KAHIBARO
Table of Contents
Intrinsic Angular Momentum
In quantum mechanics, spin is a form of angular momentum carried by particles. It is called intrinsic angular momentum because it belongs to the particle itself. It is not caused by the particle literally spinning like a tiny ball. That classical picture is tempting, but it is misleading.
Spin is a purely quantum property. Like mass and electric charge, it is one of the basic characteristics of a particle. Electrons, protons, neutrons, and many other particles all have spin.
Why Spin Matters
Spin is important because it affects how particles behave in magnetic fields, how they combine with each other, and how quantum states are organized. Many physical phenomena depend on spin, including atomic structure, magnetism, and the arrangement of electrons in atoms.
A particle with spin behaves as if it has angular momentum even when it is not moving through space in an orbit. Orbital angular momentum and spin angular momentum are different contributions to total angular momentum.
Quantization of Spin
In classical physics, angular momentum can vary continuously. In quantum mechanics, spin is quantized. This means only certain values are allowed.
The total spin angular momentum has magnitude
$$
|\mathbf{S}| = \sqrt{s(s+1)}\,\hbar
$$
where $s$ is the spin quantum number.
Different particles have different values of $s$. For example:
| Particle | Spin quantum number $s$ |
|---|---|
| Electron | $\frac{1}{2}$ |
| Proton | $\frac{1}{2}$ |
| Neutron | $\frac{1}{2}$ |
| Photon | $1$ |
| Alpha particle | $0$ |
The component of spin along a chosen axis, usually the $z$ axis, is also quantized:
$$
S_z = m_s \hbar
$$
where $m_s$ can take the values
$$
m_s = -s, -s+1, \dots, s-1, s
$$
So a spin $\frac{1}{2}$ particle has only two possible results for a measurement of $S_z$:
$$
m_s = +\frac{1}{2}, \quad -\frac{1}{2}
$$
That gives
$$
S_z = +\frac{\hbar}{2} \quad \text{or} \quad -\frac{\hbar}{2}
$$
For a particle with spin quantum number $s$,
$$
|\mathbf{S}| = \sqrt{s(s+1)}\,\hbar
$$
and the allowed measured values along one axis are
$$
S_z = m_s \hbar, \quad m_s = -s, -s+1, \dots, s
$$
A spin $\frac{1}{2}$ particle has exactly two possible outcomes along any chosen axis.
Spin States
For a spin $\frac{1}{2}$ particle, the two basic spin states along the $z$ axis are often written as
$$
|+\rangle \quad \text{and} \quad |-\rangle
$$
or
$$
|\uparrow\rangle \quad \text{and} \quad |\downarrow\rangle
$$
These represent spin up and spin down relative to the chosen axis.
A general spin state can be a superposition of these two states:
$$
|\psi\rangle = a|\uparrow\rangle + b|\downarrow\rangle
$$
where $a$ and $b$ are complex numbers. The probabilities of measuring spin up or spin down are related to $|a|^2$ and $|b|^2$, with normalization
$$
|a|^2 + |b|^2 = 1
$$
This means the particle does not have to be only up or only down before measurement. It can exist in a quantum combination of both.
Measuring Spin
When we measure spin along one axis, we get one of the allowed quantized values. For a spin $\frac{1}{2}$ particle, a measurement along the $z$ axis gives either $+\hbar/2$ or $-\hbar/2$.
If the particle is prepared in a definite spin up state along $z$, then measuring along $z$ again gives spin up with certainty. But if we measure along a different axis, such as $x$, the result is not definite. Quantum mechanics predicts probabilities.
This shows that spin components along different axes are not simultaneously known with exact certainty.
Spin and Magnetic Moment
Charged particles with spin often have a magnetic moment. This means they interact with magnetic fields. For an electron, the spin magnetic moment is proportional to its spin:
$$
\boldsymbol{\mu} \propto \mathbf{S}
$$
Because of this, a magnetic field can distinguish different spin orientations. This is one reason spin can be measured experimentally.
The details of magnetic moment and magnetic interactions are important in atomic physics and electromagnetism, but here the key point is simple. Spin is not just an abstract label. It has real physical effects.
Spin One Half as a Two-State System
Spin $\frac{1}{2}$ is one of the simplest quantum systems. It is a two-state system. This makes it very useful for understanding quantum ideas.
