The examples thus far have been exclusively for 1 dimensional systems with 1 particle. We can construct Hilbert Spaces for 2-D or 3-D systems and we can consider multi-particle systems. For multi-particle systems we have to consider the concept of identical particles. We’ll take a look of that when we cover spin. The extensions to 2-D or 3-D follow similarly from 1-D. Inner products now are double or triple sums or integrals. For basis vectors we typically label use a label for each dimension. For instance, $\ket{n l m}$ describes a basis set for the Hydrogen atom with three discrete labels. In these higher dimensional spaces with have orthonormality relations for each dimension. Let’s see how this extension to 3D works in position space:
$$ \bold{\hat p} = -i\hbar\bold\nabla\\ \hat H=-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi $$
The wave function and potential energy are functions of three position variables (and time). Often this is Cartesian, but other coordinate systems like spherical and cyclindrical coordinates work as well (with the appropriate adjustments to the gradient and laplacian operators).
$$ \text{Probability of finding particle within }\bold r \text{ to }\bold r+d\bold r =|\Psi(\bold r,t)|^2d^3\bold r\\ \text{Normalization: } \int |\Psi|^2 d^3\bold r =1 $$
For a time independent potential energy function, there will exist a complete set of stationary states
$$ \Psi_n(\bold r,t)=\psi_n(\bold r)e^{-iE_n t/\hbar} $$
where the spatial wave function satisfies the TISE
$$ -\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi $$
and the general solution is a superposition of the stationary states (sum for a discrete spectrum and integral for a continuous spectrum).
$$ \Psi(\bold r,t)=\sum_n c_n\psi_n(\bold r) e^{-i E_nt/\hbar} $$
Note that the spatial wave functions may involve a degeneracy of states at the same energy which introduces more sums (or integrals) to this superposition. For instance, the Hydrogen atom has $n$, $l$, and $m$ quantum numbers and the general solution would be a triple sum over the allowed quantum numbers.