Quantum numbers | Electronic structure of atoms | Chemistry | Khan Academy

The principal quantum number, symbolized by n and defined as a positive integer, sets the electron's energy level or shell in a hydrogen atom. Quantum mechanics replaces the outdated Bohr model by describing the electron as occupying an orbital—a region where it is most likely to be found—rather than following a fixed circular path. As n increases, the electron's average distance from the nucleus expands and its energy rises, emphasizing the intrinsic link between energy levels and spatial distribution.

The angular momentum quantum number, denoted as l, determines the shape of an electron's orbital and ranges from 0 up to n-1 based on the principal energy level n. In the first energy level (n=1), l is exclusively 0, indicating a spherical s orbital with electrons predominantly found within its volume. When n equals 2, l can be 0 or 1, corresponding to a spherical s orbital or a uniquely shaped p orbital with a dumbbell or bow-tie form that represents distinct subshells.

S Orbital Simplicity: A Unified Orientation The magnetic quantum number, mₗ, designates the orientation of an electron orbital around the nucleus. When the angular momentum quantum number l is zero, there is only one allowable mₗ value—zero—reflecting the perfect symmetry of a spherical s orbital. This singular orientation aligns with the concept that a sphere has only one way to be positioned in space.

P Orbitals: Three Unique Spatial Alignments For an orbital where l equals one, the magnetic quantum number mₗ can take the integral values -1, 0, and 1, each describing a distinct spatial direction. This range of values corresponds to the electron cloud’s dumbbell shape found in p orbitals. The different mₗ values establish three unique orientations along the x, y, and z axes, illustrating the directional diversity inherent in atomic orbitals.

The spin quantum number, denoted as mₛ, introduces two distinct electron states rather than a literal rotation. Visual aids often depict electrons as spinning like tops, but this illustration is merely a metaphor to explain the two possible values: +½ (spin up) and -½ (spin down). This abstraction, rather than a physical spin, plays a crucial role in differentiating electrons in orbitals using the four quantum numbers.