Spin One-Half

In one sentence: An electron has a built-in spin that, measured along any direction, gives only one of two values, called up or down.

$S_z = m_s\,\hbar, \qquad m_s = \pm\frac{1}{2}, \qquad |\mathbf{S}| = \frac{\sqrt{3}}{2}\,\hbar$

Read it out loud: “S sub z equals m sub s times h-bar, where m sub s is plus or minus one half. Here S sub z is the component of the spin along the z axis, and h-bar is the reduced Planck constant. The length of the spin, written as S with bars around it, equals the square root of three over two, times h-bar. Plus or minus means two answers, one with each sign.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Angular momentum, $\hbar$ and the unit J s: Angular momentum is the measure of turning, the quantity that tells how much something spins or orbits. Its unit is joule seconds, J s. The reduced Planck constant $\hbar$ , said h-bar, is $1.054571817 \times 10^{-34}$ J s, and it is the step size for angular momentum. An electron in an atom has angular momentum from its orbit, and it also has a separate built-in kind called spin. This page is about the built-in kind.
Spin as a name for a property: The word spin is a name for a built-in property, not a literal picture. Physicists chose the name because the property acts like a turning quantity. A tiny ball spinning on its axis is only a picture, and it is wrong when taken literally. An electron has spin even when it is at rest, and the spin does not come from any motion we can see.
Plus or minus, $\pm$ , and fractions: The symbol $\pm$ means plus or minus, and it stands for two answers, one with each sign. So $\pm\frac{1}{2}$ means $+\frac{1}{2}$ or $-\frac{1}{2}$ . A fraction $\frac{1}{2}$ means one divided by two, which is one half, or 0.5. Two halves make a whole, so one half is half of a whole.
Square roots, like $\sqrt{3}$: The square root of 3, written $\sqrt{3}$ , is the number that multiplied by itself gives 3. It is about 1.732, since $1.732 \times 1.732$ is about 3. Dividing by 2 gives $\frac{\sqrt{3}}{2} \approx 0.866$ . A square root of a fraction works the same way, so $\sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$ .
The electron: An electron is a tiny particle with a negative electric charge. Electrons surround the nucleus of an atom and move in its orbitals. An electron’s mass is about 1/1836 of a proton’s mass, and it is one of the basic particles that make up matter. Its spin is one of its basic properties, like its charge and its mass.
Magnetic field: A magnetic field is the region around a magnet or an electric current where a magnet feels a push or a turn. Its strength is measured in tesla, written T. A tiny magnet inside an atom lines up with a magnetic field. If the field is stronger on one side than the other, it pushes the tiny magnets toward one side or the other, depending on their direction.
Cosine and inverse cosine: The cosine of an angle, written $\cos\theta$ , is a ratio that always lies between $-1$ and 1. Its inverse, written $\arccos$ (arc cosine), turns a ratio back into an angle. For example $\cos 60^\circ = 0.5$ , so $\arccos(0.5) = 60^\circ$ . On a calculator, the arccos key finds the angle from the ratio.
Powers of ten, like $10^{-34}$ and $10^{-35}$: Very small numbers are written as a short number times a power of ten. The exponent says how many times to divide by 10. The power $10^{-35}$ is a decimal point followed by 34 zeros and then a 1. Multiplying powers adds exponents, and dividing subtracts them. For example, $1.054 \times 10^{-34}$ divided by 2 is $0.527 \times 10^{-34}$ , which tidies to $5.27 \times 10^{-35}$ .

What each symbol means

Symbol	Meaning
$S_z$	The component of the spin along the chosen z axis. It is measured in joule seconds (J s), like $\hbar$ . Only this component has a sharp, measurable value, and that value is either $+\frac{\hbar}{2}$ or $-\frac{\hbar}{2}$ . Its size is $5.27 \times 10^{-35}$ J s.
$m_s$	The spin projection number. For an electron it is $+\frac{1}{2}$ or $-\frac{1}{2}$ . It has no unit, because it is a pure number. Its sign tells you whether the spin points along the positive or the negative z direction.
$\|\mathbf{S}\|$	The length of the spin vector, in joule seconds (J s). It equals $\frac{\sqrt{3}}{2}\hbar$ , which is about $0.866\hbar$ , or about $9.13 \times 10^{-35}$ J s. The bars mean length, so this is a size with no direction.
$\hbar$	The reduced Planck constant, $h/2\pi = 1.054571817 \times 10^{-34}$ J s. It is the step size for spin, and its unit is joule seconds, the same unit as angular momentum. It is tiny because quantum effects are tiny for everyday objects.

