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  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
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  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Quantum mechanics → Angular momentum and spin ↓


Clebsch–Gordan coefficient


In quantum mechanics, angular momentum plays an important role, especially in the study of atomic and subatomic particles. One of the essential aspects of understanding angular momentum is the summation of angular momenta, which often leads us to the Clebsch-Gordan coefficients. Named after physicists Alfred Clebsch and Paul Gordon, these coefficients are important when combining angular momenta in quantum systems, such as when considering the spin of two particles. This explanation will explore the concept, provide visual examples, and explain the calculation and significance of these coefficients.

Understanding angular momentum in quantum mechanics

To understand the Clebsch-Gordan coefficient, we first need a solid understanding of angular momentum in quantum mechanics. Angular momentum in quantum mechanics is quantized, which means that it can only take on certain discrete values. It is described by two quantum numbers: the magnitude of the angular momentum, given by j, and its projection along a chosen axis, usually z axis, given by m.

The quantum number j can take values such as 0, 1/2, 1, 3/2, etc., while m can range from -j to +j in integer steps. Thus, for a particle with angular momentum j = 1, the possible values of m will be -1, 0, and 1.

Adding angular momentum

When we deal with systems of many particles, each of which has its own angular momentum, we often need to calculate the total angular momentum. This situation arises in many areas of quantum mechanics, such as when combining the intrinsic spins of particles or when interacting with orbital angular momentum in atomic systems.

To add angular momenta, we use the rule that if you have two angular momenta, j_1 and j_2, their combined angular momentum, J, can take values between |j_1 - j_2| and j_1 + j_2, in integer steps. The projection quantum number M of the combined system is simply the sum of the projections: M = m_1 + m_2, where m_1 and m_2 are the projection quantum numbers of the individual angular momenta.

For example, if you combine two particles with spin j = 1/2, the possible values of the total angular momentum J will be 1 and 0.

Introduction to the Clebsch-Gordan coefficient

Clebsch-Gordan coefficients come in handy when you want to express the combined angular momentum states in terms of the individual angular momentum states. Specifically, they provide the weights needed to express a particular total angular momentum state as a linear combination of the product states of the individual motions.

Visual representation

Consider a visual representation of the combination of two angular momenta, where j_1 = 1 and j_2 = 1/2:

j = 3/2 j = 1/2 j = 1/2

Mathematical formulas

The general formula for combining two angular momentum states is expressed using Clebsch–Gordan coefficients as follows:

    |j_1, j_2; j, m⟩ = ∑ |j_1, m_1; j_2, m_2⟩

Here, |j_1, j_2; J, M⟩ is the combined angular momentum state, and |j_1, m_1; j_2, m_2⟩ are the individual angular momentum states. The Clebsch-Gordan coefficients <j_1, j_2; m_1, m_2 | J, M> are numerical factors that tell us how to weight the product states to produce the total angular momentum state.

Examples of Clebsch–Gordan coefficients

Let's look at a more explicit example. Suppose you have two particles, one with j_1 = 1/2 and the other with j_2 = 1/2. The possible total angular momentum J values are 1 and 0.

For J = 1, we have:

    |1, 1⟩ = |+1/2⟩|+1/2⟩
    |1, 0⟩ = (|+1/2⟩|-1/2⟩ + |-1/2⟩|+1/2⟩) / √2
    |1, -1⟩ = |-1/2⟩|-1/2⟩

For J = 0, we have:

    |0, 0⟩ = (|+1/2⟩|-1/2⟩ - |-1/2⟩|+1/2⟩) / √2

Here, the coefficient of each term before the kets represents the Clebsch–Gordan coefficient for that particular term. For example, the position |1, 0⟩ is a symmetric combination of |+1/2⟩|-1/2⟩ and |-1/2⟩|+1/2⟩ with equal weights, so the Clebsch–Gordan coefficient is 1/√2.

Properties of the Clebsch–Gordan coefficient

  • The Clebsch–Gordan coefficients are real numbers, due to the real nature of the angular momentum operators.
  • Their magnitude is limited between 0 and 1, giving a sense of probability amplitude.
  • Orthogonality: Different combinations of m_1 and m_2 for the same total J are orthogonal.

Orthogonality example

For orthogonality, if we consider two states |J, M⟩ and |J', M'⟩ which have different total momenta or projections, then their inner product must be zero.

    ⟨J, M|J, M'⟩ = 0 if M ≠ M'

Finding the Clebsch-Gordan coefficient

Although tables and software are available to find Clebsch-Gordan coefficients easily, calculating them by hand helps to understand their nature in depth. Recursive properties, symmetry considerations, and normalization conditions are applied in their derivation.

Recursive strategy

A standard method for calculating the Clebsch-Gordan coefficients involves using iterative formulas derived from angular momentum algebra. The main idea is to start with situations where the calculations are simple and then build up to more complicated situations.

    √((J + M)(J - M + 1)) * ⟨j_1, m_1; j_2, m_2|J, M – 1⟩ +
    √((j - m)(j + m + 1)) * ⟨j_1, m_1; j_2, m_2|J, M + 1⟩ = 
    m_1 * ⟨j_1, m_1; j_2, m_2|J, M⟩

Applications in physics

In practical applications, Clebsch–Gordan coefficients simplify the task of evaluating matrix elements where systems are coupled. For example, they appear in calculations of transition probabilities when particles with spin interact with electromagnetic radiation, such as in atomic orbital transitions or in photon scattering experiments.

Examples in particle physics

In particle physics, knowing the Clebsch-Gordan coefficients is important for calculating amplitudes where particles, such as quarks, interact and form composite particles such as protons and neutrons. Combination rules help predict the mixing of flavor and color states according to quantum chromodynamics.

Consider hadronization, where quarks combine to form hadrons: by applying the Clebsch-Gordan coefficients, one can predict which combinations of quark spins and colours are viable solutions in the confinement scenario.

Conclusion

Clebsch-Gordan coefficients provide an essential mathematical formalism for handling the complexities of quanta dealing with angular momentum. As the foundation of quantum mechanical systems that combine spin, their role extends from theoretical constructions to experimental predictions where they directly impact measurements and interpretations. Understanding these coefficients, their derivation and application provides a better understanding not only of quantum mechanics but also of broader physical theories at a fundamental level.


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