The Exercise

Two particles of spin $1/2$, with gyromagnetic ratios $\gamma_1$ and $\gamma_2$, sit a distance $r$ apart, and the unit vector $\hat n$ points along the line from one to the other. Their magnetic dipoles interact through the dipolar coupling $H^D_{1,2}$, an operator on the pair of spins that depends on $\hat n$. Exercise 7.35 asks to show that this coupling vanishes on average when the direction $\hat n$ ranges over the whole sphere, every direction counted with equal weight.

The library states the exercise for the coupling written out in Pauli matrices, with the average over directions as an integral over the polar and azimuthal angles of $\hat n$.

Two Spins

The Pauli matrices $X$, $Y$ and $Z$ are the complex $2\times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(1) also written $\sigma_1 = X$, $\sigma_2 = Y$, $\sigma_3 = Z$, the three components of the Pauli vector $\vec\sigma$. For a real vector $\vec n = (n_1, n_2, n_3)$ in three dimensions, $\vec n \cdot \vec\sigma$ is the complex $2\times 2$ matrix $$\vec n \cdot \vec\sigma = n_1 X + n_2 Y + n_3 Z.$$(2)^✦

The two spins are two qubits, and an operator on the pair is a complex $4\times 4$ matrix acting on $\mathbb{C}^2 \otimes \mathbb{C}^2$. In a tensor product $A \otimes B$ of $2\times 2$ matrices, $A$ acts on spin 1 and $B$ on spin 2; $\vec\sigma_1$ and $\vec\sigma_2$ denote the Pauli vector acting on spin 1 and on spin 2.

The isotropic coupling $\vec\sigma_1 \cdot \vec\sigma_2$ of the two spins is the operator $$\vec\sigma_1 \cdot \vec\sigma_2 = \sum_{a=1}^{3} \sigma_a \otimes \sigma_a = X \otimes X + Y \otimes Y + Z \otimes Z$$(3)^✦ on the pair. For a real vector $\vec n$, the anisotropic coupling $(\vec\sigma_1 \cdot \vec n)(\vec\sigma_2 \cdot \vec n)$ along $\vec n$ is the tensor product of $\vec n \cdot \vec\sigma$ with itself: $$(\vec\sigma_1 \cdot \vec n)(\vec\sigma_2 \cdot \vec n) = (\vec n \cdot \vec\sigma) \otimes (\vec n \cdot \vec\sigma).$$(4)^✦

The dipolar bracket $D(\vec n)$ along a real vector $\vec n$ is the operator on the pair $$D(\vec n) = \vec\sigma_1 \cdot \vec\sigma_2 - 3\,(\vec\sigma_1 \cdot \vec n)(\vec\sigma_2 \cdot \vec n).$$(5)^✦ For real numbers $\gamma_1$, $\gamma_2$, $\hbar$ and $r$, the dipolar coupling $H^D_{1,2}(\vec n)$ along $\vec n$ is the real multiple of the bracket $$H^D_{1,2}(\vec n) = \frac{\gamma_1 \gamma_2 \hbar}{4 r^3}\, D(\vec n).$$(6)^✦ Evaluated at the unit vector $\hat n$ joining the spins, it is the coupling of the exercise.

Averaging Over Directions

The unit vector $\hat n(\theta,\phi)$ of polar angle $\theta$ and azimuth $\phi$ is the point of the unit sphere with Cartesian coordinates $$\hat n(\theta,\phi) = (\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta).$$(7)^✦

The spherical average $\langle f \rangle$ of a complex function $f$ of a real vector in three dimensions is the mean of $f$ over the unit sphere, $$\langle f \rangle = \frac{1}{4\pi} \int_0^{\pi} \left( \int_0^{2\pi} f\bigl(\hat n(\theta,\phi)\bigr)\, d\phi \right) \sin\theta \, d\theta .$$(8)^✦ The spherical average $\langle F \rangle$ of a function $F$ that assigns a complex matrix to every real vector in three dimensions is the matrix of the same shape whose entry in row $j$ and column $k$ is the spherical average of the corresponding entry of $F$: $$\langle F \rangle_{jk} = \bigl\langle \vec n \mapsto F(\vec n)_{jk} \bigr\rangle .$$(9)

The Claim

The dipolar coupling is such a function, $\vec n \mapsto H^D_{1,2}(\vec n)$, with values in the $4\times 4$ matrices on the pair, and its spherical average is the average over the directions $\hat n$ that the exercise asks for.

What Is Checked

The theorem is stated over the dipolar coupling (6), built from the Pauli matrices (1) through the couplings (3) and (4), and over the spherical average of a matrix (9), with the integral (8) over the angles of (7).

The library proves of these constructions that each of the sixteen entries of $H^D_{1,2}\bigl(\hat n(\theta,\phi)\bigr)$, a function of the two angles, integrates to zero over the sphere.