Total Angular Momentum

Total angular momentum is the generator of rotations of a whole system. For two angular momenta acting on different factors, it is

$$\mathbf J=\mathbf J_1+\mathbf J_2.$$

That formula is compact, but it carries an important operator meaning: $\mathbf J_1$ acts only on the first factor, $\mathbf J_2$ acts only on the second factor, and the sum generates a simultaneous rotation of both factors. The tensor-product representation point of view is developed in Tensor Product Representations; this page is the canonical operator-level account.

Operator Definition

Let

$$\mathcal H = \mathcal H_{j_1}\otimes\mathcal H_{j_2}.$$

The first angular momentum acts as $\mathbf J_1\otimes I_2$, and the second acts as $I_1\otimes\mathbf J_2$. Component by component, the total angular momentum is

$$J_i = J_{1i}\otimes I_2 + I_1\otimes J_{2i}, \qquad i=x,y,z.$$

The identity operators are not decoration. They specify which factor each operator acts on. In compact notation one often writes

$$\mathbf J=\mathbf J_1+\mathbf J_2,$$

but the tensor-product meaning is always present.

For a system with several angular momenta, the same idea gives

$$\mathbf J = \mathbf J_1+\mathbf J_2+\cdots+\mathbf J_n,$$

where each term acts on its own factor and as the identity on all the others.

Generator of Simultaneous Rotations

If $U_1(R)$ and $U_2(R)$ represent the same spatial rotation $R$ on the two factors, the rotation of the composite system is

$$U(R) = U_1(R)\otimes U_2(R).$$

For a rotation by angle $\theta$ about a unit vector $\hat{\mathbf n}$,

$$U(R) = \exp\left( -\frac{i}{\hbar} \theta\,\hat{\mathbf n}\cdot\mathbf J \right).$$

Thus the total operator $\mathbf J$ is not merely a convenient sum of observables. It is the infinitesimal generator of joint rotations of the composite state. This is the physical reason total angular momentum, rather than the separate angular momenta, is the relevant conserved quantity in rotationally invariant coupled systems.

Commutation Relations

Operators on different tensor factors commute:

$$[J_{1i}\otimes I_2,\ I_1\otimes J_{2j}] =0.$$

Assume each factor separately satisfies the angular momentum algebra,

$$[J_{ai},J_{aj}] = i\hbar\sum_k\epsilon_{ijk}J_{ak}, \qquad a=1,2.$$

Then the total components satisfy the same algebra:

$$\begin{aligned} [J_i,J_j] &= [J_{1i},J_{1j}]\otimes I_2 + I_1\otimes[J_{2i},J_{2j}] \\ &= i\hbar\sum_k\epsilon_{ijk} \left( J_{1k}\otimes I_2 + I_1\otimes J_{2k} \right) \\ &= i\hbar\sum_k\epsilon_{ijk}J_k. \end{aligned}$$

Total angular momentum is therefore an angular momentum in its own right. It has a Casimir operator

$$J^2=J_x^2+J_y^2+J_z^2$$

and can be labeled using simultaneous eigenstates of $J^2$ and one component, conventionally $J_z$.

J Squared and Jz

The $z$ component is additive:

$$J_z=J_{1z}+J_{2z}.$$

Therefore an uncoupled product state

$$\lvert j_1,m_1\rangle \lvert j_2,m_2\rangle$$

is an eigenstate of $J_z$ with

$$M=m_1+m_2.$$

By contrast, the same product state is not usually an eigenstate of $J^2$. The square of the total angular momentum is

$$J^2 = J_1^2+J_2^2 + 2\,\mathbf J_1\cdot\mathbf J_2.$$

The cross term is what mixes product states with the same $M$ but different individual projections. The coupled basis is built precisely to diagonalize

$$J_1^2,\quad J_2^2,\quad J^2,\quad J_z.$$

The basis dictionary is developed in Coupled and Uncoupled Bases.

