Combining Two Angular Momenta

Two angular momenta in one system

Almost every interesting quantum system carries more than one source of angular momentum. An electron in an atom has orbital angular momentum $\mathbf{L}$ from its motion around the nucleus and an intrinsic spin $\mathbf{S}$ . Two electrons each carry their own spin. A nucleus and an electron each contribute. In every such case we face the same question: given two angular momenta, how do they combine into a single total angular momentum?

We write the two angular momentum operators as $\mathbf{J}_1$ and $\mathbf{J}_2$ . Each separately satisfies the angular-momentum algebra,

[J_{1x}, J_{1y}] = i\hbar\, J_{1z}, \qquad [J_{2x}, J_{2y}] = i\hbar\, J_{2z},

and any cyclic permutation thereof. Because the two systems are physically distinct, operators belonging to different subsystems commute:

[J_{1a}, J_{2b}] = 0 \quad \text{for all components } a, b.

Defining the total angular momentum

The natural candidate for the combined angular momentum is the component-wise sum

\mathbf{J} = \mathbf{J}_1 + \mathbf{J}_2, \qquad J_a = J_{1a} + J_{2a}.

A short calculation shows that $\mathbf{J}$ is itself a bona-fide angular momentum. For example,

[J_x, J_y] = [J_{1x}+J_{2x},\, J_{1y}+J_{2y}] = [J_{1x},J_{1y}] + [J_{2x},J_{2y}] = i\hbar(J_{1z}+J_{2z}) = i\hbar\, J_z,

where the cross terms vanish because operators from different subsystems commute. So $\mathbf{J}$ obeys exactly the same commutation relations as $\mathbf{J}_1$ and $\mathbf{J}_2$ . This is why the total angular momentum is a meaningful, conserved quantity for an isolated combined system.

The addition problem

Each subsystem is described by a quantum number: $j_1$ fixes the magnitude of $\mathbf{J}_1$ through $\mathbf{J}_1^2 = \hbar^2 j_1(j_1+1)$ , and likewise $j_2$ for $\mathbf{J}_2$ . The addition of angular momenta problem asks:

Given $j_1$ and $j_2$ , what values of the total quantum number $j$ are possible, and how is the combined state space organized into states of definite total $j$ and definite $J_z$ ?

Answering this is the entire content of Module 8. The headline result — which we will build up carefully — is that the allowed totals run from $|j_1 - j_2|$ up to $j_1 + j_2$ in integer steps,

j \in \{\, |j_1 - j_2|,\; |j_1 - j_2| + 1,\; \ldots,\; j_1 + j_2 \,\}.

Why it matters

The answer to the addition problem governs atomic spectra (term symbols and fine structure), the rules for how light is emitted and absorbed (selection rules), and the classification of two-qubit states into the singlet and triplet that appear throughout quantum information. The next lesson introduces the two natural ways to label the combined states — the uncoupled and coupled bases — which is the language we use for everything that follows.