# what is the magnetic quadrupole operator?

To find magnetic or electrical moments in quantum theory we must calculate the expectation value of an appropriate operator. the dipoles operator are similar and is easy to find but the magnetic quarupole moment, I couldn't find.

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The magnetic quadrupole moment tensor is given by $$m_{ij}=\left\langle \frac{2}{3}\left(\mathbf{r}\times\mathbf{J}\right)_i r_j \right\rangle,$$ in analogy with the magnetic dipole moment vector $$m_i=\left\langle \frac{1}{2}\left(\mathbf{r}\times\mathbf{J}\right)_i \right\rangle.$$ The magnetic field at a point $\mathbf{R}$ is then, up to quadrupole order, $$B_i(\mathbf{R})=\frac{\mu_0}{4\pi}\left[ m_j \frac{R_iR_j-3R^2\delta_{ij}}{R^5} +\frac{3m_{jk}}{2R^7}\left(5R_iR_jR_k -R^2(R_i\delta_{jk}+R_j\delta_{ki}+R_k\delta_{ij}) \right) \right].$$ This link has what looks like a good exposition of the subject (including the formulae above) but for the life of me I can't figure out what book it's a part of. The list of references looks pretty comprehensive too.
The current $\mathbf{J}$ is proportional to the particle's velocity and this may or may not be proportional to the canonical momentum (in which case, for example, $\mathbf{m}\propto\mathbf{L}$), depending on what gauge you're working in. Be careful!