We can represent the two basis states using column vectors:
$$
|\uparrow\rangle =
\begin{pmatrix}
1 \\ 0
\end{pmatrix},
\qquad
|\downarrow\rangle =
\begin{pmatrix}
0 \\ 1
\end{pmatrix}
$$
Then a general state is
$$
|\psi\rangle =
\begin{pmatrix}
a \\ b
\end{pmatrix}
$$
This matrix form is very convenient for calculations.
Spin Operators and Pauli Matrices
For spin $\frac{1}{2}$ particles, the spin operators can be written using the Pauli matrices. These are
$$
\sigma_x =
\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix},
\qquad
\sigma_y =
\begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix},
\qquad
\sigma_z =
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
$$
The spin operators are
$$
S_x = \frac{\hbar}{2}\sigma_x,\qquad
S_y = \frac{\hbar}{2}\sigma_y,\qquad
S_z = \frac{\hbar}{2}\sigma_z
$$
For example, applying $S_z$ to the basis states gives
$$
S_z |\uparrow\rangle = \frac{\hbar}{2} |\uparrow\rangle
$$
and
$$
S_z |\downarrow\rangle = -\frac{\hbar}{2} |\downarrow\rangle
$$
So these basis states are eigenstates of $S_z$.
For spin $\frac{1}{2}$ particles,
$$
S_x = \frac{\hbar}{2}\sigma_x,\qquad
S_y = \frac{\hbar}{2}\sigma_y,\qquad
S_z = \frac{\hbar}{2}\sigma_z
$$
The states $|\uparrow\rangle$ and $|\downarrow\rangle$ are eigenstates of $S_z$ with eigenvalues $+\hbar/2$ and $-\hbar/2$.
Noncommuting Spin Components
An important quantum feature is that spin components in different directions do not commute. Their operators satisfy relations such as
$$
[S_x, S_y] = i\hbar S_z
$$
and similarly for cyclic permutations.
This means that knowing one component exactly prevents exact knowledge of another component. This is not due to poor measurement tools. It is a basic property of quantum systems.
Visualizing Spin
Spin is difficult to picture classically, but we can still build intuition. For spin $\frac{1}{2}$, imagine that any measurement along a chosen axis gives one of two answers, up or down. Changing the axis changes the probabilities.
A useful modern picture is the Bloch sphere, where a spin $\frac{1}{2}$ state is represented by a point on a sphere. The north and south poles correspond to $|\uparrow\rangle$ and $|\downarrow\rangle$ along one axis. Other points correspond to superpositions. This is a geometric way to think about spin states.
Example of Allowed Spin Values
The table below shows how many possible values of $m_s$ exist for several spins.
| Spin quantum number $s$ | Allowed $m_s$ values | Number of possible values |
|---|---|---|
| $0$ | $0$ | 1 |
| $\frac{1}{2}$ | $-\frac{1}{2}, +\frac{1}{2}$ | 2 |
| $1$ | $-1, 0, +1$ | 3 |
| $\frac{3}{2}$ | $-\frac{3}{2}, -\frac{1}{2}, +\frac{1}{2}, +\frac{3}{2}$ | 4 |
In general, the number of allowed spin projection values is
$$
2s + 1
$$
A particle with spin quantum number $s$ has
$$
2s + 1
$$
possible values of the spin component along any chosen axis.
A Simple Measurement Picture
Suppose an electron is in the state
$$
|\psi\rangle = \frac{1}{\sqrt{2}} |\uparrow\rangle + \frac{1}{\sqrt{2}} |\downarrow\rangle
$$
If we measure $S_z$, the probability of getting $+\hbar/2$ is
$$
\left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}
$$
and the probability of getting $-\hbar/2$ is also
$$
\frac{1}{2}
$$
So the outcomes are equally likely.
Spin Compared with Orbital Angular Momentum
Both spin and orbital angular momentum follow quantum rules and are described by angular momentum operators. But they are physically different.
Orbital angular momentum comes from motion through space, such as an electron moving around a nucleus. Spin does not come from spatial motion of that kind. It is intrinsic.
Still, both have quantized magnitudes and quantized components, and both contribute to total angular momentum.
Simple Diagram of Spin Measurement
Key Idea to Remember
Spin is an intrinsic quantum angular momentum. It is quantized, it has discrete measured components, and for spin $\frac{1}{2}$ particles it leads to a fundamental two-state system. This makes spin one of the clearest and most important examples of how quantum mechanics differs from classical physics.