Table of Contents

1. Introduction
2. Motivation for Combining Angular Momenta
3. Coupled and Uncoupled Bases
4. What Are Clebsch–Gordan Coefficients?
5. Mathematical Definition
6. Orthogonality and Normalization
7. How to Read and Use CG Coefficients
8. Clebsch–Gordan Series
9. Explicit Examples
10. CG Table for \( j_1 = \frac{1}{2} \) and \( j_2 = \frac{1}{2} \)
11. Symmetry Properties
12. Recursive Relations and Computation
13. Physical Applications
14. Generalization: Wigner Symbols and Beyond
15. Conclusion

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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.

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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

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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 \( m \).

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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.

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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:

\[
\langle j_1, m_1; j_2, m_2 | j, m \rangle = C_{j_1 m_1, j_2 m_2}^{j m}
\]

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6. Orthogonality and Normalization

They satisfy orthonormality:

\[
\sum_{j, m} C_{j_1 m_1, j_2 m_2}^{j m} C_{j_1 m_1′, j_2 m_2′}^{j m} = \delta_{m_1 m_1′} \delta_{m_2 m_2′}
\]

\[
\sum_{m_1, m_2} C_{j_1 m_1, j_2 m_2}^{j m} C_{j_1 m_1, j_2 m_2}^{j’ m’} = \delta_{j j’} \delta_{m m’}
\]

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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

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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 \( m \in [-j, j] \).

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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 \( j = \frac{1}{2}, \frac{3}{2} \)

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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

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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.

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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.

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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

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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.

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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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Table of Contents

1. Introduction
2. Why Add Angular Momenta?
3. Individual Angular Momenta
4. Total Angular Momentum Operator
5. Commutation Relations and Conservation
6. Allowed Values of Total Angular Momentum \( J \)
7. Clebsch–Gordan Coefficients
8. Coupled vs Uncoupled Basis
9. Examples: Adding Two Spin-1/2 Particles
10. Triplet and Singlet States
11. Angular Momentum Addition Rules
12. Clebsch–Gordan Table: A Quick Look
13. Physical Applications
14. Extensions to Higher Spins
15. Conclusion

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1. Introduction

In quantum mechanics, systems often involve multiple particles, each with their own angular momentum. To understand the behavior of the whole system—whether in atoms, molecules, or nuclei—we must learn how to add angular momentum operators. This leads to a new set of quantum numbers and states that represent the combined system.

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2. Why Add Angular Momenta?

Reasons include:

• Combining orbital \( \vec{L} \) and spin \( \vec{S} \) into total angular momentum \( \vec{J} \)
• Describing multi-particle systems, such as electrons in atoms
• Understanding selection rules and spectral line splitting

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3. Individual Angular Momenta

For two particles, we label:

• \( \vec{J}_1 \) with quantum number \( j_1 \), and \( m_1 \)
• \( \vec{J}_2 \) with quantum number \( j_2 \), and \( m_2 \)

Their respective states:
\[
|j_1, m_1\rangle \quad \text{and} \quad |j_2, m_2\rangle
\]

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4. Total Angular Momentum Operator

The total angular momentum is:

\[
\vec{J} = \vec{J}_1 + \vec{J}_2
\]

And satisfies:

\[
\hat{J}^2 = (\hat{J}_1 + \hat{J}_2)^2 = \hat{J}_1^2 + \hat{J}_2^2 + 2\hat{J}_1 \cdot \hat{J}_2
\]

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5. Commutation Relations and Conservation

Total angular momentum components obey the same algebra:

\[
[\hat{J}i, \hat{J}_j] = i\hbar \epsilon{ijk} \hat{J}_k
\]

If the Hamiltonian is rotationally invariant, then \( \hat{J}^2 \) and \( \hat{J}_z \) are conserved quantities.

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6. Allowed Values of Total Angular Momentum \( J \)

The resulting quantum number \( j \) from adding \( j_1 \) and \( j_2 \) can take on values:

\[
j = |j_1 – j_2|, |j_1 – j_2| + 1, \dots, j_1 + j_2
\]

For example, \( j_1 = 1 \), \( j_2 = \frac{1}{2} \) ⇒ \( j = \frac{1}{2}, \frac{3}{2} \)

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7. Clebsch–Gordan Coefficients

These coefficients define how to convert between:

• The uncoupled basis \( |j_1, m_1\rangle |j_2, m_2\rangle \)
• The coupled basis \( |j, m\rangle \), where \( m = m_1 + m_2 \)

\[
|j, m\rangle = \sum_{m_1, m_2} C_{j_1 m_1, j_2 m_2}^{j m} |j_1, m_1\rangle |j_2, m_2\rangle
\]

These coefficients are tabulated and obey orthogonality and normalization.