Total-J Eigenstates

A coupled state is written

$$\lvert j_1,j_2;J,M\rangle.$$

It satisfies

$$J^2\lvert j_1,j_2;J,M\rangle = \hbar^2J(J+1) \lvert j_1,j_2;J,M\rangle,$$

and

$$J_z\lvert j_1,j_2;J,M\rangle = \hbar M \lvert j_1,j_2;J,M\rangle.$$

For fixed $j_1$ and $j_2$, the allowed total angular momenta are

$$J = |j_1-j_2|, |j_1-j_2|+1, \ldots, j_1+j_2.$$

For each allowed $J$,

$$M=-J,-J+1,\ldots,J.$$

The endpoint $J=j_1+j_2$ corresponds to maximal alignment of the two angular momenta. Lower values correspond to states in which the angular momenta are combined less constructively, down to the minimum allowed by the triangle inequality.

The dimensions match:

$$(2j_1+1)(2j_2+1) = \sum_{J=|j_1-j_2|}^{j_1+j_2}(2J+1).$$

For two angular momenta, each allowed $J$ occurs once. With three or more angular momenta, the same total $J$ may occur with multiplicity, and one needs additional coupling labels or recoupling coefficients.

Why Product States Usually Are Not Total-J States

Because $J_z=J_{1z}+J_{2z}$, a product state has a definite total projection $M=m_1+m_2$. But $J^2$ contains the scalar product $\mathbf J_1\cdot\mathbf J_2$, so it can connect product states with the same $M$.

For two spin-$1/2$ systems, the states

$$\lvert\uparrow\downarrow\rangle \quad \text{and} \quad \lvert\downarrow\uparrow\rangle$$

both have $M=0$. The eigenstates of total spin are instead the symmetric and antisymmetric combinations:

$$\begin{aligned} \lvert1,0\rangle &= \frac{1}{\sqrt2} \left( \lvert\uparrow\downarrow\rangle + \lvert\downarrow\uparrow\rangle \right), \\ \lvert0,0\rangle &= \frac{1}{\sqrt2} \left( \lvert\uparrow\downarrow\rangle - \lvert\downarrow\uparrow\rangle \right). \end{aligned}$$

The full derivation is given in Two Spin-1/2 Particles. The lesson is general: product labels are excellent for separate components, while total labels are excellent for joint rotational symmetry.

Conserved Total Angular Momentum

If the Hamiltonian is invariant under the joint rotations generated by $\mathbf J$, then

$$[H,J_i]=0, \qquad i=x,y,z.$$

It follows that

$$[H,J^2]=0, \qquad [H,J_z]=0.$$

In that case $J$ and $M$ can be used as good quantum numbers. The separate angular momenta may or may not be conserved, depending on the Hamiltonian.

For example, a central spin–orbit interaction has the angular form

$$H_{\mathrm{SO}} = \xi(r)\,\mathbf L\cdot\mathbf S.$$

It does not generally conserve $L_z$ and $S_z$ separately, but it is invariant under simultaneous rotations generated by

$$\mathbf J=\mathbf L+\mathbf S.$$

The conserved quantity is total angular momentum. The detailed angular application is Spin–Orbit Coupling.

Scalar Products and Energy Splittings

The identity

$$\mathbf J_1\cdot\mathbf J_2 = \frac12 \left( J^2-J_1^2-J_2^2 \right)$$

is one of the most useful consequences of the definition $\mathbf J=\mathbf J_1+\mathbf J_2$. If a Hamiltonian contains

$$H_{\mathrm{int}} = A\,\mathbf J_1\cdot\mathbf J_2,$$

then the coupled basis diagonalizes the interaction for fixed $j_1$ and $j_2$. On a state of total $J$,

$$\mathbf J_1\cdot\mathbf J_2 \longrightarrow \frac{\hbar^2}{2} \left[ J(J+1)-j_1(j_1+1)-j_2(j_2+1) \right].$$

This single formula underlies singlet–triplet splittings, spin–orbit splittings, hyperfine structure, and effective exchange interactions. The coefficient $A$ and the physical interpretation depend on the system; the angular algebra is universal.

