Quantum Number Calculator

The quantum number calculator allows you to list all possible values of quantum numbers for a given electron shell.

Orbitals in this subshell52l+1 values of m_l, from −2 to +2 — this is the 3d subshell

Electrons this subshell holds10two per orbital, one for each spin

Allowed l values for n = 30 to 2giving the subshells 3s, 3p, 3d

Allowed m_l values−2 to +25 orientations of the orbital angular momentum in space

Allowed m_s values±½always exactly two, for every orbital there has ever been

Orbitals in the whole shell9n² — the sum of 2l+1 over l from 0 to n−1

Electrons in the whole shell182n²: 2, 8, 18, 32 for the first four shells

Orbital angular momentum2.44949 ħ√(l(l+1))ħ, NOT lħ — the magnitude always exceeds the largest measurable component of 2ħ, so the vector can never point fully along any axis

Largest measurable component2 ħsmaller than the magnitude above, which has no classical analogue at all

Number of radial nodes0n − l − 1

Number of angular nodes2always l, and they total n − 1 with the radial ones

Why the letters s, p, d, fsharp, principal, diffuse, fundamentalnamed for the appearance of spectral lines decades before anyone knew what orbitals were. After f they simply continue alphabetically, skipping j

The formula

l = 0…n−1; m_l = −l…+l; m_s = ±½; states = 2n²

Each number constrains the next

The principal number n can be any positive integer. Given n, the azimuthal number l runs from 0 to n−1 — which is why there is no 1p orbital and no 2d. Given l, the magnetic number m_l takes 2l+1 values from −l to +l. Spin is always ±½. Multiplying it out gives n² orbitals and 2n² electrons per shell: 2, 8, 18, 32.

The angular momentum magnitude is √(l(l+1))ħ while the largest measurable component is only lħ. The magnitude always exceeds the biggest component, so the vector can never point entirely along the axis you measure — a purely quantum feature with no classical counterpart.

The letters s, p, d and f come from "sharp", "principal", "diffuse" and "fundamental", describing how spectral lines looked decades before orbitals were understood. After f they continue alphabetically, skipping j.