Book: Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra

Overview
Wigner's 1931 monograph lays out a systematic use of group representation theory to classify atomic energy levels and to derive transition rules in quantum mechanics. The book translates abstract group-theoretic concepts into concrete tools for the spectral analyst, showing how symmetry constrains possible energy degeneracies and electromagnetic transitions. It bridges rigorous mathematics and practical spectroscopy at a formative moment for quantum theory.
Wigner develops the connection between physical symmetry operations and unitary representations acting on quantum states, emphasizing how irreducible representations label and organize the states of atoms. The exposition makes clear that group theory is not a mere mathematical curiosity but an organizing principle that reduces complicated spectroscopic problems to algebraic ones.

Core mathematical tools
Central mathematical ideas include the representation theory of the rotation group and its double cover, the role of characters and orthogonality relations, and the construction and manipulation of tensor operators. Wigner introduces and employs the D-matrices that represent rotations on angular-momentum eigenstates, and he develops the algebra of coupling coefficients that later appear as 3-j and 6-j symbols. The monograph explains how these symbols encode the rules for combining angular momenta and how they simplify matrix elements.
Permutation symmetry and the use of symmetric and antisymmetric representations for identical particles receive careful attention. Wigner shows how group characters and projection operators extract components of definite symmetry, providing a systematic method to account for exchange effects and to build properly (anti)symmetrized multi-electron states.

Applications to atomic spectra
The book applies the formalism to classification of atomic terms, spectral multiplets, and splitting patterns under various coupling schemes. Wigner analyzes LS coupling and other schemes by decomposing total wavefunctions into irreducible pieces under the relevant symmetry groups, thereby predicting allowed term multiplicities and degeneracies. The treatment clarifies how spectroscopic term symbols arise from group-theoretic selection of irreducible representations.
Calculations of line intensities and fine-structure patterns are recast in representation-theoretic language, showing how matrix elements factor according to symmetry. This makes it straightforward to identify which transitions vanish by symmetry and which survive, streamlining the derivation of many selection rules familiar to spectroscopists.

Consequences for selection rules and spectroscopy
Symmetry considerations lead directly to selection rules for electric and magnetic multipole transitions. Wigner demonstrates that many apparent rules follow from orthogonality and tensor-decomposition properties rather than from ad hoc physical arguments. The Wigner-Eckart relationship between reduced matrix elements and Clebsch-Gordan-type coefficients appears as a central organizing principle, separating geometric coupling from dynamical quantities.
The monograph also clarifies degeneracy lifting under perturbations that break symmetry, providing a framework to predict level splitting patterns when external fields or smaller interactions reduce the symmetry group. This gives spectroscopists a powerful predictive language for interpreting experimental spectra.

Legacy and influence
Wigner's exposition established group theory as an indispensable tool in quantum physics, influencing atomic, molecular, nuclear, and solid-state theory. The notation and techniques introduced became standard: D-matrices, coupling coefficients, and the Wigner-Eckart approach remain core elements of modern quantum mechanics education and practice. The book seeded widespread adoption of symmetry methods, enabling more systematic and algebraic treatment of complex quantum problems.
Beyond immediate spectroscopic applications, the conceptual shift toward symmetry-based classification shaped later theoretical developments, from particle physics to crystallography. The monograph stands as a foundational reference that codified how symmetry constrains both the structure and dynamics of quantum systems.