What is the general prescription for constructing the quantum mechanical momentum operator conjugate to a given coordinate?

Question

Obviously, if $x$ is a Cartesian coordinate, then the corresponding momentum operator is $-i \hbar \partial_x$. But what if $x$ is something more complicated, like some sort of curvilinear coordinate in 3D space?

There is this question elsewhere on this site. Quantum mechanical analogue of conjugate momentum But it's not clear to me that the accepted answer there is really answering the question I am asking. Nothing in the question or answer specifically addresses non-Cartesian coordinates, certainly not in any detail.

Answer 1

If an $n$-dimensional configuration manifold $M$ of some physical system is orientable and endowed with a positive volume form $$\Omega~=~\rho(x)\mathrm{d}x^1\wedge\ldots\wedge\mathrm{d}x^n, \qquad \rho(x)~>~0,$$ we can define a sesquilinear form $$\langle \phi | \psi\rangle~:=~\int_M \! \Omega ~\phi^{\ast} \psi. $$ The Schroedinger representation of the self-adjoint phase space operators is $$ \hat{x}^j~=~x^j,\qquad \hat{p}_k~=~\frac{\hbar}{i\sqrt{\rho(x)}} \frac{\partial}{\partial x^k} \sqrt{\rho(x)}, \qquad [\hat{x}^j,\hat{p}_k]~=~i\hbar\delta^j_k \mathbb{1}. $$ See also this related Phys.SE post.