# Geometric visualization of addition of angular momenta

Introduction

Consider the Hilbert space of two $$\frac{1}{2}$$ spin particles (electrons, for instance), spanned by

$$\begin{equation} {|\alpha\rangle, \, \alpha = \uparrow,\downarrow} \end{equation}$$

$$\begin{equation} {|\beta\rangle, \, \beta = \uparrow,\downarrow} \end{equation}$$

In order to study the system of two particles, we usually take the tensorial product of both spaces, such that $$\mathcal{E} = \mathcal{E}_1\otimes \mathcal{E}_2$$. If $$\vec{J} = \vec{S_1} + \vec{S_2}$$, $$\mathcal{E}$$ has two invariant subspaces spanned by

$$\begin{equation} \left\{|J = 1, M_J = 1\rangle, |J = 1, M_J = 0\rangle, |J = 1, M_J = -1\rangle \right\} \end{equation}$$

and

$$\begin{equation} \left\{|J = 0, M_J = 0\rangle \right\} \end{equation}$$

which is related to the tensorial product basis:

$$\begin{equation} |1,1\rangle = |\uparrow, \uparrow \rangle \end{equation}$$

$$\begin{equation} |1, 0\rangle = \frac{1}{\sqrt{2}} \left(|\uparrow, \downarrow \rangle + |\downarrow, \uparrow \rangle \right) \end{equation}$$

$$\begin{equation} |1,-1\rangle = |\downarrow , \downarrow \rangle \end{equation}$$

$$\begin{equation} |0, 0\rangle = \frac{1}{\sqrt{2}} \left(|\uparrow, \downarrow \rangle - |\downarrow, \uparrow \rangle \right) \end{equation}$$

My questions

• Is is possible to visualize this relations geometrically? For example, is $$|1,1\rangle = |\uparrow, \uparrow \rangle$$ because we have to aligned spins in the $$z$$ direction?

• If if it is possible, what's the geometric difference between $$|1, 0\rangle$$ and $$|0, 0\rangle$$?