From Spin Experiments to Spin Formalism

Spin entered quantum mechanics through evidence that did not fit ordinary orbital motion: discrete Stern–Gerlach splitting, anomalous Zeeman patterns, atomic doublets, and Pauli’s two-valuedness. The modern result is a compact formalism: a spin-$1/2$ system is described by a two-dimensional Hilbert space, spin components are represented by Pauli matrices, and rotations act through $SU(2)$.

Begin with Stern–Gerlach Experiment for the apparatus, use What Spin Is and Is Not for its physical meaning, and study Spin-1/2 Hilbert Space.

Spin Projection and Beam Deflection

The Stern–Gerlach experiment sends a neutral atomic beam through an inhomogeneous magnetic field. A classical distribution of magnetic-moment orientations suggests a continuous smear of deflections. The original silver beam resolved two components. In the modern electronic ground-state description, a closed shell has zero net angular momentum and the outer electron gives $J=1/2$. This is an atomic magnetic-moment measurement, not a beam of free electrons.

An inhomogeneous magnetic field converts magnetic-moment projection into spatial deflection. In the spin-$1/2$ idealization, a single analyzer has two output beams associated with two eigenvalues of the measured spin component.

In modern notation, an apparatus oriented along a unit vector $\hat{\mathbf n}$ measures the spin component

$$S_{\hat{\mathbf n}} = \hat{\mathbf n}\cdot\mathbf S.$$

For spin-$1/2$,

$$S_{\hat{\mathbf n}} = \frac{\hbar}{2} \hat{\mathbf n}\cdot\boldsymbol\sigma,$$

so the possible outcomes are

$$\pm\frac{\hbar}{2}.$$

The sign of the spatial deflection also depends on $\boldsymbol\mu=\gamma\mathbf S$ and the field gradient. For an electron $\gamma<0$, so “upper beam” need not mean positive spin. The key change from a classical picture is not merely that two values appear. It is that each apparatus axis defines a different observable. A state with a sharp $z$ projection is generally not sharp along $x$. Sequential Stern–Gerlach experiments therefore reveal both discreteness and noncommuting measurement axes.

Two-Level Systems

A spin-$1/2$ Hilbert space is two-dimensional. Choosing the $S_z$ eigenbasis,

$$\lvert +z\rangle = \begin{pmatrix}1\\0\end{pmatrix}, \qquad \lvert -z\rangle = \begin{pmatrix}0\\1\end{pmatrix}.$$

A general pure state is

$$\lvert\psi\rangle = \alpha\lvert +z\rangle + \beta\lvert -z\rangle,$$

with

$$\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1.$$

The probabilities for a $z$-axis measurement are

$$P(+z)=\lvert\alpha\rvert^2, \qquad P(-z)=\lvert\beta\rvert^2.$$

The relative phase between $\alpha$ and $\beta$ is physically important. It affects measurements along axes other than $z$, even though an overall global phase does not change the physical ray. This is why a two-level system is richer than a classical bit with two possible labels.

Spin is the cleanest historical entrance, but the same two-dimensional mathematics appears in many places: two atomic levels, two localized wells, polarization qubits, superconducting circuit states, and effective avoided crossings. Explore other two-level realizations in Two-Level Systems and Spin-1/2 as a Canonical System.

Spin Projection

For any measurement direction $\hat{\mathbf n}$, the spin-up and spin-down projectors can be written

$$P_\pm(\hat{\mathbf n}) = \frac12 \left( I \pm \hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

Given a normalized state $\lvert\psi\rangle$, the probabilities are

$$P(\pm\hat{\mathbf n}) = \langle\psi| P_\pm(\hat{\mathbf n}) |\psi\rangle.$$

This formula is the formal version of the two-spot experiment. The magnet orientation chooses $\hat{\mathbf n}$; the spin state determines the probabilities; the measurement output is one of the two eigenvalues.

For example, the $x$-axis eigenstates are

$$\lvert +x\rangle = \frac{ \lvert +z\rangle+\lvert -z\rangle }{\sqrt2}, \qquad \lvert -x\rangle = \frac{ \lvert +z\rangle-\lvert -z\rangle }{\sqrt2}.$$

Thus a state prepared as $\lvert +z\rangle$ gives equal probabilities for $+x$ and $-x$. This is not ignorance about a hidden classical $x$ label inside the same formalism. It is the Born rule applied to a different measurement basis.

To calculate the effect of measuring and then selecting an output, use Projective Measurement and Sequential Measurements.

Pauli Matrices

The Pauli Matrices obey

$$\sigma_i\sigma_j=\delta_{ij}I+i\sum_k\epsilon_{ijk}\sigma_k.$$

With $S_i=\hbar\sigma_i/2$, this gives the angular-momentum commutator $[S_i,S_j]=i\hbar\sum_k\epsilon_{ijk}S_k$. It also lets us expand every Hermitian two-by-two Hamiltonian as

$$H=c_0I+\mathbf b\cdot\boldsymbol\sigma,$$

with real $c_0$ and $\mathbf b$, all with units of energy.

