Chapter 17 Further Reading

Primary Textbook References

Griffiths & Schroeter, Introduction to Quantum Mechanics (3rd ed., 2018)

  • Section 7.1 — Non-degenerate perturbation theory. Griffiths's presentation is the most accessible introduction at the undergraduate level. The derivation is clean, the notation is clear, and the examples (infinite well with delta-function perturbation, harmonic oscillator with linear and cubic terms) are well-chosen. His treatment of the Stark effect is brief but correct. This is the recommended first-reading reference.
  • Problem 7.1–7.6 — Straightforward applications that build confidence with the formulas before moving to more challenging problems.

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

  • Section 5.1 — Time-independent perturbation theory in Dirac notation, at a somewhat higher level than Griffiths. Sakurai's presentation emphasizes the operator structure and is closer to the formalism used in this chapter. His discussion of the resolvent operator approach provides a more sophisticated perspective that connects to many-body physics.
  • Section 5.2 — The hydrogen atom in an electric field (Stark effect), including both the ground-state quadratic effect and the excited-state linear effect. More detailed than Griffiths, with explicit matrix element calculations.

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

  • Chapter 17 — The most comprehensive treatment of perturbation theory in a standard graduate text. Shankar derives the formulas carefully, works through multiple examples, and provides excellent discussion of convergence and validity. His treatment of the anharmonic oscillator goes to higher order than most texts.
  • Section 17.2 — Particularly valuable for its discussion of the "smallness" condition for the perturbation — when is $\hat{H}'$ truly "small"?

Townsend, A Modern Approach to Quantum Mechanics (2nd ed., 2012)

  • Chapter 11 — Perturbation theory presented in a matrix-mechanics framework, consistent with Townsend's Dirac-first approach. The finite-dimensional examples (2-state and 3-state systems) are particularly clear and build intuition before moving to the infinite-dimensional case.

Going Deeper

Cohen-Tannoudji, Diu & Laloe, Quantum Mechanics (2nd ed., 2020)

  • Chapter XI — The most rigorous and complete treatment of stationary perturbation theory available in a textbook. Cohen-Tannoudji derives the formulas to arbitrary order, proves convergence conditions, and includes extensive complements. The complement on the anharmonic oscillator (calculating corrections to high order) is a tour de force.
  • Complement $H_{XI}$ — The method of Dalgarno and Lewis for summing perturbation series without explicitly evaluating individual matrix elements. This is the technique used to obtain the exact hydrogen polarizability.

Merzbacher, Quantum Mechanics (3rd ed., 1998)

  • Chapter 18 — Perturbation theory with careful attention to convergence issues. Merzbacher's discussion of the Brillouin-Wigner vs. Rayleigh-Schrödinger versions of perturbation theory is more detailed than most texts and is valuable for students heading toward many-body physics.

Bransden & Joachain, Quantum Mechanics (2nd ed., 2000)

  • Chapter 8 — Good balance between formalism and application. The treatment of the Stark effect is particularly thorough, including the calculation of individual matrix elements and comparison with experiment.

The Stark Effect: Specialized References

Bethe & Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (1957, Dover reprint 2008)

  • The classic reference for atomic structure calculations. Chapter III includes the most complete analytical treatment of the hydrogen Stark effect, including the solution using parabolic coordinates. Despite its age, this book remains unsurpassed for the depth of its atomic physics.

Drake (ed.), Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed., 2023)

  • Chapter 17 (by T. Gallagher) — "Stark Effect" — A comprehensive review of both theoretical and experimental aspects, including Rydberg atoms in electric fields, Stark ionization, and applications in modern AMO physics.

Gallagher, T.F., Rydberg Atoms (Cambridge, 1994)

  • The definitive reference for the Stark effect in highly excited states, where the perturbative treatment breaks down and the physics becomes qualitatively different (ionization, chaos, intersecting Stark manifolds). Essential reading for students interested in AMO physics.

Convergence and Asymptotic Series

Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer, 1999)

  • Chapter 8 — The best introduction to asymptotic expansions and their application to physics. Bender and Orszag explain with characteristic clarity why perturbation series diverge, how to extract useful information from divergent series, and how to go beyond perturbation theory. The treatment of the anharmonic oscillator as a paradigmatic example is superb.
  • Chapter 10 — WKB and turning-point problems, providing context for the non-perturbative effects discussed in Section 17.8.

Simon, B., "Large Orders and Summability of Eigenvalue Perturbation Theory: A Mathematical Overview," Int. J. Quantum Chem. 21, 3–25 (1982)

  • A rigorous mathematical review of the asymptotic nature of quantum-mechanical perturbation series. Simon proves that the Rayleigh-Schrödinger series for the anharmonic oscillator is Borel summable (meaning the Borel-resummed series gives the exact answer). This paper requires graduate-level mathematical sophistication but is authoritative.

Dyson, F.J., "Divergence of Perturbation Theory in Quantum Electrodynamics," Phys. Rev. 85, 631–632 (1952)

  • The one-page paper that changed how physicists think about perturbation theory. Dyson's argument — that QED perturbation theory must diverge because reversing the sign of the coupling constant ($e^2 \to -e^2$) makes the vacuum unstable — is a masterpiece of physical reasoning. Essential reading.

Non-Perturbative Physics and Resurgence

Zinn-Justin, J., Quantum Field Theory and Critical Phenomena (5th ed., Oxford, 2021)

  • Chapters 37–39 — Instantons, large-order behavior, and the connection between perturbative and non-perturbative effects. The most complete textbook treatment of these topics. Suitable for advanced graduate students.

Marino, M., Instantons and Large N (Cambridge, 2015)

  • A modern introduction to non-perturbative effects in quantum mechanics and field theory, including instantons, resurgence, and trans-series. The first two chapters (which cover quantum mechanics) are accessible to advanced undergraduates with good mathematical preparation.

Dunne, G.V. & Unsal, M., "New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifolds to Bions and Resurgence," Ann. Rev. Nucl. Part. Sci. 66, 245 (2016)

  • A review of the resurgence program, showing how perturbative and non-perturbative physics are connected through the mathematical structure of trans-series. The introduction and sections on quantum mechanics are accessible; the field theory sections require more background.

Historical References

Schrödinger, E., "Quantisierung als Eigenwertproblem (Dritte Mitteilung)," Ann. Phys. 80, 437–490 (1926)

  • Schrödinger's third paper on wave mechanics, in which he develops perturbation theory and applies it to the Stark effect of hydrogen. A historically important paper that shows how perturbation theory was present at the birth of quantum mechanics.

Rayleigh, Lord, The Theory of Sound (1877, Dover reprint 1945)

  • Remarkably, the mathematical framework of perturbation theory was developed by Rayleigh for acoustics problems fifty years before quantum mechanics. The "Rayleigh-Schrödinger" perturbation theory of this chapter is essentially Rayleigh's method applied to the Schrödinger equation.

Computational Resources

Wolfram Research, "Quantum Perturbation Theory" — Mathematica documentation and notebooks

  • High-precision computation of perturbation series to high order. Useful for numerical verification of analytical results and for exploring convergence behavior.

QuTiP (Quantum Toolbox in Python) — qutip.org

  • Open-source library for quantum mechanics simulations. The Qobj class makes it straightforward to construct Hamiltonians, diagonalize them, and compare with perturbation theory. Used in the code examples for this chapter.