- Quantum Numbers Definition: Quantum numbers are defined as values that describe the location, energy level, and spin of electrons in an atom.
- Principal Quantum Number: This number, denoted as ‘n’, represents the main energy level or shell an electron occupies.
- Orbital Quantum Number: Also known as the azimuthal quantum number, this number, denoted as ‘l’, indicates the subshell and shape of the orbital.
- Magnetic Quantum Number: This number, denoted as ‘m or ml’, describes the orientation of orbitals within a subshell and ranges from -l to +l.
- Spin Magnetic Quantum Number: This number, denoted as ‘ms’, represents the spin direction of an electron and can be either +1/2 or -1/2.
Quantum numbers label the allowed quantum states of electrons in an atom. They describe an orbital’s shell, angular momentum and angular-momentum projection, together with an electron’s spin projection. They do not specify a fixed position or classical path. Electron configurations use four quantum numbers:
- Principal quantum number (n)
- Orbital angular momentum or azimuthal quantum number (l)
- Magnetic orbital quantum number (m or ml)
- Spin projection quantum number (ms)
Principal Quantum Number (n)
The principal quantum number, n, labels the electron shell and takes positive integer values 1, 2, 3 and so on. For hydrogen-like atoms it determines the orbital energy. In multi-electron atoms it still identifies the shell, but electron-electron interactions mean that energy also depends on other quantum numbers.
Shells with n = 1, 2, 3 and 4 are sometimes called K, L, M and N. A shell contains n² orbitals. Because each orbital can hold two electrons with opposite spin projections, its maximum capacity is 2n² electrons, as shown below.
| Shell number | Shell letter | Principal quantum number n | Maximum electrons (2n2) |
| 1 | K | 1 | 2 × 12 = 2 |
| 2 | L | 2 | 2 × 22 = 8 |
| 3 | M | 3 | 2 × 32 = 18 |
| 4 | N | 4 | 2 × 42 = 32 |
As the principal quantum number increases, an orbital generally extends farther from the nucleus. For a hydrogen-like atom, the energy increases towards zero as n increases, so the electron is less tightly bound. In a multi-electron atom, both n and subshell interactions affect orbital energy.
Orbital Angular Momentum or Azimuthal Quantum Number (l)
The orbital angular momentum quantum number, l, identifies a subshell and determines the magnitude of orbital angular momentum. For each principal quantum number n, the allowed integer values are 0 through n – 1.
A subshell is a group of orbitals with the same n and l values; the terms are not interchangeable. The labels s, p, d and f correspond to l = 0, 1, 2 and 3. A shell with principal quantum number n has n allowed subshells. A subshell contains 2l + 1 orbitals and can therefore hold at most 2(2l + 1) electrons.
| Subshell number | Subshell | Quantum number l | Maximum electrons, 2(2l + 1) |
| 1 | s | 0 | 2(2 × 0 + 1) = 2 |
| 2 | p | 1 | 2(2 × 1 + 1) = 6 |
| 3 | d | 2 | 2(2 × 2 + 1) = 10 |
| 4 | f | 3 | 2(2 × 3 + 1) = 14 |
The orbital angular momentum quantum number helps determine an orbital’s probability-density shape. An s orbital has spherical symmetry, while p orbitals have two main lobes. The d and f families have more complex angular patterns. These are probability distributions, not straight or elliptical electron paths.
Each l value allows a characteristic family of orbital shapes, with radial structure also depending on n.
In electron-configuration notation, the principal quantum number precedes the subshell letter and a superscript gives the number of electrons in that subshell. Six electrons in the n = 2 p subshell are written as 2p6.
Magnetic Quantum Number (m or ml)
The magnetic orbital quantum number ml specifies the projection of orbital angular momentum along a chosen axis. For a given l, the magnetic quantum number ml takes every integer value from -l to +l. For a p subshell, ml can therefore be ml = -1, 0 or +1. The familiar real orbitals px, py and pz describe three spatial orientations and can be formed from combinations of magnetic-quantum-number states. Each l has 2l + 1 allowed ml values. Summing the orbitals across all subshells gives n2 orbitals in shell n. The table lists the allowed magnetic quantum numbers.
| Subshell | Orbital angular momentum quantum number l | Number of orbitals 2l + 1 | Magnetic quantum number (m or ml) |
| s | 0 | 1 | 0 |
| p | 1 | 3 (px, py, pz) | -1, 0, +1 |
| d | 2 | 5 (dx2-y2, dz2, dxy, dxz, dyz) | -2, -1, 0, +1, +2 |
| f | 3 | 7 (fz3, fxz2, fxyz, fx(x2-3y2), fyz2, fz(x2-y2), fy(3x2-y2)) | -3, -2, -1, 0, +1, +2, +3 |
Spin Projection Quantum Number (ms)
Electron spin names an intrinsic angular momentum and does not describe the literal rotation of a small sphere or an electron orbiting the nucleus. The spin projection quantum number, ms, has two allowed values: +1/2 and -1/2. These values affect magnetic interactions and can affect atomic energy through interactions such as spin-orbit coupling. An orbital can hold two electrons only when their spin projections are opposite, as required by the Pauli exclusion principle. No two electrons in one atom can share the same complete set of four quantum numbers. The symbol ms labels the spin projection.
