I'm reading from "Quantum Physics for Dummies", by Steven Holzner. In chapter two, entitled "Entering the Matrix: Welcome to State Vectors", the author introduces the notation for a gradient of a wave function.
I understand all of the complement, but what confuses me is the use of the relationship operators, what do they signify in this context? How can you compare a partial derivative with a basis vector?
This is typeset terribly. What it is supposed to mean is ket-vectors in Dirac notation, which are usually set with an "rangle" ($\rangle$) and "langle" ($\langle$), not > and <.
Unfortunately the author misuses the bra-ket notation. A state should be written in the form
$$ | \phi \rangle $$
instead of $|\phi >$. This way, the gradient should appear as
$$ \nabla |\psi\rangle = \frac{\partial}{\partial x}|\psi\rangle {\bf i} + \frac{\partial}{\partial y}|\psi\rangle {\bf j} + \frac{\partial}{\partial z}|\psi\rangle {\bf k} $$
