Quantum Numbers Calculator

Chemistry

m_l fill order

−l first+l first

First spin in each orbital

+½ first−½ first

Filling convention. Which m_l and m_s a given electron gets in a partly filled subshell is bookkeeping (the orbitals are degenerate), and textbooks differ. Every result that depends on it is tagged with the convention in use.

n (principal)1, 2, 3, …

l (azimuthal)0 … n − 1

m_l (magnetic)−l … +l

m_s (spin)+½ or −½

Type exactly what you want checked: nothing is rounded or clamped. Fill n alone for a shell, n and l for a subshell, n, l and m_l for an orbital.

Examples:

Allowed

3d subshell, m_l = −2 orbital, spin up (m_s = +½)

3d subshell holds

5 orbitals · 10 e⁻

n = 3 shell holds

9 orbitals · 18 e⁻

Radial nodes (n − l − 1)

Angular nodes (l)

Total nodes (n − 1)

m_l = −2 is a complex orbital. The real 3dxy and 3dx²−y² orbitals are ± combinations of m_l = +2 and −2, so neither matches m_l = −2 alone.

First ground-state element with this electron:

Scandium (Z = 21)

−l first · +½ first

The n = 3 shell

l = 0 · 3s

↑

↓

1 orbital · 2 e⁻

l = 1 · 3p

↑

↓

−1

↑

↓

↑

↓

3 orbitals · 6 e⁻

l = 2 · 3d

↑

↓

−2

↑

↓

−1

↑

↓

↑

↓

↑

↓

5 orbitals · 10 e⁻

n = 3 shell: n² = 9 orbitals · 2n² = 18 e⁻ · each half-box is one allowed electron (↑ = +½, ↓ = −½)

Radial probability P(r) — hydrogen 3d

a₀pm

P(r) = r²R²

radial node

most probable radius

Radial nodes: n − l − 1 = 0

none

Most probable radius

9.000 a₀ (476.3 pm)

Mean radius ⟨r⟩ = (3n² − l(l + 1)) / 2

10.500 a₀ (555.6 pm)

Hydrogen atom. In many-electron atoms orbitals are pulled inward, but the number of radial nodes is the same. Angular nodes: l = 2; total nodes: n − 1 = 2.

Check several at once

Quantum Numbers Calculator – Rules, Shells and the Electrons of Any Element

Every electron in an atom is described by four quantum numbers. This quantum numbers calculator checks whether a set is allowed and explains every rule it breaks. It also lists all the allowed states of a shell and gives the quantum numbers of any electron of an element or ion, with orbital box diagrams and the radial probability curve of each subshell.

The four quantum numbers and their rules

Principal, n: 1, 2, 3, … It picks the shell and sets its size.
Azimuthal, l: whole numbers from 0 to n − 1. It picks the subshell shape: 0 = s, 1 = p, 2 = d, 3 = f, then g, h, i, k (j is skipped).
Magnetic, m_l: whole numbers from −l to +l, so a subshell has 2l + 1 orbitals.
Spin, m_s: +½ or −½.

Worked example: is (3, 2, −2, +½) allowed? n = 3 permits l = 0, 1 or 2, so l = 2 (a d subshell) is fine. l = 2 permits m_l from −2 to +2, so −2 is fine, and +½ is a valid spin. It is a 3d electron. Now try (2, 2, 3, 1). It breaks three rules: l must be at most 1 when n = 2; m_l = 3 needs l ≥ 3, which needs n ≥ 4; and m_s can only be ±½. The calculator reports all three, not just the first. You can also check part of a set: n and l alone name a subshell, so 2d, 1p and 3f are impossible, while 4f exists.

Counting orbitals and electrons

A subshell holds 2(2l + 1) electrons: 2 for s, 6 for p, 10 for d and 14 for f. A shell has n² orbitals and holds 2n² electrons. Counting questions follow the same pattern. Within shell n, a given m_l appears once in every subshell with l ≥ |m_l|, so it labels n − |m_l| orbitals. For n = 3 and m_l = −1 that is 3p and 3d: 2 orbitals and 4 electrons. Fixing m_s as well halves the count. These are capacities; a real atom fills only some of them.

