Electron Spin and the Spin Quantum Number

Introduction

The Schrödinger equation for hydrogen produces three quantum numbers, n, l and mₗ, all arising from motion in three-dimensional space. Yet atomic spectra and the structure of the periodic table demand a fourth . Electrons carry an intrinsic angular momentum, called spin , that has no classical counterpart and cannot be derived from the non-relativistic Schrödinger equation. This page describes the experiment that first revealed it, the rules that govern it and how it completes the description of an electron.

Core explanation

The Stern–Gerlach experiment (1922). Otto Stern and Walther Gerlach passed a beam of silver atoms through an inhomogeneous magnetic field. A magnetic dipole in such a field feels a force proportional to the component of its magnetic moment along the field gradient. Classically, randomly oriented moments would give a continuous smear on the detector. Instead, the beam split into two distinct spots.

A silver atom has the configuration [Kr]4d¹⁰5s¹. The filled shells contribute no net angular momentum, and the 5s electron has l = 0, so there is no orbital magnetic moment. The two spots must come from an angular momentum belonging to the electron itself. Moreover, orbital angular momentum always gives 2l + 1 components, an odd number; two components require a half-integer quantum number. Uhlenbeck and Goudsmit proposed electron spin in 1925 to explain this and the doublet structure of alkali spectra.

The rules of spin. Spin behaves like any quantum angular momentum:

- Spin quantum number s = ½ for every electron (it never changes). - Magnitude S = √(s(s + 1)) ħ = (√3/2)ħ ≈ 0.866ħ. - z-component S z = mₛħ, with mₛ = +½ or −½ only.

The two states are written α (mₛ = +½, "spin up") and β (mₛ = −½, "spin down"). They are orthonormal: ∫α α dσ = ∫β β dσ = 1 and ∫α β dσ = 0.

Spin magnetic moment. The electron's spin magnetic moment has a z-component of about ∓μ B, where the Bohr magneton μ B = eħ/2mₑ ≈ 9.27 × 10⁻²⁴ J T⁻¹. The electron g-factor is about 2.0023, roughly twice the value for orbital motion, which is why a spin of only ½ produces a full Bohr magneton.

Spin-orbitals. A complete one-electron state is a spatial orbital multiplied by a spin function, for example

ψ = ψ₁ₛ(r) α or ψ = ψ₂ₚ₋₁(r) β

Each hydrogen state is now labelled by four quantum numbers: n, l, mₗ and mₛ. The Pauli exclusion principle states that no two electrons in an atom can share all four, so each spatial orbital holds at most two electrons, with opposite spins.

Where spin comes from. Spin emerges naturally from Dirac's relativistic equation for the electron (1928). In non-relativistic quantum chemistry it is simply added as an extra variable.

Formulae

s = ½; mₛ = ±½; S = √(s(s + 1)) ħ = (√3/2)ħ; S z = mₛħ; μ z ≈ −g e mₛ μ B with g e ≈ 2.0023; μ B = eħ/2mₑ = 9.27 × 10⁻²⁴ J T⁻¹.

Step-by-step reasoning

Interpreting a Stern–Gerlach result:

1. Count the spots: 2j + 1 spots correspond to an angular momentum quantum number j. 2. Two spots give 2j + 1 = 2, so j = ½. 3. Orbital angular momentum gives only odd numbers of spots (2l + 1), so a half-integer value must be spin. 4. For a ground-state atom with one unpaired s electron, the splitting is entirely due to that electron's spin.

Visual explanation

Picture the detector plate. Classical physics predicts a vertical smudge spread continuously between two extremes. The experiment shows two sharp, separate spots above and below the centre, with nothing in between. Each silver atom is found to be either "up" or "down" — never at an intermediate angle.

Real-world analogy

Spin is like a coin that, whenever you check it along any chosen direction, shows only heads or tails, never an edge or an angle in between. The analogy has limits: the coin is not literally spinning, and which direction you choose to check changes the question being asked.

Real-world example

Magnetic resonance imaging and electron spin resonance both depend on spin. In ESR a sample with unpaired electrons is placed in a magnetic field, which splits the α and β energies by g e μ B B. At a field of about 0.34 T this gap corresponds to microwaves of about 9.5 GHz, which the unpaired electrons absorb, giving information about radicals and transition-metal centres.

Why?

Why is a half-integer value allowed for spin but not for orbital angular momentum? Orbital wavefunctions must return to the same value after a 360° rotation in space, forcing integer mₗ. Spin is not a function of spatial angles, so this condition does not apply; spin-½ states actually change sign after a single 360° rotation.

Common misconception

"The electron is a tiny ball spinning on its axis." A point-like electron spinning fast enough to give ħ/2 would need a surface speed far above the speed of light. Spin is an intrinsic quantum property, like charge, and the name is only a historical label.

Worked example

Question: Calculate the magnitude of the spin angular momentum of an electron and the angle between the spin vector and the z-axis for an α electron.

Reasoning: S = √(½ × 3/2) ħ = (√3/2)ħ = 0.866ħ. S z = ½ħ. cos θ = S z/ S = 0.5/0.866 = 0.577.

Answer: S = 0.866ħ ≈ 9.13 × 10⁻³⁵ J s, and θ ≈ 54.7°: the spin can never point exactly along z.

Quick check

1. How many quantum numbers are needed to specify an electron state in hydrogen, and which one comes from spin? Answer: Four quantum numbers are needed: n, l, mₗ and mₛ, and the fourth, mₛ = ±½, comes from electron spin.

Exam focus

Know the Stern–Gerlach result and why silver (l = 0 ground state) was ideal. Quote s = ½, mₛ = ±½ and S = (√3/2)ħ, and be able to explain that two spots imply a half-integer angular momentum. Link spin to the Pauli principle and the capacity of two electrons per orbital.

Advanced insight

Spin couples to orbital angular momentum through the magnetic interaction between the electron's spin moment and the field produced by its orbital motion. This spin–orbit coupling makes l and s combine into a total angular momentum j = l ± ½ and splits levels such as 2p into 2p₁/₂ and 2p₃/₂. In heavy atoms the coupling becomes large enough to dominate chemistry, for example contributing to the inertness of the 6s² pair in mercury and thallium.

Summary

The Stern–Gerlach experiment split silver atoms into two beams, revealing an intrinsic electron angular momentum with only two orientations. The electron has s = ½, mₛ = ±½, S = (√3/2)ħ and S z = ±½ħ. Spin functions α and β multiply spatial orbitals to form spin-orbitals, adding a fourth quantum number that underlies the Pauli principle. Spin is a quantum property with no classical picture.

Practice questions

1. Why did Stern and Gerlach's use of silver atoms rule out orbital angular momentum as the cause of the splitting? Answer: Silver's single valence electron is in 5s with l = 0 and its inner shells are filled, so the atom has no orbital angular momentum; the splitting must come from spin. 2. How many spots would a Stern–Gerlach apparatus produce for atoms with total angular momentum quantum number j = 3/2? Answer: 2j + 1 = 4 spots. 3. Write two spin-orbitals describing the electrons of ground-state helium in the orbital approximation. Answer: ψ₁ₛα and ψ₁ₛβ: both electrons share the 1s spatial orbital with opposite spins. 4. Why is an inhomogeneous rather than a uniform magnetic field needed? Answer: A uniform field only exerts a torque on a magnetic dipole; a net deflecting force requires a field gradient.