Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Rotation Matrices and the Wigner–Eckart Theorem; Fine and Hyperfine Structure of Energy Levels in Hydrogen-Like Atoms

Rocco Schiavilla

Published Dec 5, 2024• 0 citations
Publisher

Abstract

Euler angles and rotation matrices; construction and properties of the rotation matrices; transformation of irreducible tensor operators under rotations; fine-structure of the hydrogen atom; hydrogen atom in a magnetic field: Zeeman and Paschen-Back effects; hyperfine structure of the hydrogen atom; tensor operators; time reversal and irreducible tensor operators.

DOI
10.1017/9781009473637.018

Related papers

The Wigner-Eckart Theorem

Igal Talmi

The use of irreducible tensor operators is very convenient for systems whose eigenstates have definite values of J and M . The transformation properties of operators and wave functions, both kinds being components of irreducible tensors, allow the efficient use of geometrical properties. This leads to great simplifications in the calculation of matrix elements as we shall see in the present section.

Routledge eBooks•Oct 6, 2017•0 citations

One-electron atoms

Luciano Colombo

We begin our study of atomic physics by relying on the phenomenological notions summarized in chapter 1 and by fully exploiting the quantum mechanical formalism outlined in chapter 2. A hierarchy of approximations will be introduced in the first instance to describe the hydrogen atom, which allows one to focus on the main features of its energy spectrum and quantum properties, including the key ‘atomic orbital’ concept. Next, these results will be extended to the more general case of hydrogenic atoms. The physical picture will be made more rigorous by introducing the magnetic interactions, included those linked to the intrinsic spin degree of freedom, leading to the full description of the fine and hyperfine structure of one-electron atoms. The effects of a static external magnetic or electric field will be eventually discussed by calculating the corresponding energy shifts and splittings of the energy levels.

Institute of Physics eBooks•Oct 1, 2023•0 citations

A SIMPLE DERIVATION OF THE WIGNER-ECKART THEOREM

Huang Shi-zhong

Based on the commutation relations between the components of an irreducible tensor operator of rank k and the angular momentum operator, a simple derivation of the Wigner-Eckart theorem is presented with the aid of the properties of angular momentum operators and their eigenstates.

Journal of Anhui Normal University•Jan 1, 2003•0 citations

Atomic and Molecular Physics: A Student's Noteboo

Shivraj Deshmukh

A comprehensive set of lecture and study notes covering fundamental atomic and molecular physics. The text systematically works through the breakdown of the classical Rutherford atom and the Bohr-Sommerfeld model, followed by a full 3D Schrödinger treatment of the hydrogen atom using spherical harmonics and Laguerre polynomials. It details relativistic corrections, spin-orbit coupling, and fine structure via Clebsch-Gordan addition and Kramers' recursion relations. The notes further explore magnetic field interactions (normal and anomalous Zeeman effect, Paschen-Back effect), time-dependent perturbation theory for light-matter transitions, electric dipole selection rules, the 21-cm hyperfine splitting of hydrogen, and two-electron systems (Helium, exchange integrals, and Slater determinants). Includes extensive reference tables and angular momentum formulas.

•Oct 1, 2026•0 citations

Perturbative Methods in Quantum Mechanics

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter we review time-independent and time-dependent perturbation theory in quantum mechanics. We begin with Rayleigh-Schrödinger perturbation theory for small corrections to the Hamiltonian for non-degenerate and degenerate spectra, which leads us to a discussion of avoided crossings. We then apply these methods to calculate the fine-structure corrections for the energy levels of the hydrogen atom, including the relativistic correction to the kinetic energy, the spin-orbit correction, and the Darwin term. We then turn to the effect of weak constant magnetic fields: for the resulting Zeeman effect we distinguish between the anomalous Zeeman effect and the Paschen-Back effect. In the context of the corresponding Stark effect for electric fields we introduce the Wigner-Eckart theorem and calculate perturbations to first and second order. We then turn to time-independent problems and discuss the Born-von Neumann series for the propagator, time-ordered integrals, the interaction picture, before explaining Fermi’s golden rule.

