General Addition of Angular Momentum: The Clebsch-Gordan Series

We have already worked several examples of addition of angular momentum. Lets work one more.

* Example: Adding $\ell =4$ to $\ell =2$.*

The result, in agreement with our classical vector model, is multiplets with $j=2,3,4,5,6$.

The vector model qualitatively explains the limits.

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In general, $j$ takes on every value between the maximum an minimum in integer steps.

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

The maximum and minimum lengths of the sum of the vectors makes sense physically. Quantum Mechanics tells up that the result is quantized and that, because of the uncertainty principle, the two vectors can never quite achieve the maximum allowed classically. Just like the z component of one vector can never be as great as the full vector length in QM.

We can check that the number of states agrees with the number of product states.

We have been expanding the states of definite total angular momentum $j$ in terms of the product states for several cases. 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. We have computed some of them here by using the lowering operator and some by making eigenstates of $J^2$.