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Clebsch–Gordan Coefficients: Combining Quantum States

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clebsch gordan coefficients

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

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 \( j_1 = \frac{1}{2} \), \( j_2 = \frac{1}{2} \)

\( m_1 \)\( m_2 \)\( j=1 \), \( m \)\( j=0 \), \( m=0 \)
10
1/√21/√2
1/√2-1/√2
10

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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Today in History – 6 August

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today in history 6 august

today in history 6 august

1787

In Philadelphia, delegates to the Constitutional Convention began debating the first complete draft of the proposed Constitution of the United States.

1890

At Auburn Prison in New York, the first execution by electrocution in history was carried out against William Kemmler, who had been convicted of murdering his lover, Matilda Ziegler, with an axe.

1905

Chatranjan Das, freedom fighter and advocate, started his newspaper “Vande Mataram”.

1920

Film Censor Board at Bombay started working. The first film censored and granted a Censor Certificate No. 1 was a short film of 600 ft., produced by Gaumont Company and it was titled as ‘Gaumont Graphic No. 963-964’. The Regional Film Censor Boards were also set-up in Calcutta, Madras and Rangoon.

1925

Surendra Nath Banerjee died. He was one of the founders of modern India and proponent of autonomy within the British Commonwealth. He was President of Indian National CongressFormation of Indian National Congress twice, in 1895 at Pune and in 1902 at Ahmedabad.

1926

On this day in 1926, on her second attempt, 19-year-old Gertrude Ederle became the first woman to swim the 21 miles from Dover, England, to Cape Griz-Nez across the English Channel, which separates Great Britain from the northwestern tip of France.

1930

On this day in 1930, New York Supreme Court judge Joseph Force Crater vanished on the streets of Manhattan near Times Square. The dapper 41-year-old’s disappearance launched a massive investigation that captivated the nation, earning Crater the title of “the missingest man in New York.”

1945

On this day in 1945, at 8:16 a.m. Japanese time, an American B-29 bomber, the Enola Gay, droped the world’s first atom bomb, over the city of Hiroshima. Approximately 80,000 people were killed as a direct result of the blast, and another 35,000 were injured. At least another 60,000 would be dead by the end of the year from the effects of the fallout.

1951

Rukmini Lakshmipathi, great social reformer and leader, died.

1965

On this day in 1965, President Lyndon Baines Johnson signed the Voting Rights Act, guaranteeing African Americans the right to vote. The bill made it illegal to impose restrictions on federal, state and local elections that were designed to deny the vote to blacks.

1965

Indian troops invaded Pakistan.

1997

Cabinet approved a proposal to amend the Hindu Marriage Act and the Special Marriage Act, with a view to removing epilepsy as a ground for annulling marriage or for declaring marriage as null and void.

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Today in History – 5 August

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Addition of Angular Momentum in Quantum Mechanics

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addition of angular momentum

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

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.


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

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


4. Total Angular Momentum Operator

The total angular momentum is:

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

And satisfies:

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


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.


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


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.


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.


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):
    \[
    |0, 0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle – |\downarrow\uparrow\rangle)
    \]

10. Triplet and Singlet States

StateTotal Spin \( j \)Symmetry
Triplet1Symmetric
Singlet0Antisymmetric

Important in quantum statistics and entanglement.


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 \( j \)

12. Clebsch–Gordan Table: A Quick Look

\( j_1 \)\( j_2 \)Resulting \( j \) Values
1/21/20, 1
11/21/2, 3/2
110, 1, 2
3/21/21, 2

These determine allowed couplings and spectra.


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

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

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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Today in History – 5 August

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today in history 5 august

today in history 5 august

1604

John Eliot, “Apostle to Indians,” Bible translator, bapti was born.

1858

After several unsuccessful attempts, the first telegraph line across the Atlantic Ocean was completed, a feat accomplished largely through the efforts of American merchant Cyrus West Field.

1861

On this day in 1861, Lincoln imposed the first federal income tax by signing the Revenue Act. Strapped for cash with which to pursue the Civil War, Lincoln and Congress agreed to impose a 3 percent tax on annual incomes over $800.

1914

The world’s first electric traffic signal was put into place on the corner of Euclid Avenue and East 105th Street in Cleveland, Ohio, on this day in 1914.

1914

On August 5, 1914, the German army launched its assault on the city of Liege in Belgium, violating the latter country’s neutrality and beginning the first battle of World War I.

1944

Hundreds of Jews were freed from forced labor in Warsaw.

1948

An earthquake hits Ecuador killing 6,000 people and injuring another 20,000 on this day in 1948. The 6.7-magnitude tremor was particularly deadly for its size.

1950

Gopinath Bardoloi, architect of modern Assam, freedom fighter and leader, passed away.

1963

Representatives of the United States, the Soviet Union, and Great Britain signed the Nuclear Test Ban Treaty, which prohibited the testing of nuclear weapons in outer space, underwater, or in the atmosphere. The treaty was hailed as an important first step toward the control of nuclear weapons.

1975

The RSS, Anand Marg, Jamat-i-Islami, and 23 other organisations were banned. Parliament approved Conservation of Foreign Exchange and Prevention of Smuggling Activities Act (COFEPOSA). Maintenance of Internal Security Act (MISA) was approved by the Parliament.

1992

Achyutrao Patwardhan, leader of ‘Quit India Movement‘, died at Varanasi at the age of 86.

1999

Nisha Millet became the first Indian swimmer to receive the International Olympic Committee’s Solidarity Scholarship, given to sportspersons with outstanding track record and possessing Olympic potential.

1999

The Global Peace March demanded demilitarisation of the entire planet.

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Spherical Harmonics: Angular Solutions in Quantum Mechanics

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spherical harmonics

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

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.


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.


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 \( \hat{L}_z \).


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:

\[ \hat{L}^2 Y_\ell^m = \hbar^2 \ell(\ell + 1) Y_\ell^m, \quad \hat{L}z Y\ell^m = \hbar m Y_\ell^m \]

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.


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.


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 \( \theta \in [0, \pi], \phi \in [0, 2\pi) \)

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.


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: \( m\hbar \)

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.


11. Real vs Complex Forms

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

\[ Y_{\ell m}^{\text{real}}(\theta, \phi) = \begin{cases} \frac{1}{\sqrt{2}}(Y_\ell^{-m} + (-1)^m Y_\ell^m), & m > 0 \ Y_\ell^0, & m = 0 \ \frac{i}{\sqrt{2}}(Y_\ell^{-m} – (-1)^m Y_\ell^m), & m < 0 \end{cases} \]

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


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)

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

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:
    \[
    Y_\ell^{-m} = (-1)^m Y_\ell^{m*}
    \]

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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