Let's take the hydrogen atom in its most simple form, i.e., non-relativistic, no spin. Its Hamiltonian is that of a proton and an electron interacting via the Coulomb potential (in the following I leave out the hats for operators; all observables are understood to be represented by self-adjoint operators, and I use Heaviside-Lorentz units for the Coulomb potential):
$$H=\frac{1}{2m_{\text{p}}} \vec{p}_1^2 + \frac{1}{2m_{\text{e}}} \vec{p}_2^2 -\frac{e^2}{4 \pi |\vec{x}_1-\vec{x}_2|}.$$
We are looking for the energy eigenvalues and eigenfunctions. For that it's always good to use all the symmetries to find a complete set of independent compatible observables, i.e., the symmetries of the problem.
Here we can borrow from the classical analogue of the problem. We have a closed system of two particles interacting via a central interaction potential. Thus the full Galileo group is a symmetry. From this it's clear that it is convenient to use the center-of-mass coordinates ##\vec{R}## and relative coordinates ##\vec{r}##
$$\vec{R}=\frac{1}{M} (m_{\text{p}} \vec{x}_1 + m_{\text{e}} \vec{x}_2), \quad \vec{r}=\vec{r}_2-\vec{r}_1$$
with ##M=m_1+m_2##.
Now we need the canonical momenta to these new position vectors. That's determined by finding ##\vec{P}## and ##\vec{p}## such that
$$[R_i,P_j]=\mathrm{i} \hbar \delta_{ij}, \quad [r_i,p_j]=\mathrm{i} \hbar \delta_{ij}, \quad [r_i,P_j]=[R_i,p_j]=0.$$
It's easy to see that from these commutation relations one has uniquely
$$\vec{P}=\vec{p}_1+\vec{p}_2, \quad \vec{p}=\frac{1}{M}(m_1 \vec{p}_2-m_2 \vec{p}_1).$$
In the new variables the Hamiltonian reads
$$H=\frac{1}{2M} \vec{P}^2+ \frac{1}{2 \mu} \vec{p}^2 -\frac{e^2}{4 \pi |\vec{r}|}.$$
Here ##\mu=m_1 m_2/M## is the reduced mass. Obviously a complete compatible set of observables is
$$\vec{P}, \quad H_{\text{rel}}, \quad \vec{L}_{\text{rel}}^2, \quad L_{\text{rel}3}$$
with
$$H_{\text{rel}}=\frac{1}{2 \mu} \vec{p}^2 -\frac{e^2}{4 \pi |\vec{r}|}, \quad \vec{L}_{\text{rel}}=\vec{r} \times \vec{p}.$$
It's clear that the diagonalization of ##H_{\text{rel}}## works as with the approximation, where the proton is just approximated as "infinitely heavy", i.e., sitting at rest in the origin and just providing an external Coulomb potential for the electron. Here ##H_{\text{rel}}## describes the motion of a quasi-particle of mass ##\mu## in such an external Coulomb potential.
The rest is nicely discussed in the following AJP article by Tommasini et al
https://arxiv.org/abs/quant-ph/9709052
It nicely discusses the energy eigenstates as entangled states between electron and proton, while it's a product state of the free center-mass motion and the relative motion.
