Quantum Chemistry || Lec # 13 || Quantum Numbers | Types & Examples | Dr. Rizwana

Quantum numbers are the four key values needed to fully describe an electron’s properties in an atom—its main energy location, the sub-shell it occupies, the orientation of its orbital in space, and the electron’s spin. The lecture frames these values as an “address” system: the principal quantum number (n) identifies the shell (the electron’s main region), while the azimuthal (orbital angular momentum) quantum number (l) narrows it to a specific sub-shell and orbital shape. The magnetic quantum number (m) then determines how that orbital is oriented under an applied magnetic field, and the spin quantum number (s) distinguishes the two possible spin states of electrons within the same orbital.

The principal quantum number n is introduced as the most fundamental identifier. For a given shell k (using the lecture’s notation), the maximum number of electrons that can fit is 2n². The examples follow directly: when n=1, a shell can hold 2 electrons; for n=2, it can hold 8; for n=3, it can hold 18; and for n=4, it can hold 32. Beyond capacity, n also links to physical size and energy: larger n corresponds to a larger “electronic cloud” (the spatial extent of the electron distribution) and higher energy, because electrons farther from the nucleus occupy higher-energy shells.

The second quantum number, l, is tied to sub-shells and orbital shapes. For each n, l can take integer values from 0 up to n−1. The lecture maps these values to familiar sub-shell labels: l=0 corresponds to an s orbital, l=1 to a p orbital, l=2 to a d orbital, and l=3 to an f orbital. It also connects l to relative sub-shell energies within the same principal shell ordering: s has the lowest energy, followed by p, then d, then f as the lecture’s hierarchy.

The magnetic quantum number m is presented as the value that determines orbital orientation. Once l is known, m can range from −l to +l in integer steps, giving the number of degenerate orbitals: s has 1, p has 3, d has 5, and f has 7. The lecture emphasizes that these orbitals share the same energy (degeneracy) but differ in spatial orientation, and m becomes meaningful when a magnetic field is applied.

Finally, the spin quantum number s accounts for the two distinct electron spin states within a single orbital. Even when electrons share the same shell, sub-shell, and orbital energy (same n, l, and m), they can still differ by spin. The lecture describes one electron as clockwise and the other as anticlockwise in the same orbital, corresponding to s=+1/2 and s=−1/2.

To consolidate the framework, the lecture works through examples like 2p3 and 3d, extracting n, l, m, and s from the orbital label and the electron’s position in the filling order. The takeaway is that each electron’s “full address” is uniquely pinned down by these four quantum numbers, allowing consistent prediction of shell capacity, sub-shell type, orbital orientation, and spin state.