Addition of Angular Momentum

It is often required to add angular momentum from two (or more) sources together to get states of definite total angular momentum. For example, in the absence of external fields, the energy eigenstates of Hydrogen (including all the fine structure effects) are also eigenstates of total angular momentum. This almost has to be true if there is spherical symmetry to the problem.

As an example, lets assume we are adding the orbital angular momentum from two electrons, $\vec{L_1}$ and $\vec{L_2}$ to get a total angular momentum $\vec{J}$. We will show that the total angular momentum quantum number takes on every value in the range

$$\bgroup\color{black}\vert\ell_1-\ell_2\vert\leq j\leq\ell_1+\ell_2.\egroup$$

We can understand this qualitatively in the vector model pictured below. We are adding two quantum vectors.

The length of the resulting vector is somewhere between the difference of their magnitudes and the sum of their magnitudes, since we don't know which direction the vectors are pointing.

The states of definite total angular momentum with quantum numbers $j$ and $m$, can be written in terms of products of the individual states (like electron 1 is in this state AND electron 2 is in that state). The general expansion is called the Clebsch-Gordan series:

$$\bgroup\color{black}\psi_{jm}=\sum\limits_{m_1m_2}\langle \el... ...2\vert jm\ell_1\ell_2\rangle Y_{\ell_1m_1}Y_{\ell_2m_2}\egroup$$

or in terms of the ket vectors

$$\bgroup\color{black}\vert jm\ell_1\ell_2\rangle=\sum\limits_{... ...rt jm\ell_1\ell_2\rangle \vert\ell_1m_1\ell_2m_2\rangle\egroup$$

The Clebsch-Gordan coefficients are tabulated although we will compute many of them ourselves.

When combining states of identical particles, the highest total angular momentum state, $s=s_1+s_2$, will always be symmetric under interchange.The symmetry under interchange will alternate as $j$ is reduced.

The total number of states is always preserved. For example if I add two $\ell=2$ states together, I get total angular momentum states with $j=0,1,2,3$ and 4. There are 25 product states since each $\ell=2$ state has 5 different possible $m$s. Check that against the sum of the number of states we have just listed.

$$\bgroup\color{black}5\otimes 5=9_S \oplus 7_A \oplus 5_S \oplus 3_A \oplus 1_S\egroup$$

where the numbers are the number of states in the multiplet.

We will use addition of angular momentum to:

• Add the orbital angular momentum to the spin angular momentum for an electron in an atom $\vec{J}=\vec{L}+\vec{S}$;
• Add the orbital angular momenta together for two electrons in an atom $\vec{L}=\vec{L_1}+\vec{L_2}$;
• Add the spins of two particles together $\vec{S}=\vec{S_1}+\vec{S_2}$;
• Add the nuclear spin to the total atomic angular momentum $\vec{F}=\vec{J}+\vec{I}$;
• Add the total angular momenta of two electrons together $\vec{J}=\vec{J_1}+\vec{J_2}$;
• Add the total orbital angular momentum to the total spin angular momentum for a collection of electrons in an atom $\vec{J}=\vec{L}+\vec{S}$;
• Write the product of spherical harmonics in terms of a sum of spherical harmonics.