Magnetic Quantum Number (ml)

In quantum mechanics, every electron in an atom is described by a unique set of four quantum numbers. The magnetic quantum number, symbolized m_l, is the third of these four numbers and specifies the spatial orientation of an orbital within a given subshell. While the principal quantum number (n) tells us the energy level and the azimuthal quantum number (l) tells us the shape of the orbital, m_l tells us how that orbital is pointed in three-dimensional space relative to an external magnetic field or reference axis. The allowed values of m_l range from −l to +l in integer steps, including zero. This means that for a p subshell (l = 1), there are three possible values: m_l = −1, 0, and +1, corresponding to the three p orbitals (p_x, p_y, and p_z). Each unique value of m_l describes a distinct orbital. Because electrons in orbitals with the same n and l but different m_l values have identical energies in the absence of a magnetic field, these orbitals are said to be degenerate. The concept of m_l becomes physically significant when an atom is placed in an external magnetic field. The field breaks the degeneracy — orbitals that previously had the same energy split into slightly different energy levels, a phenomenon known as the Zeeman effect. This splitting was historically crucial evidence that quantum numbers are real physical quantities and not just mathematical abstractions, and it remains important in modern applications such as nuclear magnetic resonance (NMR) spectroscopy and MRI scanning.

Key takeaways
  • Recap: The Quantum Number Hierarchy
  • Allowed Values of m_l
  • Physical Meaning: Spatial Orientation
  • Degeneracy in the Absence of a Magnetic Field
  • Breaking Degeneracy: The Zeeman Effect
  • Counting Orbitals and Electrons per Subshell

Recap: The Quantum Number Hierarchy

Every electron is described by four quantum numbers: n (principal), l (azimuthal/angular momentum), m_l (magnetic), and m_s (spin). They must be assigned in order. First choose n (any positive integer: 1, 2, 3, …). Then choose l (an integer from 0 to n−1). Only then can you determine the allowed values of m_l.

Allowed Values of m_l

For a given value of l, m_l can take any integer value from −l to +l, inclusive. The total number of allowed values is (2l + 1). For example: s subshell (l = 0) → m_l = 0 only → 1 orbital. p subshell (l = 1) → m_l = −1, 0, +1 → 3 orbitals. d subshell (l = 2) → m_l = −2, −1, 0, +1, +2 → 5 orbitals. f subshell (l = 3) → m_l = −3, −2, −1, 0, +1, +2, +3 → 7 orbitals.

Physical Meaning: Spatial Orientation

Each value of m_l corresponds to one orbital of a given shape oriented differently in space. For p orbitals, the three orientations point along the x, y, and z axes (p_x, p_y, p_z). For d orbitals, the five orientations include lobes between and along axes (d_xy, d_xz, d_yz, d_x²−y², d_z²). The label m_l = 0 for p orbitals conventionally corresponds to the p_z orbital, oriented along the z-axis (the reference magnetic field axis).

Degeneracy in the Absence of a Magnetic Field

Orbitals within the same subshell (same n and l) that differ only in m_l have exactly the same energy in an isolated atom with no external field. This equal-energy condition is called degeneracy. All three 2p orbitals are degenerate; all five 3d orbitals are degenerate. This is why Hund's rule matters: when filling degenerate orbitals, electrons spread out to maximize unpaired spins, minimising repulsion and lowering total energy.

Breaking Degeneracy: The Zeeman Effect

When an external magnetic field B is applied, the energy of each orbital depends on m_l. Orbitals with different m_l values experience slightly different interactions with the field, causing energy splitting. An orbital with m_l = +1 is raised in energy, m_l = 0 is unchanged, and m_l = −1 is lowered. This splitting of spectral lines in a magnetic field is the Zeeman effect, discovered by Pieter Zeeman in 1896. The energy shift is given by ΔE = m_l · μ_B · B, where μ_B is the Bohr magneton (9.274 × 10⁻²⁴ J T⁻¹).