The formula piece by piece

$s = \frac{1}{2}$: The spin number of an electron. It is one half, a fixed property of the particle, and it is the same for every electron.
$m_s$: The projection number, which takes only the two values $\pm\frac{1}{2}$ . Nothing in between is allowed, and that is what makes spin different from everyday turning.
$S_z = m_s\,\hbar$: The z component, which is the projection number times $\hbar$ . With $m_s = +\frac{1}{2}$ it is $+\frac{\hbar}{2}$ , and with $m_s = -\frac{1}{2}$ it is $-\frac{\hbar}{2}$ .
$\sqrt{s(s+1)}$: The square root of $s$ times $s$ plus one. For an electron, $s(s+1) = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}$ , and the square root of $\frac{3}{4}$ is $\frac{\sqrt{3}}{2}$ .
$|\mathbf{S}| = \frac{\sqrt{3}}{2}\hbar$: The total length of the spin, in steps of $\hbar$ . It is about $0.866\hbar$ , which is larger than the largest z value, $0.5\hbar$ , so the spin can never point exactly along z.
$S_z = m_s\,\hbar, \qquad m_s = \pm\frac{1}{2}, \qquad |\mathbf{S}| = \frac{\sqrt{3}}{2}\,\hbar$: An electron’s spin has only two sharp values along any chosen axis, plus or minus half of $\hbar$ , and its total length is $\frac{\sqrt{3}}{2}$ times $\hbar$ .

What the formula is doing, and why

What spin is. Electrons carry an intrinsic angular momentum called spin. The spin number $s = \frac{1}{2}$ for an electron is a fixed property, like its charge and its mass. The word spin is a label. It does not mean a tiny ball turning on its axis, because that picture does not match the measured values. Spin is part of the electron even when the electron is at rest.

Why only two values? Measured along any chosen axis, the z axis for example, spin gives only two values: $S_z = +\frac{\hbar}{2}$ or $S_z = -\frac{\hbar}{2}$ . These are the two values of $m_s$ multiplied by $\hbar$ . The count works like this: a spin number $s$ gives $2s + 1$ values along an axis, and $2 \times \frac{1}{2} + 1 = 2$ . Electrons have only two values, and there is no value in between.

Why the length is bigger than the z value. The total length of the spin is $\sqrt{s(s+1)}\,\hbar = \frac{\sqrt{3}}{2}\hbar \approx 0.866\,\hbar$ . This is larger than the largest z value, $\frac{\hbar}{2} = 0.5\,\hbar$ . The same rule gives the length of orbital angular momentum, with $\ell$ in place of $s$ . So spin can never point exactly along the chosen axis, because the full length is bigger than its z part.

The experiment. In 1922 Otto Stern and Walther Gerlach sent a beam of silver atoms through a magnetic field that was stronger on one side than on the other. A tiny magnet inside each atom is pushed up or down by that field. A classical spinning magnet would have given a smooth spread of directions. Instead the beam split into two separate spots. The two spots came from the spin of the outermost electron of each silver atom, which can point in one of two directions.

Why one half? The Dirac equation, the relativistic wave equation for the electron, gives spin one-half naturally. Particles with half-integer spin, including electrons, protons and neutrons, are called fermions. Fermions follow the Pauli exclusion principle, which means two identical electrons cannot share the same full set of quantum labels.

An everyday picture

Think of a light switch with exactly two positions, up and down. Checking spin along an axis is like checking the switch: you always find one of the two positions, and nothing in between. The switch can be thought of as pointing in any direction, but each check along one chosen direction still gives only up or down.

