The Exercise

Two particles of spin $\tfrac12$, here two qubits, have a total angular momentum, the sum of their two spins. Its components $j_x$, $j_y$, $j_z$ and its square $J^2$ act on the state space of the pair, which has dimension four. The exercise is to write $J^2$ and $j_z$ as matrices in the computational basis, and to show that the three triplet states and the singlet are eigenvectors of both: of $J^2$ with eigenvalue $j(j+1)$, where the total spin is $j = 1$ on the triplet and $j = 0$ on the singlet, and of $j_z$ with eigenvalue $m$, the $z$ component of the total spin.

Two Spins

An operator on two qubits is written as a $4 \times 4$ complex matrix in the computational basis, with rows and columns in the order $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$: the basis state $|ab\rangle$ stands in position $2a + b$, counting from $0$.^✦

With $\hbar = 1$, the spin of a single qubit is half its Pauli matrices $X$, $Y$, $Z$. The components $j_x$, $j_y$, $j_z$ of the total angular momentum of two qubits are the operators $$j_x = \tfrac{1}{2} \left( X \otimes I + I \otimes X \right),$$(1) $$j_y = \tfrac{1}{2} \left( Y \otimes I + I \otimes Y \right),$$(2) $$j_z = \tfrac{1}{2} \left( Z \otimes I + I \otimes Z \right),$$(3) on the pair, where $I$ is the $2 \times 2$ identity. The square $J^2$ of the total angular momentum is the operator $$J^2 = j_x^2 + j_y^2 + j_z^2 .$$(4)^✦

Four states of the pair, the coupled states, carry the labels $|j, m\rangle_J$ of a total spin $j$ and its $z$ component $m$; the subscript $J$ marks them as eigenstates of the total angular momentum, to tell them from the product states $|ab\rangle$. The triplet states, with $j = 1$, are $$|1, 1\rangle_J = |00\rangle,$$(5) $$|1, 0\rangle_J = \frac{1}{\sqrt{2}} \left( |01\rangle + |10\rangle \right),$$(6) $$|1, -1\rangle_J = |11\rangle,$$(7) and the singlet, with $j = 0$, is $$|0, 0\rangle_J = \frac{1}{\sqrt{2}} \left( |01\rangle - |10\rangle \right).$$(8)

The Claim

What Is Checked

The claim is stated over the operators $j_z$ and $J^2$ of (3) and (4) and over the coupled states (5) to (8). For these operators and states the library shows that $J^2$ and $j_z$ have the matrices (9) and (10) in the computational basis, and that each of the triplet states and the singlet is an eigenvector of both, with the eigenvalues (11) to (18).