Counting Orbitals and Electrons per Subshell

The number of orbitals in a subshell equals (2l + 1). Since each orbital holds a maximum of 2 electrons (differing in m_s), the maximum number of electrons in a subshell is 2(2l + 1). For s: 2 electrons; p: 6 electrons; d: 10 electrons; f: 14 electrons. These numbers directly explain the block structure of the periodic table.

The Key Equation

ml∈{−l,  −(l−1),  …,  0,  …,  +(l−1),  +l}m_l \in \{-l,\; -(l-1),\; \ldots,\; 0,\; \ldots,\; +(l-1),\; +l\}
  • m_l — Magnetic quantum number (dimensionless (integer)): Specifies the spatial orientation of the orbital; ranges from −l to +l in integer steps.
  • l — Azimuthal (angular momentum) quantum number (dimensionless (integer)): Defines the subshell and orbital shape; ranges from 0 to n−1.
  • 2l+1 — Number of orbitals per subshell (dimensionless (integer)): The total count of distinct spatial orientations (orbitals) available within a subshell of angular momentum quantum number l.

Worked Example

Problem. An electron is known to be in the 4d subshell of an atom. (a) What are the allowed values of m_l for this electron? (b) How many orbitals does the 4d subshell contain? (c) What is the maximum number of electrons that can occupy the 4d subshell? (d) Which value of m_l conventionally corresponds to the d_z² orbital?

ml={−2, −1, 0, +1, +2};5 orbitals;max 10 electrons;dz2↔ml=0m_l = \{-2,\,-1,\,0,\,+1,\,+2\};\quad \text{5 orbitals};\quad \text{max }10\text{ electrons};\quad d_{z^2} \leftrightarrow m_l = 0

Answer. Allowed m_l values: −2, −1, 0, +1, +2. The 4d subshell contains 5 degenerate orbitals and can hold a maximum of 10 electrons. The d_z² orbital corresponds to m_l = 0.

Figure
Diagram
Orbital Orientations and m_l Valuess (l=0)m_l = 01sp (l=1)m_l = −1, 0, +1p_xp_yp_z (m_l=0)d (l=2)m_l = −2,−1,0,+1,+2d_xyEnergy Levels: Degeneracy and Zeeman SplittingNo field2p (3 degenerate)With B fieldm_l = +1m_l = 0m_l = −1Zeeman effect: external magnetic field B lifts degeneracy
Origins · 1 minPieter Zeeman and how this was discovered
PZ
Pieter Zeeman
1865–1943 · Dutch

Pieter Zeeman discovered in 1896 that spectral lines broaden and split when atoms are placed in a strong magnetic field — a phenomenon now called the Zeeman effect. This observation provided the first experimental evidence that atomic electrons possess angular momentum and magnetic properties, directly supporting the quantum mechanical framework of magnetic quantum numbers. He shared the 1902 Nobel Prize in Physics with Hendrik Lorentz for this discovery. Zeeman first noticed the splitting of the sodium D-line by placing a sodium flame between the poles of a powerful electromagnet, using diffraction gratings to resolve the effect — all in a basement laboratory at Leiden University.

  • Principal Quantum Number (n) — n is the first quantum number in the hierarchy and determines the energy level and size of the orbital; m_l can only be assigned after n and l are known.
  • Azimuthal Quantum Number (l) — l defines the subshell and orbital shape, and directly determines the range of allowed m_l values through the relationship m_l = −l to +l.
  • Spin Quantum Number (m_s) — m_s is the fourth quantum number; together with m_l it completes the unique quantum address of each electron as required by the Pauli exclusion principle.
  • Orbital Shapes (s, p, d, f) — The physical shapes and orientations of orbitals (p_x, p_y, p_z; d_xy, etc.) are the real-space representations of the different m_l values within each subshell.
  • Hund's Rule and Pauli Exclusion Principle — Hund's rule governs how electrons fill degenerate orbitals (same m_l options), and the Pauli exclusion principle requires that no two electrons share all four quantum numbers including m_l.