Are there any other useful representations of state space apart from position and momentum?

As I understand, $$|\psi\rangle\in\mathcal{H}$$, where $$\mathcal{H}$$ is the Hilbert state space, is a general representation of the wave function of a system. It is a vector that, in itself, is independent of the basis you choose to represent your wave function in. The choice of basis leads to the different representations of state space.

So far, the two representations that have been explained to me in class are:

• the position representation, that you obtain like this: $$\langle \vec{r} |\psi\rangle = \psi(\vec{r})$$.
• the momentum representation, that you obtain like this: $$\langle \vec{p} |\phi\rangle = \phi(\vec{p})$$.

Assuming I got all that right, are there any other representations that are used in quantum mechanics?

• This site is strict about asking one question at a time. Your three questions are actually quite separate from eachother, so you should pick your favorite and delete the other two. Otherwise I'm sure the question will be quickly closed. Also $\langle p|\psi\rangle=\phi(p)$, not $\psi(p)$. Writing $\psi(p)$ would seem to imply that the position-space wavefunction is for some reason exactly the same function as $\psi(x)$. Like for some reason the wavefunction's value at $x=1$ m is the same as the position-wavefunctions value at $p=1$ kg m/s. (of course the units being different makes this silly) Sep 15, 2023 at 17:51
• Thanks for the suggestions @AXensen, I've updated the question. Sep 15, 2023 at 18:17
• @Axensen You need to have some way of denoting that the position-space and momentum-space wave functions are representation of the same state. There are multiple ways to do this. I sometimes use $\langle x |\psi\rangle = \psi(x)$ and $\langle p |\psi\rangle = \tilde{\psi}(p)$. Sometimes I'll use a hat (but that gets mixed up with operator notation), and sometimes I'll use a superscript, like $\psi^{\textrm{mom}}$ or something, but it seems reasonable to want $\psi$ to appear in both wave function names, since they both represent $|\psi\rangle$. Sep 15, 2023 at 19:47
• @AXensen I'll follow from march to say you can even overload functions if you are sure about units. $\psi(x)$ and $\psi(p)$ could be different functions if you ensure that the function of position uses one definition and the function of momentum uses another. Best to avoid confusion though, I agree Sep 15, 2023 at 20:55