# Explanation of 6j Symbols (Tetrahedron)

• Tilde90
In summary, the conversation discusses the 6-j symbols and their role in coupling three angular momenta. The 6-j symbols describe a unitary transformation between different ways of coupling and involve six angular momenta. A book recommendation is also given: "Angular Momentum in Quantum Mechanics" by A.R. Edmonds, which includes a discussion of the tetrahedron diagram.
Tilde90
Hello all! (I'm new to the forum)
I'd like to ask if you could give me a simple explanation regarding the 6j symbols: I don't understand their formulation in terms of Clebsch-Gordan coefficients with three angular momenta and the related "tetrahedron rule".
Alternatively, could you please suggest me a book to consult?
Thank you very much for your help!

Last edited:
Tilde90, Angular momentum can be a really complicated subject. As you probably know (or you wouldn't be asking this!) the 6-j symbols are involved in coupling three angular momenta. You can couple j1 and j2 to form j12, then j12 and j3 to form J. Or, you can couple j2 and j3 to form j23, then j23 and j1 to form J. The 6-j symbols describe a unitary transformation between the two ways of doing things. And naturally they involve six angular momenta, j1, j2, j3, j12, j23 and J.

A book reference, including a discussion of the tetrahedron diagram, is the monograph, "Angular Momentum in Quantum Mechanics" by A.R. Edmonds, section 6.1.

I would second the recommendation for Edmonds.

## 1. What are 6j symbols and how are they used in explaining tetrahedrons?

6j symbols are mathematical expressions used in quantum mechanics to describe the coupling of angular momenta. They are used in the explanation of tetrahedrons by representing the possible orientations of four angular momenta vectors in a tetrahedral configuration.

## 2. How do 6j symbols relate to the concept of tetrahedral symmetry?

6j symbols are closely related to the concept of tetrahedral symmetry because they represent the possible ways in which angular momenta can be combined in a tetrahedral configuration, which is a form of symmetry. They also follow specific rules and symmetries in their mathematical formulation.

## 3. Can 6j symbols be used to explain other geometrical shapes besides tetrahedrons?

Yes, 6j symbols can be used to explain other geometrical shapes and configurations, such as octahedrons and hexagons. They can also be applied to various physical systems, including atoms, nuclei, and molecules, to describe their symmetry and behavior.

## 4. Are there any limitations or assumptions in using 6j symbols to explain tetrahedrons?

One limitation of using 6j symbols to explain tetrahedrons is that they assume a static configuration of angular momenta, whereas in reality, these vectors are constantly changing. Additionally, the mathematical formulation of 6j symbols may become more complex for larger systems with higher numbers of angular momenta.

## 5. How do 6j symbols contribute to our understanding of tetrahedrons in physics?

6j symbols provide a quantitative approach to understanding the symmetries and orientations of angular momenta in tetrahedrons, which are crucial in many physical systems. They also allow for predictions and calculations of physical properties related to tetrahedral symmetry, aiding in our understanding of various phenomena in physics.

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