Table of Contents
- Introduction
- Motivation for Combining Angular Momenta
- Coupled and Uncoupled Bases
- What Are Clebsch–Gordan Coefficients?
- Mathematical Definition
- Orthogonality and Normalization
- How to Read and Use CG Coefficients
- Clebsch–Gordan Series
- Explicit Examples
- CG Table for \( j_1 = \frac{1}{2} \) and \( j_2 = \frac{1}{2} \)
- Symmetry Properties
- Recursive Relations and Computation
- Physical Applications
- Generalization: Wigner Symbols and Beyond
- Conclusion
1. Introduction
When two quantum systems, each with angular momentum, are combined into a composite system, the resulting angular momentum states are described in terms of Clebsch–Gordan (CG) coefficients. These coefficients allow us to switch between product states and total angular momentum eigenstates, forming the foundation of coupled quantum systems.
2. Motivation for Combining Angular Momenta
We combine angular momenta to:
- Understand multi-particle systems
- Construct total spin/orbital states in atoms
- Predict allowed transitions and spectral splitting
3. Coupled and Uncoupled Bases
Uncoupled Basis:
\[
|j_1, m_1\rangle \otimes |j_2, m_2\rangle
\]
Describes individual angular momenta.
Coupled Basis:
\[
|j, m\rangle
\]
Describes total angular momentum \( \vec{J} = \vec{J}_1 + \vec{J}_2 \) with quantum numbers \( j \) and \( m \).
4. What Are Clebsch–Gordan Coefficients?
They are the expansion coefficients in:
\[
|j, m\rangle = \sum_{m_1, m_2} C_{j_1 m_1, j_2 m_2}^{j m} |j_1, m_1\rangle |j_2, m_2\rangle
\]
These coefficients describe how to reconstruct a total angular momentum state from the tensor product of individual states.
5. Mathematical Definition
Given two angular momenta \( j_1 \) and \( j_2 \), the CG coefficients \( C_{j_1 m_1, j_2 m_2}^{j m} \) are real (in most common phase conventions) and satisfy:
\[
\langle j_1, m_1; j_2, m_2 | j, m \rangle = C_{j_1 m_1, j_2 m_2}^{j m}
\]
6. Orthogonality and Normalization
They satisfy orthonormality:
\[
\sum_{j, m} C_{j_1 m_1, j_2 m_2}^{j m} C_{j_1 m_1′, j_2 m_2′}^{j m} = \delta_{m_1 m_1′} \delta_{m_2 m_2′}
\]
\[
\sum_{m_1, m_2} C_{j_1 m_1, j_2 m_2}^{j m} C_{j_1 m_1, j_2 m_2}^{j’ m’} = \delta_{j j’} \delta_{m m’}
\]
7. How to Read and Use CG Coefficients
Use tabulated values to:
- Construct total angular momentum states
- Convert back to product basis
- Analyze symmetries and selection rules in interactions
8. Clebsch–Gordan Series
For two angular momenta \( j_1 \) and \( j_2 \), the total \( j \) runs from:
\[
j = |j_1 – j_2|, …, j_1 + j_2
\]
Each \( j \) has \( 2j + 1 \) values of \( m \in [-j, j] \).
9. Explicit Examples
Example 1: \( j_1 = j_2 = \frac{1}{2} \)
\[
|1, 1\rangle = |\uparrow\uparrow\rangle
\]
\[
|1, 0\rangle = \frac{1}{\sqrt{2}} (|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle)
\]
\[
|1, -1\rangle = |\downarrow\downarrow\rangle
\]
\[
|0, 0\rangle = \frac{1}{\sqrt{2}} (|\uparrow\downarrow\rangle – |\downarrow\uparrow\rangle)
\]
Example 2: \( j_1 = 1, j_2 = \frac{1}{2} \)
Yields \( j = \frac{1}{2}, \frac{3}{2} \)
10. CG Table for \( j_1 = \frac{1}{2} \), \( j_2 = \frac{1}{2} \)
| \( m_1 \) | \( m_2 \) | \( j=1 \), \( m \) | \( j=0 \), \( m=0 \) |
|---|---|---|---|
| +½ | +½ | 1 | 0 |
| +½ | -½ | 1/√2 | 1/√2 |
| -½ | +½ | 1/√2 | -1/√2 |
| -½ | -½ | 1 | 0 |
11. Symmetry Properties
\[
C_{j_1 m_1, j_2 m_2}^{j m} = (-1)^{j_1 + j_2 – j} C_{j_2 m_2, j_1 m_1}^{j m}
\]
\[
C_{j_1 -m_1, j_2 -m_2}^{j -m} = (-1)^{j_1 + j_2 – j} C_{j_1 m_1, j_2 m_2}^{j m}
\]
These help reduce computation and derive unknown coefficients.
12. Recursive Relations and Computation
CG coefficients can be computed using recursive relations, Racah formulas, or software packages like:
- Wolfram Mathematica
- SymPy
- QuantumToolbox.jl
They are also encoded in Wigner 3-j symbols, which are closely related.
13. Physical Applications
- Spectroscopy: term splitting and selection rules
- Atomic physics: shell structure and configurations
- Quantum information: entanglement and spin addition
- Nuclear and particle physics: isospin and SU(2) symmetry
14. Generalization: Wigner Symbols and Beyond
CG coefficients are part of a larger framework including:
- Wigner 3-j symbols
- 6-j and 9-j symbols
- Tensor operators and spherical tensor formalism
These are used in more complex coupling schemes in many-body systems.
15. Conclusion
Clebsch–Gordan coefficients are indispensable tools for adding angular momenta in quantum mechanics. They form the bridge between individual and total angular momentum descriptions, enabling detailed predictions about the structure, behavior, and interaction of quantum systems. Mastery of CG coefficients is essential in atomic physics, quantum theory, and modern applications like quantum computing.
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