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8. Coupled vs Uncoupled Basis

Uncoupled:

\[
|j_1, m_1\rangle \otimes |j_2, m_2\rangle
\]

Good for describing independent spins.

Coupled:

\[
|j, m\rangle
\]

Better for symmetric Hamiltonians and when total angular momentum is conserved.

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9. Examples: Adding Two Spin-1/2 Particles

Each particle has \( s = \frac{1}{2} \), so:

• \( s_{\text{total}} = 0 \) or \( 1 \)

Uncoupled basis:
\[
|\uparrow\uparrow\rangle, \quad |\uparrow\downarrow\rangle, \quad |\downarrow\uparrow\rangle, \quad |\downarrow\downarrow\rangle
\]

Coupled basis:

• Triplet (spin-1):
  \[
  |1, 1\rangle = |\uparrow\uparrow\rangle
  \]
  \[
  |1, 0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle)
  \]
  \[
  |1, -1\rangle = |\downarrow\downarrow\rangle
  \]
• Singlet (spin-0):
  \[
  |0, 0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle – |\downarrow\uparrow\rangle)
  \]

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10. Triplet and Singlet States

State	Total Spin \( j \)	Symmetry
Triplet	1	Symmetric
Singlet	0	Antisymmetric

Important in quantum statistics and entanglement.

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11. Angular Momentum Addition Rules

Key rules:

• Resultant angular momentum values are discrete
• Total \( m \): \( m = m_1 + m_2 \)
• States of the same \( j \) but different \( m \) span a multiplet
• \( 2j + 1 \) total states for each \( j \)

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12. Clebsch–Gordan Table: A Quick Look

\( j_1 \)	\( j_2 \)	Resulting \( j \) Values
1/2	1/2	0, 1
1	1/2	1/2, 3/2
1	1	0, 1, 2
3/2	1/2	1, 2

These determine allowed couplings and spectra.

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13. Physical Applications

• Fine and hyperfine splitting in atoms
• Spectroscopic term symbols
• Nuclear spin configurations
• Coupling rules in multi-electron atoms
• Entangled spin states in quantum computing

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14. Extensions to Higher Spins

Addition works similarly for higher spins:

• Use CG coefficients and angular momentum algebra
• Combine multiple \( j \)’s sequentially
• Important for atoms with multiple electrons or complex nuclei

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15. Conclusion

Adding angular momenta in quantum mechanics allows us to describe the collective behavior of complex systems. By transitioning between coupled and uncoupled bases using Clebsch–Gordan coefficients, we build a complete picture of how particles with spin or orbital angular momentum combine. Mastery of this topic is essential for atomic physics, spectroscopy, and quantum information science.

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Table of Contents

1. Introduction
2. Why Spherical Harmonics Matter
3. Definition and Notation
4. The Angular Part of the Schrödinger Equation
5. Associated Legendre Polynomials
6. Mathematical Expression of Spherical Harmonics
7. Properties and Symmetries
8. Orthogonality and Normalization
9. Quantum Numbers \( \ell \) and \( m \)
10. Visualization of Spherical Harmonics
11. Real vs Complex Forms
12. Applications in Quantum Mechanics
13. Applications Beyond Physics
14. Summary of Key Formulas
15. Conclusion

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1. Introduction

Spherical harmonics are special functions defined on the surface of a sphere. In quantum mechanics, they arise as the angular part of the wavefunction when solving the Schrödinger equation in spherical coordinates. They are essential in understanding the structure of atoms, orbital shapes, and the behavior of angular momentum.

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2. Why Spherical Harmonics Matter

Spherical harmonics are crucial for:

• Solving central potential problems like the hydrogen atom
• Describing angular momentum eigenstates
• Visualizing atomic orbitals
• Forming a complete basis on the sphere

They also appear in mathematical physics, computer graphics, and geophysics.

