Atomic Quantum Numbers

Equivalent forms

E_n = -\frac{13.6\,\text{eV}}{n^2}

\text{states in shell } n = 2n^2

Four integers and one exclusion rule generate the entire shape of the periodic table.

Symbol	Name	SI unit	Dimension	Range
$n$	Principal quantum number Sets the shell and energy scale	1	dimensionless	1 – 7
$l$	Azimuthal quantum number Orbital shape (s,p,d,f = 0,1,2,3), 0..n-1	1	dimensionless	0 – 6
$capacity$	Shell capacityoutput Number of electrons the shell n can hold, 2n^2	1	dimensionless	2 – 98

Unit systems

Symbol	SI	CGS	Imperial
$n$	1	1	1
$l$	1	1	1
$capacity$	1	1	1

Where it holds

Exact labels for the hydrogen atom; in multi-electron atoms n and l remain good approximate labels while electron-electron interaction reorders subshell energies.

Dimensional analysis

n, l, m_l, m_s are integers or half-integers

counting rule 2n^2

All quantum numbers are dimensionless labels; the capacity is a pure count.

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Discovery

Bohr, Sommerfeld, Pauli & Goudsmit-Uhlenbeck · 1925

Bohr's n (1913) and Sommerfeld's l gained a magnetic number m_l; then in 1925 Pauli's exclusion principle demanded a fourth label, which Goudsmit and Uhlenbeck identified as electron spin m_s. Together they explained the periodic table's structure.

Try this

Why does the periodic table have exactly the rows and blocks it does?

Four integers label every electron in an atom: n, l, m_l and m_s. Their allowed ranges — and the rule that no two electrons share all four — dictate how orbitals fill, and therefore the entire architecture of the periodic table.

Research status: stable

Real-world applications

Electron configurations that explain chemical periodicity and bonding
Selection rules $(\Delta l = +/- 1)$ governing spectral lines and lasers
Shell structure behind X-ray emission lines used in materials analysis

Common misconceptions

l runs only up to n-1, so a 1p or 2d orbital does not exist
The energy of hydrogen depends only on n; l-degeneracy is lifted only in multi-electron atoms
m_s is not orbital motion — it is intrinsic spin with just two values

Experimental verification

The periodic table's period lengths (2, 8, 8, 18, ...) and X-ray/optical spectral series (Lyman, Balmer) directly confirm the allowed quantum numbers and shell capacities. Zeeman splitting resolves the 2l+1 m_l states in a magnetic field.

Derivation

Solving the hydrogen Schrodinger equation separates into radial and angular parts.

The angular part (spherical harmonics) quantizes

l = 0

,1,...,n-1 and

m_l = -l

,...,+l; the radial part quantizes

n = 1

,2,3,... with energy

E_n = -13.6\,\mathrm{eV/n}^2

.

Electron spin adds

m_s = +/- 1/2

.

Pauli's exclusion principle forbids two electrons from sharing all four numbers, so each orbital (fixed n, l, m_l) holds exactly two electrons.

Counting gives 2(2l+1) per subshell and

sum_{l=0}^{n-1} 2(2l+1) = 2n^2

per shell — the pattern 2, 8, 18, 32 of the periodic table's rows.

Limiting cases

n = 1

⟶ only

l = 0 (1s)

, 2 electronsThe ground shell holds a single s-orbital with two spins.

subshell of given l⟶ 2(2l+1) electrons2l+1 orbitals times two spin states.

shell n⟶ 2n^2 electronsSumming 2(2l+1) over

l = 0

..n-1 gives the shell capacity.

What if…

What if electrons ignored the Pauli principle?

Every electron would collapse into the 1s orbital; chemistry, the periodic table, and solid matter as we know it would vanish.

What if l could equal n?

It cannot — the radial equation has no acceptable solution; l is capped at n-1, which is why certain orbitals never appear.

1

Subshells in the n = 3 shell

Given ·

n:: 3

Find · allowed l and capacity

Steps

$l = 0$ .. $n-1 = 0$ ,1,2
capacities $2(2l+1) = 2$ ,6,10
total $18 = 2n^2$

Result ·

l = 0

,1,2 (3s, 3p, 3d) holding

2 + 6 + 10 = 18 = 2*3^2

electrons.

2

How many electrons in a d-subshell?

Given ·

l:: 2

Find · capacity

Steps

$m_l = -2$ ..2 -> 5 orbitals
times 2 spins
$= 10$

Result ·

2(2l+1) = 2*5 = 10

electrons — the ten columns of the transition-metal block.

Pauli Exclusion Principle→hydrogen wavefunctionsBohr Hydrogen Energy Levels→Spherical Harmonics→