The vector $\mathbf b$ behaves like an effective field in the two-level Hilbert space. That form appears in spin magnetic resonance, avoided crossings, driven atoms, and qubit Hamiltonians. The detailed matrix treatment belongs to Pauli Matrices and Two-State Hamiltonians.

SU(2) Rotations

Spin is not merely a two-outcome measurement label. It is tied to spatial rotations. A spin-$1/2$ rotation by angle $\theta$ about $\hat{\mathbf n}$ is represented by

$$U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i}{\hbar} \theta\,\hat{\mathbf n}\cdot\mathbf S \right) = \exp\left( -\frac{i}{2} \theta\,\hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

The half-angle is a signature of the relation between $SU(2)$ and $SO(3)$. A $2\pi$ rotation multiplies a spinor by $-1$, while a $4\pi$ rotation returns it to itself. For a single isolated ray, the sign is not observable. If a rotation acts on only one arm of a coherent interferometer, however, the minus sign becomes a relative phase between arms and can shift the output fringes.

The angular-momentum algebra is the shared structure behind orbital angular momentum, spin, and rotations. The spinor-specific treatment is Spin Rotations, and the broader algebraic home is Angular Momentum Algebra.

Qubit Language

A qubit is an abstract two-dimensional quantum system. Spin-$1/2$ supplies the prototype, but the word “qubit” does not mean “electron spin” by definition. It means the system has a chosen two-dimensional computational subspace with coherent superpositions, controlled operations, and measurements.

The Bloch-sphere parameterization makes the connection explicit. A pure spinor can be written, up to global phase, as

$$\lvert\psi\rangle = \cos\frac{\theta}{2}\lvert0\rangle + e^{i\phi} \sin\frac{\theta}{2}\lvert1\rangle.$$

The corresponding Bloch vector is

$$\mathbf r = (\sin\theta\cos\phi,\, \sin\theta\sin\phi,\, \cos\theta).$$

For density operators,

$$\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right), \qquad \lvert\mathbf r\rvert\le 1.$$

This language unifies spin measurements, two-level atoms, driven systems, and quantum-information gates. Its danger is overfamiliarity: the Bloch sphere is a representation of a two-dimensional Hilbert space, not a literal sphere on which a microscopic arrow always points. Use Bloch Sphere for spinors and Bloch Sphere: Wave-Mechanics Perspective for broader two-level systems.

Combining the density operator with the spin projector gives a useful measurement formula:

$$P(\pm\hat{\mathbf n})=\operatorname{Tr}(\rho P_\pm) =\frac{1\pm\mathbf r\cdot\hat{\mathbf n}}2.$$

Thus $\mathbf r=0$ gives equal outcomes on every axis, whereas a unit Bloch vector gives one certain outcome along that vector. The same formula handles pure and mixed states without introducing a hidden classical spin direction.

Exercises

1. A spin-$1/2$ state is $\lvert +z\rangle$. What are the probabilities for measuring $S_x=\pm\hbar/2$?

Solution

Use

$$\lvert +z\rangle = \frac{ \lvert +x\rangle+\lvert -x\rangle }{\sqrt2}.$$

The amplitudes for $+x$ and $-x$ are both $1/\sqrt2$, so

$$P(+x)=\frac12, \qquad P(-x)=\frac12.$$

2. Show that $P_+(\hat z)=(I+\sigma_z)/2$ projects onto $\lvert +z\rangle$ in the standard basis.

Solution

In the standard basis,

$$I+\sigma_z = \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix} + \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} = \begin{pmatrix} 2&0\\ 0&0 \end{pmatrix}.$$

Therefore

$$P_+(\hat z) = \frac12(I+\sigma_z) = \begin{pmatrix} 1&0\\ 0&0 \end{pmatrix},$$

which leaves $\begin{pmatrix}1\\0\end{pmatrix}$ unchanged and annihilates $\begin{pmatrix}0\\1\end{pmatrix}$.

3. Why is a qubit not the same thing as a classical bit with an unknown value?

Solution

A classical bit has one of two values, even if an observer is ignorant of which one. A qubit state can be a coherent superposition $\alpha\lvert0\rangle+\beta\lvert1\rangle$ with a relative phase. That phase affects measurements in other bases and enables interference. A mixed state can represent ignorance, but a pure superposition is not merely an unknown classical bit value.

4. A state has Bloch vector $\mathbf r=(0,0,0.6)$. Find its $+z$ and $+x$ probabilities. Is it pure?

Solution

The probabilities are $(1+0.6)/2=0.8$ and $(1+0)/2=0.5$. Its Bloch-vector length is less than one, so it is mixed. Its purity is $\operatorname{Tr}\rho^2=(1+0.6^2)/2=0.68$.

References

• W. Gerlach and O. Stern, “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld,” Zeitschrift für Physik 9, 349-352, 1922.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• W. Pauli, “Über den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren,” Zeitschrift für Physik 31, 765-783, 1925.
• W. Pauli, “Zur Quantenmechanik des magnetischen Elektrons,” Zeitschrift für Physik 43, 601-623, 1927.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• G. E. Uhlenbeck and S. Goudsmit, “Spinning Electrons and the Structure of Spectra,” Nature 117, 264-265, 1926.