4. Addition of Angular Momenta

When dealing with composite quantum systems, we must understand how to combine angular momenta. This is essential for understanding atomic structure, nuclear physics, and particle interactions.

The Problem

Consider two angular momentum operators $\vec{J}_1$ and $\vec{J}_2$ acting on different spaces:

\begin{align*} [\hat{J}_{1i}, \hat{J}_{1j}] &= i\hbar\epsilon_{ijk}\hat{J}_{1k} \\ [\hat{J}_{2i}, \hat{J}_{2j}] &= i\hbar\epsilon_{ijk}\hat{J}_{2k} \\ [\hat{J}_{1i}, \hat{J}_{2j}] &= 0 \end{align*}

Define total angular momentum:

$$\vec{J} = \vec{J}_1 + \vec{J}_2$$

Question: What are the eigenvalues and eigenstates of $\hat{J}^2$ and $\hat{J}_z$?

Uncoupled vs. Coupled Basis

Uncoupled basis: Eigenstates of $\hat{J}_1^2, \hat{J}_{1z}, \hat{J}_2^2, \hat{J}_{2z}$

$$|j_1, m_1; j_2, m_2\rangle = |j_1, m_1\rangle \otimes |j_2, m_2\rangle$$

Dimension: $(2j_1 + 1)(2j_2 + 1)$
Good quantum numbers: $j_1, m_1, j_2, m_2$
Natural for non-interacting particles

Coupled basis: Eigenstates of $\hat{J}_1^2, \hat{J}_2^2, \hat{J}^2, \hat{J}_z$

$$|j_1, j_2; j, m\rangle$$

Same dimension: $(2j_1 + 1)(2j_2 + 1)$
Good quantum numbers: $j_1, j_2, j, m$
Natural when interaction couples the two systems

Triangle Inequality

The allowed values of total angular momentum $j$ satisfy:

$$|j_1 - j_2| \leq j \leq j_1 + j_2$$

With $j$ taking integer steps

Geometric interpretation: Classical angular momentum vectors form a triangle

$$j = |j_1 - j_2|, |j_1 - j_2| + 1, \ldots, j_1 + j_2 - 1, j_1 + j_2$$

Total of $2\min(j_1, j_2) + 1$ possible values

For each $j$:

$$m = -j, -j+1, \ldots, j-1, j$$

Total of $2j + 1$ states

Verify dimension:

$$\sum_{j=|j_1-j_2|}^{j_1+j_2} (2j+1) = (2j_1+1)(2j_2+1) \quad \checkmark$$

Example: Two Spin-1/2 Particles

For $j_1 = j_2 = 1/2$:

$$j = |1/2 - 1/2|, \ldots, 1/2 + 1/2 \quad \Rightarrow \quad j \in \{0, 1\}$$

Uncoupled basis (4 states):

$$|\uparrow\uparrow\rangle, \quad |\uparrow\downarrow\rangle, \quad |\downarrow\uparrow\rangle, \quad |\downarrow\downarrow\rangle$$

Coupled basis:

Triplet states ($j=1$, 3 states):

\begin{align*} |1, 1\rangle &= |\uparrow\uparrow\rangle \\ |1, 0\rangle &= \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle + |\downarrow\uparrow\rangle) \\ |1, -1\rangle &= |\downarrow\downarrow\rangle \end{align*}

Singlet state ($j=0$, 1 state):

$$|0, 0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle)$$

Symmetry:

Triplet: symmetric under particle exchange
Singlet: antisymmetric under particle exchange

Example: Spin-Orbit Coupling

Electron with orbital angular momentum $\vec{L}$ and spin $\vec{S}$:

$$\vec{J} = \vec{L} + \vec{S}$$

For orbital quantum number $l$ and spin $s = 1/2$:

$$j = l \pm \frac{1}{2}$$

Example: p-orbital ($l = 1$):

$j = 3/2$: 4 states (notation: $p_{3/2}$)
$j = 1/2$: 2 states (notation: $p_{1/2}$)
Total: 6 states = $(2l+1)(2s+1) = 3 \times 2$

Spin-orbit Hamiltonian:

$$\hat{H}_{SO} = \xi(r)\vec{L} \cdot \vec{S} = \frac{\xi(r)}{2}[\hat{J}^2 - \hat{L}^2 - \hat{S}^2]$$

