Addition of Angular Momenta
When a quantum system carries two or more angular momenta, such as orbit and spin, they combine into a total angular momentum whose allowed values follow a simple rule; the change between the separate and combined descriptions is encoded by Clebsch-Gordan coefficients.
Definition
The addition of angular momenta is the procedure for combining two or more commuting angular-momentum operators into a total angular momentum, whose eigenstates form the coupled basis related to the product basis by Clebsch-Gordan coefficients.
Scope
The topic covers the coupling of two angular momenta into a total, the triangle rule giving the allowed total quantum numbers, the uncoupled and coupled bases, the Clebsch-Gordan coefficients that connect them, the construction of coupled states with raising and lowering operators, and applications such as spin-orbit coupling and the addition of multiple spins.
Core questions
- What total angular-momentum values can result from combining two given angular momenta?
- How do the coupled and uncoupled bases differ?
- What role do Clebsch-Gordan coefficients play in the change of basis?
- How does angular-momentum addition explain spin-orbit coupling and multiplet structure?
Key concepts
- total angular momentum
- triangle rule
- uncoupled basis
- coupled basis
- Clebsch-Gordan coefficients
- spin-orbit coupling
Key theories
- Triangle rule and coupled basis
- Two angular momenta combine to give total quantum numbers ranging from their sum down to the absolute value of their difference in integer steps, and the simultaneous eigenstates of the total magnitude and projection form the coupled basis appropriate when the two momenta interact.
- Clebsch-Gordan coefficients
- Each coupled state is a specific superposition of product states whose weights are the Clebsch-Gordan coefficients; these coefficients express the unitary change of basis and encode the selection rules and intensities of transitions in atomic and nuclear spectra.
Clinical relevance
Angular-momentum addition organizes the structure of atoms and nuclei: it produces the fine-structure splitting from spin-orbit coupling, the term symbols and multiplets seen in atomic spectra, and the coupling rules used to interpret molecular and nuclear energy levels and their selection rules.
History
The coupling coefficients trace back to nineteenth-century invariant theory of Clebsch and Gordan; Wigner and Racah developed the modern quantum theory of angular-momentum coupling in the 1930s and 1940s, providing the algebraic machinery for atomic and nuclear spectroscopy.
Key figures
- Eugene Wigner
- Giulio Racah
- Alfred Clebsch
- Paul Gordan
Related topics
Seminal works
- edmonds1957
- sakurai2017
Frequently asked questions
- Why does combining two angular momenta give a range of possible totals?
- The two momenta can be relatively aligned, anti-aligned, or anywhere in between subject to quantization, so the total quantum number runs from the sum, when fully aligned, down to the absolute difference, when most opposed, in integer steps.
- What are Clebsch-Gordan coefficients used for?
- They give the amplitudes for writing a state of definite total angular momentum as a superposition of product states, which is needed to compute transition rates, selection rules, and the structure of coupled systems such as spin-orbit-coupled atoms.