The Four Quantum Numbers: n, l, ml, ms

Four quantum numbers describe the state of an electron in an atom: the principal quantum number n (shell, size and energy), the angular momentum quantum number l (subshell, shape), the magnetic quantum number ml (orientation) and the spin quantum number ms (spin direction, +½ or −½). In brief: the allowed values are n = 1, 2, 3, …; l = 0 to n − 1; ml = −l to +l; ms = ±½, and no two electrons in an atom can have the same set of all four numbers.

The quantum numbers are the “address” of an electron and explain the structure of the periodic table: why the shells hold 2, 8, 18 and 32 electrons and why the s, p, d and f blocks have the sizes they do. This article defines each number, shows the allowed combinations for the first four shells in a table, explains the Pauli exclusion principle and shows how to decide whether a set of quantum numbers is valid.

What is the principal quantum number?

The principal quantum number n can be any positive integer: 1, 2, 3, … It determines the size of the orbital and, for hydrogen, the energy: En = −2.18 × 10−18 J / n2. A larger n means that the electron is on average further from the nucleus and has a higher energy. Electrons with the same n are in the same shell (K, L, M, N for n = 1, 2, 3, 4).

What is the angular momentum quantum number?

The quantum number l can have the integer values from 0 to n − 1. It determines the shape of the orbital and the angular momentum. Electrons with the same n and l are in the same subshell, named by letters: l = 0 is s, 1 is p, 2 is d and 3 is f (the letters come from the early spectroscopic terms sharp, principal, diffuse and fundamental). A subshell is written with n and the letter: 2p means n = 2, l = 1. In an atom with several electrons, the energy depends on both n and l.

What is the magnetic quantum number?

The quantum number ml can have the integer values from −l to +l, including zero, so there are 2l + 1 values. It defines the orientation of the orbital in space (its component of the angular momentum along an axis). Each combination of n, l and ml defines one orbital. A p subshell has three orbitals (ml = −1, 0, +1, oriented along the x, y and z axes), a d subshell five and an f subshell seven. In the absence of a magnetic field, orbitals of the same subshell have the same energy (they are degenerate).

What is the spin quantum number?

The electron has an intrinsic angular momentum, called spin, that can have two orientations, represented by ms = +½ (spin up, ↑) or −½ (spin down, ↓). It does not depend on n, l or ml. The spin gives the electron a magnetic moment. Atoms with unpaired electrons are paramagnetic and are attracted to a magnet, and atoms with all electrons paired are diamagnetic.

Allowed combinations for n = 1 to 4

Subshells, orbitals and electron capacity
n l Subshell ml values Orbitals Maximum electrons
101s012
202s012
212p−1, 0, +136
303s012
313p−1, 0, +136
323d−2, −1, 0, +1, +2510
404s012
414p−1, 0, +136
424d−2 to +2510
434f−3 to +3714

A shell with principal quantum number n has n2 orbitals and can hold 2n2 electrons: 2, 8, 18 and 32 for n = 1 to 4. This pattern is the origin of the lengths of the periods in the periodic table.

n = 1 0 1s n = 2 0 2s −1 0 +1 2p n = 3 0 3s −1 0 +1 3p −2 −1 0 +1 +2 3d n = 4 0 4s −1 0 +1 4p −2 −1 0 +1 +2 4d −3 −2 −1 0 +1 +2 +3 4f

What is the Pauli exclusion principle?

Wolfgang Pauli stated in 1925 that no two electrons in an atom can have the same set of four quantum numbers. As a result, an orbital (defined by n, l and ml) can hold at most two electrons, and they must have opposite spins (paired). Together with the rule that electrons fill the lowest-energy orbitals first (the Aufbau principle) and Hund’s rule (electrons occupy the orbitals of a subshell singly with parallel spins before they pair up), the exclusion principle determines the electron configuration of all elements. Without it all electrons would collapse into the 1s orbital, and chemistry would not exist.

Worked examples

Example 1: are these sets of quantum numbers allowed?

Example 2: counting orbitals and electrons

How many orbitals and electrons are in the n = 3 shell? The subshells are 3s (1 orbital), 3p (3 orbitals) and 3d (5 orbitals): 1 + 3 + 5 = 9 = n2 orbitals, which hold 2 × 9 = 18 electrons.

Example 3: naming a subshell from the quantum numbers

An electron has n = 4 and l = 2. It is in a 4d subshell, which has five orbitals (ml = −2 to +2) and can hold 10 electrons.

Example 4: the quantum numbers of the valence electron of sodium

Sodium (Z = 11) has the configuration 1s22s22p63s1. The outer electron is in a 3s orbital: n = 3, l = 0, ml = 0 and ms = +½ or −½ (conventionally +½ for the first electron in an orbital).

Example 5: the quantum numbers of the electrons in carbon’s 2p subshell

Carbon (1s22s22p2) has two 2p electrons. According to Hund’s rule they are in different orbitals with parallel spins, for example (2, 1, −1, +½) and (2, 1, 0, +½). They have the same n, l and ms, but different ml, so the Pauli principle is not violated.

C −1 0 +1 ml

How were the quantum numbers introduced?

The principal quantum number n came from Bohr’s model of 1913. Arnold Sommerfeld introduced l (as an additional quantum number for elliptical orbits) in 1916, and the magnetic quantum number was introduced to explain the Zeeman effect, the splitting of spectral lines in a magnetic field. Otto Stern and Walther Gerlach showed in 1922 that a beam of silver atoms splits into two in an inhomogeneous magnetic field, which was later understood as evidence for the electron spin. George Uhlenbeck and Samuel Goudsmit proposed electron spin in 1925, and in the same year Pauli formulated the exclusion principle, for which he received the Nobel Prize in Physics in 1945.

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