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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 $|0, 0\rangle$?

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