Question

Explain the significance of quantum numbers?

Answer 1

The set of four numbers that is used to describe the energy and position of electrons in an atom are the quantum numbers. They are also used to understand characteristics of atoms like ionization and atomic radii.

Complete step by step answer:
Within the context of the $\to$ atomic model by Niels Bohr (1885 -1962), observable spectrum lines of frequency v are described as$\to$ quantum jumps of bond$\to$ electrons between quantized energy levels ${E_n}$ according to the rule: ${v_{nm}} = \dfrac{{\left( {{E_n} - {E_m}} \right)}}{h}$ , with h = a quantum of action, $\to$ Planck's constant. With the small correction for the so-called reduced mass, the energy ${E_n}$ of each electron orbit around the atomic core is given as:
${E_n} = \dfrac{{2{\pi ^2}{e^4}mM}}{{{h^2}(m + M)}}.\dfrac{1}{{{n^2}}}$
$m =$ the mass of the electrons; $M =$ the mass of the atomic core; $n$ is the first (or "main" quantum number) mainly determining the energy level of each electron ,aside from small corrections mostly relevant to precision$\to$ spectroscopy and described by other subsequently introduced quantum numbers.
In order to describe this Sommerfeld introduced another azimuthal quantum number I (sometimes also called k or${n_\varphi }$), describing the degree of eccentricity of the electron orbit, with the largest and b the smallest diameter (see, for instance, Sommerfeld 1919).
Classically, all eccentricities $\varepsilon = \dfrac{b}{a}$ are permissible, but within the early$\to$ quantum theory another quantization condition is imposed and only certain orbits are allowed.
When an external field is imposed on the atom, these ellipses can orient themselves in various ways with respect to the field (for instance a magnetic field causing the $\to$ Zeeman effect or an electric field leading to the $\to$ Stark effect). Again, classically, all angles between orbit and external field would be permissible, but in quantum theory only certain angular orientations $\alpha$ are allowed ($\to$ space quantization, see also $\to$Stern-Gerlach experiment and $\to$vector model). Systematic analysis of data from the $\to$spectroscopy of Zeeman multiples showed that all permissible orientations could be labelled with one additional magnetic quantum number $m$ , with $|m| \leqslant l$ , thus $m = - l, - l\, + 1, - l\, + 2....,0,\,1\,,\,2.....l\, - 2,l\, - 1,l;$
and for the angle $\alpha :\,\cos \,\alpha \, = m/1$ and $|m|\, \leqslant \,|l|\, \leqslant \,|n|.$
as is explained in more detail in the article on $\to$spin, in January 1925 Pauli first expressed this mechanically indescribable ambiguity as a new quantum number $\mu$ , later redubbed x = $\pm \dfrac{1}{2}$ (for doublets). Each electron was described by a set of four$\to$ quantum numbers.
With this set of four different quantum numbers$n,\,I,\,m$ and $\mu$ (sometimes alternatively $n,\,l,\,j\,$and$s$), It was possible to classify all electrons in bound states around an atom's positively charged core. In order to achieve a perfect fit with the number of atoms in each row of the periodic table, Pauli had to introduce another constraint on the shell structure; no two electrons of an atom may have all the four quantum numbers in common, the Paul principle (exclusion principle).

Note: There can never be two or more equivalent electrons in the atom in which the values of all (Four) quantum numbers….. concur within a strong field…. If in the atom there is an electron for which these quantum numbers………. Have specific values, then this state is occupied.