Addition of Angular Momentum in Quantum Mechanics

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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