Common Mistakes

• Writing $\mathbf J=\mathbf J_1+\mathbf J_2$ while forgetting that the two terms act on different tensor factors.
• Assuming a product state with definite $m_1$ and $m_2$ automatically has definite $J$. It automatically has definite $M=m_1+m_2$, not usually definite $J$.
• Treating $J$ as always equal to $j_1+j_2$. All values from $|j_1-j_2|$ to $j_1+j_2$ can occur.
• Confusing conservation of total angular momentum with conservation of each separate angular momentum.
• Interpreting $\mathbf J_1\cdot\mathbf J_2$ as an ordinary geometric dot product of classical vectors rather than a quantum operator.
• Dropping the labels $j_1,j_2$ in problems where the same total $J$ can arise from different constituent angular momenta.

Cross-Links

• Tensor Product Representations in Angular Momentum
• Coupled and Uncoupled Bases
• Two Spin-1/2 Particles
• Singlet and Triplet States
• Clebsch–Gordan Coefficients
• Spin–Orbit Coupling
• Hyperfine Structure
• Angular Momentum Algebra
• Orbital Angular Momentum
• Spin as Intrinsic Angular Momentum
• Simultaneous Eigenstates and Good Quantum Numbers
• Tensor Products
• Complete Sets of Commuting Observables
• Toolkit Tensor Product Representations

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Springer, 2015.

Exercises

1. Show that $J_z$ has eigenvalue $\hbar(m_1+m_2)$ on an uncoupled product state.

Solution

Use

$$J_z=J_{1z}+J_{2z}.$$

Then

$$\begin{aligned} J_z \lvert j_1,m_1\rangle \lvert j_2,m_2\rangle &= (J_{1z}+J_{2z}) \lvert j_1,m_1\rangle \lvert j_2,m_2\rangle \\ &= \hbar m_1 \lvert j_1,m_1\rangle \lvert j_2,m_2\rangle + \hbar m_2 \lvert j_1,m_1\rangle \lvert j_2,m_2\rangle \\ &= \hbar(m_1+m_2) \lvert j_1,m_1\rangle \lvert j_2,m_2\rangle. \end{aligned}$$

2. For $j_1=3/2$ and $j_2=1$, list the allowed total $J$ values and check the dimension count.

Solution

The allowed total angular momenta are

$$J = \left|\frac32-1\right|, \left|\frac32-1\right|+1, \ldots, \frac32+1.$$

Thus

$$J=\frac12,\frac32,\frac52.$$

The tensor-product dimension is

$$(2j_1+1)(2j_2+1) = 4\cdot3 = 12.$$

The coupled multiplet dimensions are

$$(2\cdot\tfrac12+1) + (2\cdot\tfrac32+1) + (2\cdot\tfrac52+1) = 2+4+6 = 12.$$

3. Compute the eigenvalue of $\mathbf J_1\cdot\mathbf J_2$ for two spin-$1/2$ particles in the triplet and singlet sectors.

Solution

Use

$$\mathbf J_1\cdot\mathbf J_2 = \frac12 \left( J^2-J_1^2-J_2^2 \right).$$

For two spin-$1/2$ particles,

$$J_1^2=J_2^2=\frac34\hbar^2.$$

In the triplet sector, $J=1$, so

$$\mathbf J_1\cdot\mathbf J_2 = \frac12 \left( 2\hbar^2-\frac34\hbar^2-\frac34\hbar^2 \right) = \frac14\hbar^2.$$

In the singlet sector, $J=0$, so

$$\mathbf J_1\cdot\mathbf J_2 = \frac12 \left( 0-\frac34\hbar^2-\frac34\hbar^2 \right) = -\frac34\hbar^2.$$

4. Explain why a Hamiltonian can conserve $\mathbf J=\mathbf L+\mathbf S$ without conserving $L_z$ and $S_z$ separately.

Solution

Conservation follows from commutation with the Hamiltonian. A rotationally invariant interaction such as $\xi(r)\mathbf L\cdot\mathbf S$ commutes with the total rotation generators $\mathbf J=\mathbf L+\mathbf S$, so total angular momentum is conserved.

However, the same interaction contains terms that can exchange projection between orbital and spin degrees of freedom. The separate labels $m_\ell$ and $m_s$ need not be conserved. The protected projection is their sum,

$$m_j=m_\ell+m_s.$$