PeroK said:
It's just a different set of basis bras and kets based on a different coordinate system. It's the same idea.
This is correct. However, the whole Dirac operator formalism has its nuances in curvilinear coordinates, specially for angular coordinates.
For example, it is problematic to define a ##\widehat{\theta}## and ##\widehat{\phi}## operators. The easiest thing to do is to define sine and cosine operators
$$\widehat{x} =\widehat{r}\sin\widehat{\theta}\cos\widehat{\phi},$$
$$\widehat{y} =\widehat{r}\sin\widehat{\theta}\sin\widehat{\phi},$$
$$\widehat{z} =\widehat{r}\cos\widehat{\theta}.$$
$$\widehat{r} =\sqrt{\widehat{x}^{2}+\widehat{y}^{2}+\widehat{z}^{2}}$$
$$\cos\widehat{\theta} =\frac{\widehat{z}}{\widehat{r}},\qquad\quad\quad\quad\;\:\sin\widehat{\theta}=\frac{\sqrt{\widehat{x}^{2}+\widehat{y}^{2}}}{\widehat{r}},
\cos\widehat{\phi} =\frac{\widehat{x}}{\sqrt{\widehat{x}^{2}+\widehat{y}^{2}}},\qquad\sin\widehat{\phi}=\frac{\widehat{y}}{\sqrt{\widehat{x}^{2}+\widehat{y}^{2}}}.$$
With Eigenvalues equation of the form
$$\widehat{r}\left|r,\theta,\phi\right\rangle =r\left|r,\theta,\phi\right\rangle, $$
$$\cos\widehat{\theta}\left|r,\theta,\phi\right\rangle =\cos\theta\left|r,\theta,\phi\right\rangle, $$
$$\cos\widehat{\phi}\left|r,\theta,\phi\right\rangle =\cos\phi\left|r,\theta,\phi\right\rangle. $$
However, this is just 1/3 of the issue. There are two things remaining (1) we have to define the corresponding momentum operators, (2) the bracket ##\left\langle r',\theta',\phi'\right|\left|r,\theta,\phi\right\rangle ## is given in terms of a
generalized delta function.
In a paper I'm writing I call the above procedure a "quantum point transformation", as given by DeWitt
https://journals.aps.org/pr/abstract/10.1103/PhysRev.85.653
In any case, the situation is not free of difficulties. For example, the angles are undefined for r=0.