In the Feynman Lectures the modified Stern-Gerlach thought experiment uses high-gradient magnets. I'm confused as to what a high-gradient magnet would be. From what I understand about the Stern-Gerlach apparatus magnets are oriented in a certain direction (such as the z direction). This means that the magnetic field produced by the two magnets is in the z direction and the gradient of the magnetic field is also in the z direction? If this is the case then how are the magnetic fields inhomogeneous? From this Stack Exchange post it seems like if a magnetic field is only in one direction then it would be homogeneous.
A homogeneous magnetic field is one which is spatially uniform, that is one with zero gradient. You may recall from magnetostatics that the force on a magnetic dipole in a magnetic field is proportional to the dot product of the dipole moment with the gradient of the magnetic field. If the magnets in the Stern-Gerlach experiment generated homogeneous fields the atoms would not be deflected at all. Only in the presence of a magnetic field with a gradient in the z-direction will you see splitting due to the z-component of the angular momentum of the atoms.