In a comment elsewhere you write that you're interested in understanding how quantum-mechanical theory describes the radiation that a hydrogen atom does and does not emit.
In your question you ask about another answer that suggests some significance to the electron having zero total momentum; I think that's a feature of the coordinate system choice rather than something physically interesting.
Here's a second answer to hopefully address that concern.
In Schrödinger's quantum mechanics the probability density $\psi$ for finding the electron in some small volume near the nucleus (charge $Z$, mass $m_\text{nuc}^{-1} = \mu^{-1} - m_e^{-1}$), obeys the differential equation
$$
\left( \frac{\hbar^2}{2\mu} \vec\nabla^2 - \frac{Z\alpha\hbar c}r
\right) \psi = E\psi.
\tag 1
$$
It turns out that this equation has bound solutions with $E<0$ if, and only if, you introduce some integer parameters $n,\ell,m$ subject to some constraints: $1\leq n$, $\ell<n$, and $|m|\leq \ell$. The energies associated with these quantum numbers are
$$
E_{n\ell m} = -\frac{\mu c^2\alpha^2Z^2}{2n^2} = Z^2 \cdot \frac{-13.6\rm\,eV}{n^2}.
\tag 2
$$
Critically for our discussion, this means that there is a state with $n=1$ that has the minimum possible energy for an electron interacting with a proton.
This is totally different from the unbound case, or the interaction between two like-charged particles, in which you can give your mobile particle any (positive) total energy that you like and inquire about its motion.
If the total energy doesn't satisfy (2), it's simply impossible for the system to obey the equation of motion (1).
You compute transition rates in quantum mechanics using Fermi's Golden Rule: a transition between an initial state $i$ and a final state $f$ occurs in some time interval $\tau_{if} = 1/\lambda_{if}$ with probability $1/e$, where the decay constant $\lambda_{if}$ is
$$
\lambda_{if} = \frac{2\pi}\hbar
\left| M_{if} \right|^2
\rho_f.
$$
The density of final states $\rho_f$ is interesting if there are multiple final states with the same energy. (For instance, in hydrogen there are generally several degenerate final states with given $n,\ell$ but varying $m$.) The matrix element measures the overlap of the initial and final state given some interaction operator $U$:
$$
M_{if} = \int d^3x\ \psi_f^* U \psi_i
$$
For electric dipole radiation the operator is $U_{E1} = e\vec r$; for magnetic dipole radiation, $U_{M1} = {e}\vec L/{2\mu}$; for quadrupole etc. radiation there are other operators.
You could also couple to multiple photons: for instanced the $n=2,\ell=0$ state cannot decay to the ground state by emitting a single photon, since the photon carries angular momentum, but can decay by emitting two dipole photons at the same time. This forbidden transition has lifetime $\sim 0.1\rm\,s$, compared with nanoseconds for the $n=2,\ell=1$ states at the same energy.
Computing matrix elements gives you some hairy integrals, so generally you let someone else do them.
You can in principle use these arguments and the Golden Rule to calculate the radiation emitted in three cases:
From a free electron with $E_i>0$ to a free electron traveling in a different direction with a different energy $E_f>0$. This should give a result most similar to the classical case, where you can get continuous radiation from an accelerating charge.
From a free electron with $E_i>0$ transitioning to a bound electron with $E_f<0$.
From one bound electron state to another.
It's this final option, transitions between bound states, that interests you. The salient feature, unique to quantum mechanics, is that the energies of the bound states are quantized. Unlike in classical mechanics, in quantum theory the equation of motion has no solutions with $E<E_1$. Even if you made up some trial sub-ground-state wavefunction to compute the matrix element for the transition (which can't be done, since the existing wavefunctions form a complete set), you'd find that the density of states at your hypothetical lower energy is $\rho_f=0$, so the time before the transition occurs is, on average, infinitely long.
The classical theory predicts radiation when a charge accelerates from one continuum momentum to another.
So does the quantum theory. But the quantum theory also predicts bound states with quantized energies.
Non-transitions from a state to itself have zero matrix element, therefore never occur; transitions from one state to another can only occur if there's a final state available.
