A quantum number is a number used to parametrise certain properties of particles or other systems in quantum mechanics. Combinations of quantum numbers can be used to identify eigenstates of the system.
Each quantum number represents a specific degree of freedom that any particle can occupy. In this viewpoint, they can be seen as analogous to properties in classical systems. For example, the principal quantum number of an electron in an atom is roughly analogous to its orbital distance in classical mechanics. However, the defining characteristic of quantum mechanics is that these are quantised: that is, there is a specific discrete set of values that are allowed for each quantum number. This quantization property is the source of the word "quantum" in "quantum mechanics".
The nature of the sets of allowed values is quite fundamental. In the case of electrons in an atom, the restrictions on the allowed values of l and ml arise from the nature of the boundary condition imposed by the shape of the potential generated by the nucleus, and the restriction on ms is due to the electron itself.
Quantumnumbers are the four numbers used to describe not only the distribution of electrons in atoms and molecular systems but also the allowable values of certain physical quantities of an electron's behavior.
It is an intrinsic quantumnumber that is unrelated to the s-shaped orbital.
However, it should be understood that the elementary particles are quantum states of the standard model of particle physics, and hence the quantumnumbers of these particles bear the same relation to the Hamiltonian of this model as the quantumnumbers of the Bohr atom does to its Hamiltonian.
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