In quantum mechanics, understanding how individual angular momenta combine to form a total angular momentum is essential for analyzing composite systems, such as multi-electron atoms or proton-neutron pairs. This post details the mathematical framework of tensor products, deriving the relationship between the independent "Uncoupled Basis" and the interacting "Coupled Basis" via Clebsch-Gordan coefficients.
Direct Product States and the Addition of Angular Momentum
1. The Concept of Composite Systems
When we deal with a system consisting of two distinct components—labeled 1 and 2—with angular momentum operators \( \mathbf{J}_1 \) and \( \mathbf{J}_2 \), the total Hilbert space is defined as the tensor product (direct product) of the individual Hilbert spaces. The total angular momentum operator \( \mathbf{J} \) is given by:
Here, \( \mathbf{J}_1 \) acts only on the subspace of particle 1, and \( \mathbf{J}_2 \) acts only on particle 2. The core problem of adding angular momenta is determining the eigenvalues and eigenstates of this total operator \( \mathbf{J} \).
2. The Uncoupled Basis (Direct Product State)
In the absence of interaction between the two parts, the operators \( \mathbf{J}_1^2, \mathbf{J}_{1z}, \mathbf{J}_2^2, \) and \( \mathbf{J}_{2z} \) all commute. They form a Complete Set of Commuting Observables (CSCO). The basis states defined by these operators are called the Uncoupled Basis.
These states are simply the direct product of the individual eigenstates:
Eigenvalue relations:
- \( \mathbf{J}_1^2 |j_1 m_1 j_2 m_2\rangle = \hbar^2 j_1(j_1+1) |j_1 m_1 j_2 m_2\rangle \)
- \( \mathbf{J}_{1z} |j_1 m_1 j_2 m_2\rangle = \hbar m_1 |j_1 m_1 j_2 m_2\rangle \)
- \( \mathbf{J}_{Total, z} |j_1 m_1 j_2 m_2\rangle = \hbar (m_1 + m_2) |j_1 m_1 j_2 m_2\rangle \)
The dimension of this basis is the product of the dimensions of the subspaces: \( N = (2j_1 + 1)(2j_2 + 1) \).
3. The Coupled Basis
When there is a coupling interaction (e.g., Spin-Orbit coupling \( H \propto \mathbf{L} \cdot \mathbf{S} \)), the individual z-components \( m_1 \) and \( m_2 \) are no longer conserved quantities. However, the total angular momentum \( \mathbf{J}^2 \) and its z-component \( \mathbf{J}_z \) are conserved.
The new CSCO is \( \{ \mathbf{J}^2, \mathbf{J}_z, \mathbf{J}_1^2, \mathbf{J}_2^2 \} \). The eigenstates of this set form the Coupled Basis, denoted as:
Selection Rules (Triangle Inequality)
Not all values of total angular momentum \( J \) are possible. They are constrained by the vector addition rules:
The z-component is additive: \( M = m_1 + m_2 \).
4. Clebsch-Gordan Coefficients
Since both bases span the same Hilbert space, we can express the coupled states as a linear combination of the uncoupled states (direct product states). This unitary transformation is given by:
The coefficients \( C \) are known as the Clebsch-Gordan (CG) Coefficients. They represent the probability amplitude that a system prepared in a coupled state \( |J, M\rangle \) will be found in the uncoupled state \( |m_1, m_2\rangle \).
5. Example: Addition of Two Spin-1/2 Particles
Consider two electrons, each with spin \( s = 1/2 \). The total dimension is \( 2 \times 2 = 4 \). The possible values for total spin \( S \) are \( 0 \) and \( 1 \).
The Triplet State (S = 1)
These states are symmetric under particle exchange:
The Singlet State (S = 0)
This state is anti-symmetric: