How the Four Quantum Numbers Work
In the quantum mechanical model of the atom, an electron's state falls out of solving the Schrödinger equation for the Coulomb potential of the nucleus, and the solution is labeled by four quantum numbers. The principal quantum number n sets the electron's shell and, for a single-electron atom, its energy. The azimuthal quantum number l sets the shape of the orbital and its orbital angular momentum. The magnetic quantum number mₗ sets the orbital's orientation in space, and the spin quantum number mₛ describes the electron's intrinsic angular momentum, independent of its motion around the nucleus. This calculator checks whether a given (n, l, mₗ) combination is an allowed quantum state, then reports the orbital notation, orbital angular momentum, subshell capacity, and hydrogen-like energy level that go with it.
Allowed values for each quantum number
- n (principal): any positive integer — 1, 2, 3, ... Determines the shell and, for hydrogen-like atoms, the energy.
- l (azimuthal / angular momentum): integer from 0 to n−1. Determines the subshell shape and is labeled s, p, d, f, g, ... for l = 0, 1, 2, 3, 4, ...
- mₗ (magnetic): integer from −l to +l, so 2l+1 possible values. Determines the orbital's orientation and its angular momentum component along a chosen axis, Lz = mₗħ.
- mₛ (spin): always +1/2 or −1/2, independent of n, l, and mₗ. Every orbital (fixed n, l, mₗ) holds exactly two electrons, one of each spin — the origin of the "2" in the 2(2l+1) subshell capacity.
Orbital angular momentum, subshell capacity, and energy
The magnitude of an electron's orbital angular momentum is L = ħ√(l(l+1)), where ħ = h/2π ≈ 1.0546 × 10⁻³⁴ J·s is the reduced Planck constant — a quantum result, not simply mvr as in the old Bohr picture. A subshell with quantum number l has 2l+1 degenerate orbitals (one per allowed mₗ) and, by the Pauli exclusion principle, holds at most 2(2l+1) electrons. Summing over every l from 0 to n−1 gives n² orbitals and 2n² electrons in a full shell n (2 for n=1, 8 for n=2, 18 for n=3). For a hydrogen-like atom or ion — one electron orbiting a nucleus of charge +Ze — the Bohr model gives the shell energy as Eₙ = −13.6 eV × Z²/n², negative because the electron is bound, and independent of l and mₗ in this simplified one-electron case.
Where this simplified picture breaks down
Eₙ = −13.6 eV × Z²/n² is exact for hydrogen and hydrogen-like ions (He⁺, Li²⁺, ...) with a single electron, and the shapes, orientations, and capacities derived from l and mₗ apply to any atom. It does not give correct energies for multi-electron atoms: electron-electron repulsion and uneven shielding make the energy depend on l as well as n (which is why, for example, 4s fills before 3d), and finer effects like spin-orbit coupling split levels further. Treat the energy figure here as exact only for one-electron systems, and as an order-of-magnitude reference otherwise.