The Coupled and Uncoupled Bases

Two ways to label the same space

Combine two angular momenta with quantum numbers $j_1$ and $j_2$ . The combined Hilbert space is the tensor product of the two subsystem spaces, with dimension

(2j_1 + 1)(2j_2 + 1).

There are two natural complete sets of commuting observables we can use to label a basis for this space, and the whole subject is really the relationship between them.

The uncoupled basis

The first choice describes each subsystem separately. We simultaneously diagonalize the four mutually commuting operators

\mathbf{J}_1^2, \quad J_{1z}, \quad \mathbf{J}_2^2, \quad J_{2z}.

Their joint eigenstates are simple tensor products, written

|j_1, m_1\rangle \otimes |j_2, m_2\rangle \equiv |j_1, m_1; j_2, m_2\rangle,

with $m_1 \in \{-j_1, \ldots, j_1\}$ and $m_2 \in \{-j_2, \ldots, j_2\}$ . Each state tells you the $z$ -projection of each angular momentum independently. There are exactly $(2j_1+1)(2j_2+1)$ of them, so they form a complete basis. Since $j_1$ and $j_2$ are fixed for the problem, we often abbreviate these as $|m_1, m_2\rangle$ .

The coupled basis

The second choice describes the total angular momentum. The operators

\mathbf{J}_1^2, \quad \mathbf{J}_2^2, \quad \mathbf{J}^2, \quad J_z

also mutually commute (we will verify the key commutators in later lessons), so they too can be diagonalized simultaneously. Their joint eigenstates are written

|j, m\rangle,

with definite total quantum number $j$ and total projection $m$ , where $J_z = J_{1z} + J_{2z}$ so $m = m_1 + m_2$ . These are the states of definite total angular momentum.

Why not just use one?

Each basis diagonalizes a different physical question. The uncoupled basis is convenient when the two subsystems are independent — for example, two spins in separate magnetic fields, where the energy depends on $m_1$ and $m_2$ separately. The coupled basis is convenient whenever the interaction depends on the total angular momentum — for example, a spin–orbit coupling term $\propto \mathbf{L}\cdot\mathbf{S}$ , whose energy is fixed by $j$ .

The crucial point is that these are two complete bases for the same space, so each coupled state is a specific linear combination of uncoupled states:

|j, m\rangle = \sum_{m_1 + m_2 = m} C^{\,j m}_{m_1 m_2}\, |j_1, m_1; j_2, m_2\rangle.

The expansion coefficients $C^{\,j m}_{m_1 m_2}$ are the Clebsch–Gordan coefficients. Building and using them — first for two spin- $\tfrac12$ particles, then in general — is the work of the rest of this module.

What we will reuse

The notation $m = m_1 + m_2$ is more than bookkeeping: because $J_z$ is diagonal in both bases with the same eigenvalues, the change of basis only mixes uncoupled states that share the same total $m$ . That single observation, developed in the next few lessons, makes the entire construction tractable.