Circular orbit → elliptical orbit
By improvement of the spectroscopy, the fine structures, which are not possible to explain using Bohr model are discovered.
The angular momentum quantization of the Bohr’s model is given as,
$$ L = rp = \hbar n : \quad 2\pi L = 2\pi p r = n h $$
Bohr’s arguement is generalized by Sommerfeld.
$$ \oint_{\rm period} p_r dr = n_r h \\ \oint_{\rm period} p_\phi d\phi = n_\phi h
$$
where $p_r$ and $p_\phi$ is canonical moemum of $r$ and $\phi$ each. The $n_r$ is radial quantum number and $n_\phi$ is azimuthal quantum number.
$$ p_r = m\dot{r} \\ p_\phi = m r^2 \dot{\phi} $$
In polar coordinate, the radial vector is $\vec{r} = r \hat{r}$ and the radial velocity vector is
$$ \vec{v} = \dot{r}\hat{r} + r\dot{\phi} \hat{\phi} $$
Without radial velocity case, $\dot{r} =0$, there are no $p_r =0$, so that $n_r = 0$ .
Under the constant angular momentum $L$ assumption, where $L = |\vec{r}\times \vec{p}|=|\vec{r}\times m\vec{v}| =mr^2\dot{\phi} = p_\phi$ ,
$$ \oint_{\rm period} p_\phi d\phi = \oint_{\rm period} (mr^2\dot{\phi}) d\phi=L\oint_{\rm period} d\phi= n_\phi h \\
L = \hbar n_\phi
$$
With finite radial velocity, $p_r$ can be written as following form,
$$ p_r = \sqrt{2mE+\frac{me^2}{2\pi\epsilon_0 r}-\frac{L^2}{r^2}} $$