bohr model:
$$E_n=-\frac {\mathcal R}{n^2(1+\frac {m_e}{m_p})}$$
- can we developed bohr model with all Quantum Numbers of the Hydrogen Atom?
$R(r)$ Principal quantum
number
$$n=1,2,3,4,5,…,n$$
P(θ) Orbital quantum
number
$$\ell=n-1$$
F(φ) Magnetic quantum number
$$m_{\ell}=-\ell,\ell+1,0,\ell,\ell-1…$$
Spin quantum
number
$$m_s=+\frac {1}{2},-\frac {1}{2}$$
Best Answer
The Bohr model was extended by Arnold Sommerfeld. Sommerfeld was able to predict all the Hydrogen atom quantum numbers. This theory is referred to today by the name "Old quantum theory". Please see also, Hazhar Ghaderi's Bachelor thesis.
The basic idea of this theory is that closed classical trajectories (not necessarily circular) lead to discrete quantum numbers. The quantization is obtained through modified condition, now known by the name "Bohr-Sommerfeld condition"
The application of this theory becomes difficult for many electron atoms where exact classical solutions are not available.
This theory was abandoned after the discovery of the Schroedinger equation which gave precise predictions for many electron atoms spectra.
Now, we know that the old quantum theory is only an approximation to the correct quantum theory which is valid in general for large quantum numbers (First order in the WKB quantization).
However, for integrable systems, the Bohr-Sommerfeld is known to be exact or almost exact.
However, this is not the end of the story. In 1980 Sniatycki discovered a deep geometrical meaning of the Bohr-Sommerfeld condition within the theory of geometric quantization. Sniatycki's work is described briefly in Matthias Blau's lecture notes.
Heuristically Sniatycki's result states that there exists a coordinate system in which the wave functions are concentrated on subspaces of the phase space where the quantum numbers are fixed to their quantized values. These spaces are called "Bohr-Sommerfeld varieties".
Nowdays, This subject is under active research and was applied sussessfully to complicated quantization problems such as the quantization of the moduli space of flat connections.