In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrödinger equation is $$H\Psi(x,t) = i\hbar\frac{\partial}{\partial t}\Psi(x,t).$$ In a representation-free notations we deal instead with the ket $|\Psi\rangle =...

Not all Hilbert space vectors are generally considered physical, due to various reasons. A particular example (as found in Hall's "Quantum mechanics for mathematicians") is the classic particle in a box example : only states obeying the boundary conditions $\psi(0) = \psi(L) = 0$ are considered. ...