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3. Definition and Notation

Spherical harmonics are denoted:

\[
Y_\ell^m(\theta, \phi)
\]

Where:

• \( \ell \) is the orbital angular momentum quantum number
• \( m \) is the magnetic quantum number, \( -\ell \le m \le \ell \)

They are eigenfunctions of \( \hat{L}^2 \) and \( \hat{L}_z \).

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4. The Angular Part of the Schrödinger Equation

When solving the 3D Schrödinger equation with a central potential:

\[
\psi(r, \theta, \phi) = R(r) Y_\ell^m(\theta, \phi)
\]

The angular part satisfies:

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5. Associated Legendre Polynomials

Spherical harmonics are built from associated Legendre polynomials \( P_\ell^m(\cos\theta) \):

\[
P_\ell^m(x) = (1 – x^2)^{|m|/2} \frac{d^{|m|}}{dx^{|m|}} P_\ell(x)
\]

Where \( P_\ell(x) \) are Legendre polynomials.

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6. Mathematical Expression of Spherical Harmonics

\[
Y_\ell^m(\theta, \phi) = N_\ell^m P_\ell^m(\cos \theta) e^{i m \phi}
\]

Where the normalization factor is:

\[
N_\ell^m = \sqrt{\frac{(2\ell + 1)}{4\pi} \cdot \frac{(\ell – m)!}{(\ell + m)!}}
\]

These functions are complex-valued unless converted to real-valued combinations.

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7. Properties and Symmetries

• Parity:
  \[
  Y_\ell^m(\pi – \theta, \phi + \pi) = (-1)^\ell Y_\ell^m(\theta, \phi)
  \]
• Conjugation:
  \[
  Y_\ell^{-m} = (-1)^m Y_\ell^{m*}
  \]
• Defined over domain \( \theta \in [0, \pi], \phi \in [0, 2\pi) \)

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8. Orthogonality and Normalization

Spherical harmonics satisfy:

\[
\int_0^{2\pi} \int_0^\pi Y_\ell^m(\theta, \phi)^* Y_{\ell’}^{m’}(\theta, \phi) \sin\theta\, d\theta\, d\phi = \delta_{\ell\ell’} \delta_{mm’}
\]

This makes them a complete orthonormal basis for square-integrable functions on the sphere.

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9. Quantum Numbers \( \ell \) and \( m \)

• \( \ell \ge 0 \): integer
• \( m \in \{-\ell, …, \ell\} \)

These define:

• Total angular momentum magnitude: \( \sqrt{\ell(\ell+1)}\hbar \)
• Z-component of angular momentum: \( m\hbar \)

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10. Visualization of Spherical Harmonics

The magnitude \( |Y_\ell^m(\theta, \phi)|^2 \) represents the probability density of an electron in an orbital.

Common shapes:

• \( \ell = 0 \): spherical (s-orbitals)
• \( \ell = 1 \): dumbbells (p-orbitals)
• \( \ell = 2 \): cloverleaf (d-orbitals)

Visualization involves mapping the angular dependence onto the sphere’s surface.

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11. Real vs Complex Forms

While spherical harmonics are naturally complex, in atomic physics we often use real combinations:

These produce real-valued orbital shapes used in chemistry and visualization.

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12. Applications in Quantum Mechanics

• Angular part of the hydrogen atom wavefunctions
• Angular momentum eigenfunctions
• Selection rules for transitions
• Coupled angular momentum (Clebsch-Gordan coefficients)

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13. Applications Beyond Physics

Spherical harmonics are used in:

• Computer graphics: lighting and rendering
• Geophysics: modeling Earth’s gravitational field
• Medical imaging: diffusion MRI
• Quantum chemistry: molecular orbital construction

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14. Summary of Key Formulas

• Definition:
  \[
  Y_\ell^m(\theta, \phi) = N_\ell^m P_\ell^m(\cos\theta) e^{i m \phi}
  \]
• Orthonormality:
  \[
  \int Y_\ell^m Y_{\ell’}^{m’*} d\Omega = \delta_{\ell\ell’} \delta_{mm’}
  \]
• Complex conjugation:
  \[
  Y_\ell^{-m} = (-1)^m Y_\ell^{m*}
  \]

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15. Conclusion

Spherical harmonics are elegant mathematical tools that encode angular information in quantum systems. They provide the angular solutions to the Schrödinger equation in spherical coordinates and are foundational in understanding atomic orbitals, angular momentum, and symmetry. Their utility extends far beyond quantum mechanics, making them essential in many areas of science and engineering.

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