modern conventions, but the physics is immediately recognizable.

Spectroscopy and Astrophysical Applications

Foot, C. J. — Atomic Physics (2005)

  • Chapters 1--3 cover hydrogen spectroscopy at a level that bridges this chapter with advanced topics (fine structure, Lamb shift, hyperfine structure).
  • Excellent treatment of selection rules with experimental context.

Carroll, B. W. & Ostlie, D. A. — An Introduction to Modern Astrophysics (2nd ed., 2017)

  • Chapter 8 covers stellar spectra and classification. The connection between hydrogen energy levels and stellar spectroscopy discussed in Case Study 1 is developed in full here.
  • Chapter 5 discusses the Boltzmann and Saha equations used to compute spectral line strengths.

The Proton Radius Puzzle

Pohl, R. et al. — "The size of the proton" (2010)

  • Nature 466, 213--216. The original paper reporting the muonic hydrogen measurement that launched the proton radius puzzle. Accessible to advanced undergraduates with the background from this chapter.

Karr, J.-P. & Julien, L. — "Hydrogen atomic spectroscopy and the proton size puzzle" (2020)

  • Comptes Rendus Physique 21(5), 413--421. An excellent review of the puzzle and its (partial) resolution as of 2020. Free preprint available on arXiv.

Gao, H. & Vanderhaeghen, M. — "The proton charge radius" (2022)

  • Reviews of Modern Physics 94, 015002. The most comprehensive review of the current status, covering both spectroscopic and scattering measurements.

Computational and Visualization Resources

Thijssen, J. M. — Computational Physics (2nd ed., 2007)

  • Chapter 4 covers the numerical solution of the hydrogen atom, including shooting methods for the radial equation.

Johansson, R. — Numerical Python (2nd ed., 2019)

  • Good reference for the SciPy special functions (scipy.special.sph_harm, scipy.special.genlaguerre) used in the code examples.

NIST Atomic Spectra Database

  • https://physics.nist.gov/PhysRefData/ASD/lines_form.html
  • The definitive source for atomic energy levels and spectral lines. You can look up every hydrogen transition discussed in this chapter and compare with our calculated values.
  • Before Chapter 6: Review the operator formalism from Chapter 2 (expectation values, Hermitian operators). Chapter 6 will build on both the hydrogen atom solution and the operator framework.
  • If you want to go deeper on angular momentum now: Sakurai Chapter 3 covers the algebraic (ladder operator) approach that we will develop in Chapter 12.
  • If the mathematics felt overwhelming: Boas Chapters 12--13 will help solidify your understanding of series solutions, orthogonal functions, and special polynomials.
  • If you want more physics and less math: Feynman Lectures Volume III, Chapters 19--20 provide Feynman's characteristic physical insight into the hydrogen atom with less formalism.