•Sep 7, 2023•0 citations

A Guide to Rotations in Quantum Mechanics

Michael A. Morrison, Gregory A. Parker

To lay a foundation for the study and use of rotation operators in graduate quantum mechanics and in research, a thorough discussion is presented of rotations in Euclidean three space (R3 ) and of their effect on kets in the Hilbert space of a single particle. The Wigner D-matrices are obtained and used to rotate spherical harmonics. An extensive ready-reference appendix of the properties of these matrices, expressed in a consistent notation, is provided. Careful attention is paid throughout to various conventions (e.g. active versus passive viewpoints) that are used in the literature.

Australian Journal of Physics•Aug 1, 1987•166 citations
Euler angles and rotation matrices; construction and properties of the rotation matrices; transformation of irreducible tensor operators under rotations; fine-structure of the hydrogen atom; hydrogen atom in a magnetic field: Zeeman and Paschen-Back effects; hyperfine structure of the hydrogen atom; tensor operators; time reversal and irreducible tensor operators.
We begin our study of atomic physics by relying on the phenomenological notions summarized in chapter 1 and by fully exploiting the quantum mechanical formalism outlined in chapter 2. A hierarchy of approximations will be introduced in the first instance to describe the hydrogen atom, which allows one to focus on the main features of its energy spectrum and quantum properties, including the key ‘atomic orbital’ concept. Next, these results will be extended to the more general case of hydrogenic atoms. The physical picture will be made more rigorous by introducing the magnetic interactions, included those linked to the intrinsic spin degree of freedom, leading to the full description of the fine and hyperfine structure of one-electron atoms. The effects of a static external magnetic or electric field will be eventually discussed by calculating the corresponding energy shifts and splittings of the energy levels.
A comprehensive set of lecture and study notes covering fundamental atomic and molecular physics. The text systematically works through the breakdown of the classical Rutherford atom and the Bohr-Sommerfeld model, followed by a full 3D Schrödinger treatment of the hydrogen atom using spherical harmonics and Laguerre polynomials. It details relativistic corrections, spin-orbit coupling, and fine structure via Clebsch-Gordan addition and Kramers' recursion relations. The notes further explore magnetic field interactions (normal and anomalous Zeeman effect, Paschen-Back effect), time-dependent perturbation theory for light-matter transitions, electric dipole selection rules, the 21-cm hyperfine splitting of hydrogen, and two-electron systems (Helium, exchange integrals, and Slater determinants). Includes extensive reference tables and angular momentum formulas.
Abstract In this chapter we review time-independent and time-dependent perturbation theory in quantum mechanics. We begin with Rayleigh-Schrödinger perturbation theory for small corrections to the Hamiltonian for non-degenerate and degenerate spectra, which leads us to a discussion of avoided crossings. We then apply these methods to calculate the fine-structure corrections for the energy levels of the hydrogen atom, including the relativistic correction to the kinetic energy, the spin-orbit correction, and the Darwin term. We then turn to the effect of weak constant magnetic fields: for the resulting Zeeman effect we distinguish between the anomalous Zeeman effect and the Paschen-Back effect. In the context of the corresponding Stark effect for electric fields we introduce the Wigner-Eckart theorem and calculate perturbations to first and second order. We then turn to time-independent problems and discuss the Born-von Neumann series for the propagator, time-ordered integrals, the interaction picture, before explaining Fermi’s golden rule.
To lay a foundation for the study and use of rotation operators in graduate quantum mechanics and in research, a thorough discussion is presented of rotations in Euclidean three space (R3 ) and of their effect on kets in the Hilbert space of a single particle. The Wigner D-matrices are obtained and used to rotate spherical harmonics. An extensive ready-reference appendix of the properties of these matrices, expressed in a consistent notation, is provided. Careful attention is paid throughout to various conventions (e.g. active versus passive viewpoints) that are used in the literature.