When a paramagnetic material is placed in a strong external magnetic field, then you'll find that the magnetic momenta of the electrons in the material will align to the field, causing a net magnetic moment and magnetic field in the material. But why, instead of just aligning with the field, won't the electrons precess around the external field (i.e. Larmor precession)? The electrons have angular momentum, meaning that precession should take place. To me, it doesn't seem like QM will help us here, as even according to it, the spin (or at least the expectation value of the spin operators) will precess, just like in ordinary CM. So what is the solution here?
They do, for a short time. However, a spin which is precessing has higher magnetic potential energy than a spin which is completely aligned with the field, so the precessing state is unstable.
The mechanism by which spins "decay" from a precessing state to a state in which they are aligned with the external field is provided by the spin-phonon interaction. In essence, the precessing spin decays to an aligned spin and dumps the excess energy into lattice vibrations. As a result, the material gets slightly hotter when it is magnetized, a phenomenon which is called the magnetocaloric effect.