The Exercise

In the hyperfine structure of an atom, the spin of the nucleus couples to the spin of an electron. Take a nucleus of spin $3/2$ and an electron of spin $1/2$, with a joint spin space of dimension $4 \times 2 = 8$. The coupled states are the states of definite total spin $F$ and of definite projection $m$ of the total angular momentum on the $z$ axis.

The exercise starts from the three spin $3/2$ matrices and their commutation relations. It then writes the components of the total angular momentum and its square as $8 \times 8$ matrices, and finds the eight coupled states as combinations of product states, a nuclear spin state times an electron spin state.

Two Spins

The spin operators $i_x$, $i_y$, $i_z$ of the nucleus are the $4 \times 4$ complex matrices below, written in the basis $|m_I\rangle$ of eigenstates of $i_z$ with $m_I = -\tfrac32, -\tfrac12, \tfrac12, \tfrac32$ in that order, and with $i$ the imaginary unit: $$i_x = \frac12 \begin{pmatrix} 0 & \sqrt3 & 0 & 0 \\ \sqrt3 & 0 & 2 & 0 \\ 0 & 2 & 0 & \sqrt3 \\ 0 & 0 & \sqrt3 & 0 \end{pmatrix},$$(1) $$i_y = \frac{i}{2} \begin{pmatrix} 0 & \sqrt3 & 0 & 0 \\ -\sqrt3 & 0 & 2 & 0 \\ 0 & -2 & 0 & \sqrt3 \\ 0 & 0 & -\sqrt3 & 0 \end{pmatrix},$$(2) $$i_z = \frac12 \, \mathrm{diag}(-3, -1, 1, 3).$$(3)

For the electron, the spin operators are $\tfrac12 X$, $\tfrac12 Y$, $\tfrac12 Z$, where $X$, $Y$, $Z$ are the Pauli matrices in the basis $|{\uparrow}\rangle$, $|{\downarrow}\rangle$ of states with $m_S = +\tfrac12$ and $m_S = -\tfrac12$: $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(4)

On the joint spin space $\mathbb{C}^4 \otimes \mathbb{C}^2 = \mathbb{C}^8$ the product basis is ordered by $m_I$ first and by $m_S$ second: $|{-\tfrac32}\rangle|{\uparrow}\rangle$, $|{-\tfrac32}\rangle|{\downarrow}\rangle$, $|{-\tfrac12}\rangle|{\uparrow}\rangle$, and so on up to $|{\tfrac32}\rangle|{\downarrow}\rangle$.

Write $I$ for the identity of the factor on which it stands. The components $f_x$, $f_y$, $f_z$ of the total angular momentum are the operators on the joint space $$f_x = i_x \otimes I + I \otimes \tfrac12 X, \qquad f_y = i_y \otimes I + I \otimes \tfrac12 Y, \qquad f_z = i_z \otimes I + I \otimes \tfrac12 Z.$$(5) The square $F^2$ of the total angular momentum is the operator $$F^2 = f_x^2 + f_y^2 + f_z^2.$$(6)^✦

The Coupled States

The coupled states $|F, m\rangle$ are eight vectors of the joint space, five with $F = 2$ and $m = -2, \dots, 2$, and three with $F = 1$ and $m = -1, 0, 1$: $$|2,-2\rangle = |{-\tfrac32}\rangle|{\downarrow}\rangle,$$(7) $$|2,-1\rangle = \tfrac12 \, |{-\tfrac32}\rangle|{\uparrow}\rangle + \tfrac{\sqrt3}{2} \, |{-\tfrac12}\rangle|{\downarrow}\rangle,$$(8) $$|2,0\rangle = \tfrac{1}{\sqrt2} \bigl( |{-\tfrac12}\rangle|{\uparrow}\rangle + |{\tfrac12}\rangle|{\downarrow}\rangle \bigr),$$(9) $$|2,+1\rangle = \tfrac{\sqrt3}{2} \, |{\tfrac12}\rangle|{\uparrow}\rangle + \tfrac12 \, |{\tfrac32}\rangle|{\downarrow}\rangle,$$(10) $$|2,+2\rangle = |{\tfrac32}\rangle|{\uparrow}\rangle,$$(11) $$|1,-1\rangle = \tfrac{\sqrt3}{2} \, |{-\tfrac32}\rangle|{\uparrow}\rangle - \tfrac12 \, |{-\tfrac12}\rangle|{\downarrow}\rangle,$$(12) $$|1,0\rangle = \tfrac{1}{\sqrt2} \bigl( |{-\tfrac12}\rangle|{\uparrow}\rangle - |{\tfrac12}\rangle|{\downarrow}\rangle \bigr),$$(13) $$|1,+1\rangle = -\tfrac12 \, |{\tfrac12}\rangle|{\uparrow}\rangle + \tfrac{\sqrt3}{2} \, |{\tfrac32}\rangle|{\downarrow}\rangle.$$(14)

The matrix $U$ of the coupled basis is the $8 \times 8$ matrix whose columns are the coupled states, in the order $|2,-2\rangle$, $|2,-1\rangle$, $|2,0\rangle$, $|2,+1\rangle$, $|2,+2\rangle$, $|1,-1\rangle$, $|1,0\rangle$, $|1,+1\rangle$: $$U = \begin{pmatrix} 0 & \tfrac12 & 0 & 0 & 0 & \tfrac{\sqrt3}{2} & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \tfrac{1}{\sqrt2} & 0 & 0 & 0 & \tfrac{1}{\sqrt2} & 0 \\ 0 & \tfrac{\sqrt3}{2} & 0 & 0 & 0 & -\tfrac12 & 0 & 0 \\ 0 & 0 & 0 & \tfrac{\sqrt3}{2} & 0 & 0 & 0 & -\tfrac12 \\ 0 & 0 & \tfrac{1}{\sqrt2} & 0 & 0 & 0 & -\tfrac{1}{\sqrt2} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \tfrac12 & 0 & 0 & 0 & \tfrac{\sqrt3}{2} \end{pmatrix}.$$(15)^✦

The Claim

What Is Checked

The theorem is stated over the spin operators (1) to (3) of the nucleus and the Pauli matrices (4), the total angular momentum (5) and its square (6) built from them on the ordered product basis, and the coupled states with their matrix $U$ (15).

About these constructions the library proves every identity of the theorem.