Every electron in an atom is described by a unique set of four quantum numbers — n, l, ml, and ms — that together pin down its energy level, subshell shape, orbital orientation, and spin. This calculator checks whether a given set follows the rules, lists which values a shell allows, and reads off a subshell's orbital count and electron capacity.

How the Quantum Numbers Calculator works

The four quantum numbers are not independent — each one constrains the next. The principal quantum number n sets the shell (n = 1, 2, 3, ...). The azimuthal quantum number l, which sets the subshell shape, is restricted to integers from 0 to n−1. The magnetic quantum number ml, which sets an orbital's orientation, is restricted to integers from −l to +l. The spin quantum number ms is always either +½ or −½, independent of the other three.

The Validate tab checks each of these constraints in order and reports exactly which one fails, if any. The Allowed Values and Subshell Info tabs run the same rules forward — given n (and l), they derive every value the remaining quantum numbers are permitted to take, along with the subshell's letter, its orbital count (2l+1), and its electron capacity (2(2l+1)).

Inputs and what they mean

n (principal) is a positive integer identifying the shell — larger values mean higher energy and, on average, greater distance from the nucleus. l (azimuthal) identifies the subshell shape and must be between 0 and n−1; it maps to the familiar s/p/d/f letters. ml (magnetic) identifies a specific orbital's orientation and must be between −l and +l. ms (spin) is always +½ or −½ and does not depend on the other three numbers.

Limits and edge cases

This calculator covers subshells up to l = 6 (through the g/h/i letters), which spans every subshell used in general and inorganic chemistry — real, ground-state elements only ever populate up to f subshells (l = 3). Values beyond that are included for completeness but rarely appear outside theoretical problems.

The tool checks quantum-number validity in isolation; it does not determine which subshells actually fill with electrons in a given atom (that follows the Aufbau principle and Hund's rule, covered by the Electron Configuration calculator) or account for relativistic effects that shift orbital energies in very heavy elements.