Structure of Atom - Quantum Numbers

Quantum numbers are a set of four numbers used to completely describe the position and energy of an electron in an atom.

The Principal Quantum Number ($n$) defines the main energy shell and determines the size and energy of the orbital. It can have positive integer values: $n = 1, 2, 3, \dots$

The Azimuthal Quantum Number ($l$), also known as the orbital angular momentum quantum number, defines the three-dimensional shape of the orbital. For a given $n$, $l$ can range from $0$ to $(n-1)$. Values $0, 1, 2, 3$ correspond to $s, p, d, f$ orbitals respectively.

The Magnetic Quantum Number ($m_l$) describes the spatial orientation of the orbital with respect to a standard set of coordinate axes. For a given $l$, there are $(2l+1)$ values of $m_l$, ranging from $-l$ to $+l$.

The Spin Quantum Number ($m_s$) describes the direction of electron spin. An electron can have two spin states, represented by $+1/2$ (spin up) and $-1/2$ (spin down).

Pauli's Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers.

The $n + l$ rule (Aufbau Principle) determines the order of filling of orbitals; the orbital with the lower $(n + l)$ value is filled first. If $(n + l)$ is the same for two orbitals, the one with the lower $n$ value is filled first.