Aufbau, Hund and Pauli

To give each electron of an element its quantum numbers, the tool fills subshells in the Aufbau order (1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, …), with the known exceptions such as chromium and copper. Within a subshell, Hund's rule puts one electron in each orbital before any are paired. The Pauli exclusion principle says no two electrons in an atom share all four numbers, so a filled orbital holds one +½ and one −½ electron. The tool checks this every time: oganesson's 118 electrons give 118 unique sets.

Two electrons get special attention. The last (differentiating) electron is the final one added in filling order. For scandium it is a 3d electron. The outermost electron sits in the shell with the highest n, which for scandium is 4s. Ions start from the neutral atom: cations lose electrons from the highest n first, so Fe²⁺ is [Ar] 3d⁶.

The filling convention

The orbitals of a subshell have equal energy and electrons are indistinguishable. So which m_l and m_s "the third 2p electron" gets is bookkeeping, and books disagree. Many fill m_l from −l upwards with +½ first, but others start at +l or with −½. Scandium's last electron is (3, 2, −2, +½) under the first choice and (3, 2, +2, +½) under the second. Set both switches to match your course; every result that depends on them is tagged.

ml is not px or py

The m_l values label complex orbitals. Only m_l = 0 matches a familiar real orbital (pz, dz², fz³). The real px and py orbitals are combinations of m_l = +1 and −1, so m_l = +1 is not px.

Nodes and the radial distribution

An orbital has n − l − 1 radial nodes, l angular nodes and n − 1 nodes in total. The chart plots the hydrogen radial probability P(r) = r²R(r)², so you can see the nodes: 2s has one, at 2 a₀, and 3s has two. When l = n − 1 there are none, and the peak sits at n² a₀. In heavier atoms the curves move inward, but the node count stays the same.

Is the Quantum Numbers Calculator free?

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Can I use the Quantum Numbers Calculator offline?

Yes, you can install the webapp as PWA.

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Yes, any data related to Quantum Numbers Calculator only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

What do the four quantum numbers mean?

The principal number n (1, 2, 3, …) picks the shell. The azimuthal number l (0 to n − 1) picks the subshell shape: s, p, d, f. The magnetic number ml (−l to +l) picks one orbital in that subshell, and the spin number ms (+½ or −½) picks the electron's spin.

How does the Quantum Numbers Calculator work?

Check a set tests your n, l, ml and ms against the rules and lists every rule broken. Explore a shell lists every allowed state for n = 1 to 10 and counts matching electrons. Element electrons takes the ground-state configuration of an element or ion, numbers its electrons in filling order and gives each one its four quantum numbers.

Why do textbooks give different answers for the last electron of an element?

Within a partly filled subshell the orbitals have the same energy, so which ml and ms a particular electron gets is a convention. Some books fill ml from −l upwards, others from +l, and some give the first electron in each orbital spin −½. That is why both choices are switches here, and every result that depends on them is labelled.

Is ml = +1 the px orbital?

No. The ml values label complex orbitals. Only ml = 0 matches one familiar real orbital (pz, dz², fz³). The real px and py orbitals are combinations of ml = +1 and ml = −1, so neither ml value is px or py on its own.

How many electrons can a shell hold?

A shell with principal number n has n² orbitals and holds at most 2n² electrons: 2 for n = 1, 8 for n = 2, 18 for n = 3 and 32 for n = 4. A subshell with azimuthal number l has 2l + 1 orbitals and holds 2(2l + 1) electrons.

What are the limits of this tool?

Shells are covered up to n = 10. Configurations come from the same data as the Electron Configuration Generator, including its known exceptions, and are predictions for the superheavy elements. The radial chart is exact for hydrogen only; in heavier atoms the orbitals are pulled inward, but the number of nodes stays the same.