Spin Angular Momentum Formula: S = √(s(s+1)) ħ Explained

The spin angular momentum formula is $S = \sqrt{s(s+1)}\,\hbar$ , where s is the spin quantum number and $\hbar = \frac{h}{2\pi} \approx 1.055 \times 10^{-34}$ J s. Its component along any chosen axis, usually called z, can only take the values $S_z = m_s\hbar$ , with $m_s$ running from −s to +s in steps of one.

Spin is angular momentum that a particle carries by its very nature, not because it travels round a path. An electron, a proton and a neutron each have s = 1/2. For the electron that gives $S = \frac{\sqrt{3}}{2}\hbar$ and $S_z = \pm\frac{1}{2}\hbar$ , often called spin up and spin down. Spin has the same unit as ordinary angular momentum, the joule second.

Quick facts

Magnitude
$S = \sqrt{s(s+1)}\,\hbar$
z component
$S_z = m_s\hbar$ , $m_s = -s, \dots, +s$
Electron
$s = \frac{1}{2}$ , $S \approx 9.13 \times 10^{-35}$ J s
SI unit
joule second (J s) = kg m²/s
Dimensional formula
[M¹ L² T⁻¹]
Classical spin
A rigid body about its own axis: $L = I\omega$

Spin angular momentum formula for an electron, worked out

Example 1: size of the electron's spin

Find S and the allowed values of Sz for an electron.

s = 1/2, so $s(s+1) = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}$ .

$S = \sqrt{0.75} \times 1.055 \times 10^{-34} \approx 9.13 \times 10^{-35}\ \text{J s}$

$S_z = \pm\frac{1}{2}\hbar \approx \pm 5.27 \times 10^{-35}\ \text{J s}$ .

Notice that $S_z$ is always smaller than S. The spin vector can never point exactly along the z axis; it always leans at an angle, here $\cos^{-1}\left(\frac{1/2}{\sqrt{3}/2}\right) \approx 54.7^\circ$ from z.

Particles and their spin numbers

Spin quantum numbers of some particles
Particle	s	Allowed mₛ	Group
Electron, proton, neutron	1/2	+1/2, −1/2	Fermions
Photon	1	+1, −1 (no 0 for a massless particle)	Boson
Deuteron (nucleus of heavy hydrogen)	1	+1, 0, −1	Boson
Helium 4 nucleus	0	0	Boson

Particles with half integer spin are fermions, and no two of them in one atom may share the same full set of quantum numbers. That rule, the exclusion principle, is why each atomic orbital holds at most two electrons, one with mₛ = +1/2 and one with mₛ = −1/2.

Spin compared with orbital angular momentum

Two sources of angular momentum in an atom
Feature	Orbital	Spin
Cause	Motion of the electron round the nucleus	Built into the particle
Magnitude	$\sqrt{l(l+1)}\,\hbar$	$\sqrt{s(s+1)}\,\hbar$
Quantum number values	l = 0, 1, 2, ...	s = 1/2 for an electron
Bohr model form	$\frac{nh}{2\pi}$	Not part of the Bohr model

The total angular momentum of an electron is the vector sum of the two. The orbital angular momentum formula page covers the first column in detail.

Classical spin: a body turning on its own axis

In mechanics, spin angular momentum also means the angular momentum of a body about an axis through its own center of mass, $L_{spin} = I\omega$ . Earth has both: orbital angular momentum from its yearly path round the Sun and spin angular momentum from its daily rotation.

Example 2: Earth's spin

Treat Earth as a uniform sphere of mass 5.97 × 10²⁴ kg and radius 6.37 × 10⁶ m, turning at 7.27 × 10⁻⁵ rad/s. Estimate its spin angular momentum.

R² is about 4.06 × 10¹³ m², so $I = \frac{2}{5}MR^2 \approx 0.4 \times 5.97 \times 10^{24} \times 4.06 \times 10^{13} \approx 9.69 \times 10^{37}\ \text{kg m}^2$ .

$L = I\omega \approx 9.69 \times 10^{37} \times 7.27 \times 10^{-5} \approx 7.0 \times 10^{33}\ \text{kg m}^2/\text{s}$

Real Earth is denser at the centre, so its true I is about 17 percent lower, but the order of size is right.

Do not picture the electron as a tiny spinning ball. A classical ball that small would need its surface to move faster than light to carry ħ/2. Spin is a quantum property with no full classical picture.

Units and common errors

ħ has the unit J s, so S does too. Since 1 J = 1 kg m²/s², J s = kg m²/s, the same as $I\omega$ . In dimensions that is [M¹ L² T⁻¹], matching Planck's constant; see the dimensional formula of Planck's constant.

Writing S = sħ. That is the largest z component, not the size of the vector.
Using h in place of ħ. The two differ by a factor of 2π.
Giving the electron s = 1 or mₛ = 0. Its only values are mₛ = ±1/2.

Classical rotation results, including torque and moment of inertia, live on the rotation hub, which also connects spin back to everyday spinning bodies.

Frequently asked questions

What is the formula for spin angular momentum?

S = √(s(s + 1)) ħ for the size, and Sz = mₛħ for the component along a chosen axis, where ħ = h / 2π.

What is the spin angular momentum of an electron?

With s = 1/2, S = (√3 / 2)ħ ≈ 9.13 × 10⁻³⁵ J s, and its z component is ±ħ/2 ≈ ±5.27 × 10⁻³⁵ J s.

What unit is spin angular momentum measured in?

Its unit is J s. Expand the joule and this becomes kg m²/s, exactly the unit of Iω for a spinning wheel.

Is the electron really spinning?

Not like a top. Spin behaves like angular momentum in every measurement, but it is an intrinsic quantum property and cannot be explained by a small rotating sphere.