Worked example: How far the spin is tilted from the z axis

An electron has spin $s = \frac{1}{2}$ , so its length is $|\mathbf{S}| = \frac{\sqrt{3}}{2}\hbar$ . In the most extreme case its z component is $S_z = +\frac{\hbar}{2}$ . Using the classical vector picture, find the angle between the spin arrow and the z axis. This is a picture to help you think, not a measured direction.

Write the cosine rule for the angle. In a vector picture, the cosine of the angle between the arrow and the z axis is the z part divided by the full length: $\cos\theta = \frac{S_z}{|\mathbf{S}|}$ . Dividing a part by the whole gives a ratio that tells how much of the arrow lies along z.
Put in the values and simplify. $\cos\theta = \frac{\hbar/2}{(\sqrt{3}/2)\hbar}$ . The $\hbar$ cancels, leaving $\frac{1/2}{\sqrt{3}/2}$ . Dividing by a fraction is the same as multiplying by its flip, so this is $\frac{1}{2} \times \frac{2}{\sqrt{3}}$ . The two 2s cancel, leaving $\frac{1}{\sqrt{3}} \approx 0.5774$ .
Take the inverse cosine. To find the angle from the ratio, use arccos, the inverse cosine: $\theta = \arccos(0.5774) \approx 54.7^\circ$ . On a calculator, press the arccos key after the number 0.5774. The answer comes out in degrees when the calculator is set to degrees.
Check the ratio. The largest z value is $0.5\,\hbar$ and the length is $0.866\,\hbar$ . The z value is $\frac{0.5}{0.866} \approx 0.577$ , or about 58 percent of the length, which matches the cosine we found.
Say what the angle means. The tilt of 54.7 degrees describes the classical arrow only. The measured values are still only $+\frac{\hbar}{2}$ or $-\frac{\hbar}{2}$ along z. The components along the other axes have no sharp values at the same time.

Answer: In the vector picture the spin is tilted about 54.7 degrees from the z axis. This angle is a picture to help you think, not a measured direction. Only the z component has a sharp value, and the spin has no sharp direction.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Thinking spin is a literal spinning ball. The name is a label, and the spin is an intrinsic property with fixed values.
Using $m_s = 0$ or $m_s = 1$ for an electron. For spin one-half the only values are $+\frac{1}{2}$ and $-\frac{1}{2}$ .
Taking the length of the spin to be $\frac{\hbar}{2}$ . The length is $\frac{\sqrt{3}}{2}\hbar$ , and $\frac{\hbar}{2}$ is only the largest z value.
Assuming spin can be sharp along two axes at once. A sharp value along z means the spin along x is not sharp.

Where you will meet it

Electron arrangements in atoms, where each orbital holds at most two electrons, one for each spin value.
Magnetic resonance imaging (MRI) and nuclear magnetic resonance (NMR), which use the spin of hydrogen nuclei, the centers of hydrogen atoms, which are single protons and also have spin one-half.
Quantum computing, where a qubit is often built from a spin one-half system and drawn on the Bloch sphere.
The Stern-Gerlach experiment and other spin measurements in quantum mechanics textbooks.

Questions people ask

Does spin mean the electron rotates?

Not in the everyday sense. Spin is an angular momentum built into the electron, and its measured values are fixed. A rotating ball picture is misleading if taken literally.

Why does an electron have spin one-half and not a whole number?

The Dirac equation for the electron gives spin one-half. Particles with half-integer spin, such as $\frac{1}{2}$ , $\frac{3}{2}$ and $\frac{5}{2}$ , are called fermions, and they obey the Pauli exclusion principle, so two identical fermions cannot share the same full set of quantum labels.

Why did the Stern-Gerlach beam split into two spots?

A classical spinning magnet would give a continuous spread of directions. Two spots mean that only two values of the spin component along the field direction are possible.

Orbital Angular Momentum Quantization: An electron’s turning motion around the center of an atom can only take certain fixed amounts, like steps on a staircase.
Zeeman Effect: A magnetic field shifts an atom’s energy levels by small equal steps, so one spectral line splits into several.
Bloch Sphere: The Bloch sphere is a round picture of a qubit: every possible state is one point on its surface, and the top and bottom points are 0 and 1.
Qubit State: A qubit is a quantum bit that can be a mix of 0 and 1, and two numbers set how likely each answer is when you check it.