Splits energy levels by $j$ (fine structure)

Change of Basis Transformation

Expansion of coupled states in uncoupled basis:

$$|j_1, j_2; j, m\rangle = \sum_{m_1, m_2} \langle j_1, m_1; j_2, m_2 | j_1, j_2; j, m\rangle |j_1, m_1; j_2, m_2\rangle$$

The coefficients $\langle j_1, m_1; j_2, m_2 | j_1, j_2; j, m\rangle$ are Clebsch-Gordan coefficients

Constraint from $\hat{J}_z = \hat{J}_{1z} + \hat{J}_{2z}$:

$$m = m_1 + m_2$$

Sum simplifies to single sum over $m_1$ (or $m_2$)

Recursive Construction

Start with maximum state (unique, no superposition needed):

$$|j_1, j_2; j_{\max}, m_{\max}\rangle = |j_1, j_1; j_2, j_2\rangle$$

Where $j_{\max} = j_1 + j_2$, $m_{\max} = j_1 + j_2$

Apply lowering operator $\hat{J}_- = \hat{J}_{1-} + \hat{J}_{2-}$:

$$\hat{J}_- |j, m\rangle = \hbar\sqrt{j(j+1) - m(m-1)} |j, m-1\rangle$$

Generate all states with $j = j_1 + j_2$ by repeated lowering

For next $j = j_1 + j_2 - 1$, find state orthogonal to already constructed states with same $m$

Continue until all $(2j_1+1)(2j_2+1)$ states constructed

Example: $j_1 = 1, j_2 = 1/2$

Allowed $j$ values: $j \in \{1/2, 3/2\}$

Maximum state ($j = 3/2, m = 3/2$):

$$|3/2, 3/2\rangle = |1, 1; 1/2, 1/2\rangle$$

Apply $\hat{J}_-$ to get $m = 1/2$ state:

$$|3/2, 1/2\rangle = \sqrt{\frac{2}{3}}|1, 1; 1/2, -1/2\rangle + \sqrt{\frac{1}{3}}|1, 0; 1/2, 1/2\rangle$$

Orthogonal state with $m = 1/2$ has $j = 1/2$:

$$|1/2, 1/2\rangle = \sqrt{\frac{1}{3}}|1, 1; 1/2, -1/2\rangle - \sqrt{\frac{2}{3}}|1, 0; 1/2, 1/2\rangle$$

Continue for all 6 states (verify dimension: $3 \times 2 = 6$)

Physical Applications

1. Atomic spectroscopy:

LS coupling: $\vec{L} = \sum_i \vec{l}_i$, $\vec{S} = \sum_i \vec{s}_i$, then $\vec{J} = \vec{L} + \vec{S}$
jj coupling: $\vec{j}_i = \vec{l}_i + \vec{s}_i$ for each electron, then $\vec{J} = \sum_i \vec{j}_i$
Term symbols: $^{2S+1}L_J$ (e.g., $^2P_{3/2}$)

2. Nuclear physics:

Nuclear spin from coupling of proton and neutron spins
Deuteron: $j_p = 1/2, j_n = 1/2 \Rightarrow j_d \in \{0, 1\}$ (observed: $j_d = 1$)

3. Particle physics:

Quark model: mesons from quark-antiquark, baryons from three quarks
Isospin addition in hadronic interactions

4. Quantum information:

Entanglement of multiple qubits
Singlet states for quantum teleportation

Selection Rules

Matrix elements vanish unless certain conditions met:

For vector operator $\vec{V}$ (e.g., dipole moment):

$$\langle j', m' | \hat{V}_q | j, m \rangle \neq 0 \quad \text{only if}$$

$|j - j'| \leq 1 \leq j + j'$ (triangle inequality)
$m' = m + q$ where $q \in \{-1, 0, 1\}$

Electric dipole transitions:

$$\Delta j = 0, \pm 1 \quad (\text{but not } 0 \to 0), \quad \Delta m = 0, \pm 1$$

These rules determine allowed atomic transitions and spectral lines

Physical Insight: Addition of angular momenta reveals the quantum mechanical nature of composite systems. The triangle inequality and quantized values of total angular momentum reflect the underlying group structure (SU(2)) and have no classical analog. Understanding this is essential for atomic physics, nuclear structure, and particle interactions.

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