Adding Spin ${1\over 2}$ to Integer Orbital Angular Momentum

Our goal is to add orbital angular momentum with quantum number $\ell$ to spin ${1\over 2}$. We can show in several ways that, for $\ell\neq 0$, that the total angular momentum quantum number has two possible values $j=\ell+{1\over 2}$ or $j=\ell-{1\over 2}$.For $\ell=0$, only $j={1\over 2}$ is allowed. First lets argue that this makes sense when we are adding two vectors. For example if we add a vector of length 3 to a vector of length 0.5, the resulting vector could take on a length between 2.5 and 3.5 For quantized angular momentum, we will only have the half integers allowed, rather than a continuous range. Also we know that the quantum numbers like $\ell$ are not exactly the length of the vector but are close. So these two values make sense physically.

We can also count states for each eigenvalue of $J_z$ as in the following examples.

* Example: Counting states for $\ell =3$ plus spin ${1\over 2}$.*
* Example: Counting states for any $\ell$ plus spin ${1\over 2}$.*
As in the last section, we could start with the highest $J_z$ state, $Y_{\ell\ell}\chi_+$, and apply the lowering operator to find the rest of the multiplet with $j=\ell+{1\over 2}$. This works well for some specific $\ell$ but is hard to generalize.

We can work the problem in general. We know that each eigenstate of $J^2$ and $J_z$ will be a linear combination of the two product states with the right $m$.

$$\bgroup\color{black}\psi_{j(m+{1\over 2})}=\alpha Y_{\ell m}\chi_++\beta Y_{\ell (m+1)}\chi_-\egroup$$

audio

The coefficients $\alpha$ and $\beta$ must be determined by operating with $J^2$.

$$\begin{eqnarray*} \psi_{(\ell+{1\over 2})(m+{1\over 2})}=\sqrt{\ell+m+1\over 2\e... ...\ell m}\chi_+ -\sqrt{\ell+m+1\over 2\ell+1}Y_{\ell (m+1)}\chi_- \end{eqnarray*}$$

We have made a choice in how to write these equations: $m$ must be the same throughout. The negative $m$ states are symmetric with the positive ones. These equations will be applied when we calculate the fine structure of Hydrogen and when we study the anomalous Zeeman effect.