# Spin-1/2

**Spin-1/2** is the property of a quantum particle whose intrinsic angular momentum quantum number is \( s = \tfrac{1}{2} \), so that a measurement of the spin component along any single axis has exactly two possible outcomes, \( +\hbar/2 \) and \( -\hbar/2 \). Electrons, protons, neutrons, and the atoms whose total angular momentum is carried by an unpaired electron, such as silver, are spin-1/2 systems, and the two-state character of such particles was first demonstrated directly by [Otto Stern](https://www.edgepedia.org/otto-stern) and [Walther Gerlach](https://www.edgepedia.org/walther-gerlach) in 1922.<sup>[1](https://pages.hep.wisc.edu/~prepost/407/stern-gerlach/stern.pdf)</sup><sup> • </sup><sup>[2](https://www.physics.wisc.edu/courses/home/spring2018/407/experiments/stern-gerlach/stern_html.html)</sup>

| Key fact | Detail |
|---|---|
| Possible outcomes | A component measurement along any axis yields only \( S_z = \pm\hbar/2 \); there is no zero state for spin one-half.<sup>[3](https://www.feynmanlectures.caltech.edu/III_06.html)</sup><sup> • </sup><sup>[4](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/content/angularmomentum.html)</sup> |
| Beam splitting | A Stern-Gerlach magnet splits a spin-1/2 beam into two beams, in contrast to three beams for spin one.<sup>[3](https://www.feynmanlectures.caltech.edu/III_06.html)</sup> |
| Historical test | The 1922 experiment sent neutral silver atoms through a non-uniform magnetic field and recorded two marks on a cold plate, where classical physics predicted a continuous spread.<sup>[1](https://pages.hep.wisc.edu/~prepost/407/stern-gerlach/stern.pdf)</sup> |
| Operators | The spin operators are \( \hat{S}_i = \frac{\hbar}{2}\bm{\sigma}_i \) in terms of the Pauli matrices; \( \hat{S}^2\ket{s\,m_s} = \hbar^2 s(s+1)\ket{s\,m_s} \).<sup>[4](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/content/angularmomentum.html)</sup> |
| Statistics | Particles with half-integer spin are fermions; those with integer spin are bosons. Electrons have \( s = \tfrac{1}{2} \) and photons \( s = 1 \).<sup>[4](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/content/angularmomentum.html)</sup> |
| Cross-axis probability | A particle prepared in the \( (+z) \) state has amplitude \( 1/\sqrt{2} \) for the \( (+x) \) state, a 50 percent chance, and amplitude \( -1/\sqrt{2} \) for \( (-x) \).<sup>[3](https://www.feynmanlectures.caltech.edu/III_06.html)</sup> |

## The two-state structure

For a particle with \( s = \tfrac{1}{2} \), the magnetic quantum number \( m_s \) takes the two values \( +\tfrac{1}{2} \) and \( -\tfrac{1}{2} \), conventionally called spin up and spin down. These are colloquial labels for the two states; the state is fully classified by \( s \) and \( m_s \) together, written \( \ket{s\,m_s} \).<sup>[4](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/content/angularmomentum.html)</sup> The angular momentum algebra defined by the commutation relations between the operators requires the total angular momentum quantum number to be an integer or a half integer, and the half-integer possibility represents internal angular momentum that has no counterpart in orbital motion.<sup>[5](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup>

The two spin states form a two-dimensional vector space, so any state is a linear combination of spin up and spin down along a chosen axis, and all operators can be written as 2 × 2 matrices. With the \( z \)-axis as reference,<sup>[4](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/content/angularmomentum.html)</sup>

\[ \hat{S}_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \hat{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \hat{S}_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. \]

The three dimensionless matrices are the Pauli spin matrices. The eigenvalues of \( \hat{S}_x \) are also \( \pm\hbar/2 \), as the choice of axis is arbitrary: \( \det(\hat{S}_x - \lambda I) = \lambda^2 - (\hbar/2)^2 \).<sup>[4](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/content/angularmomentum.html)</sup>

**Measurement and preparation.** A Stern-Gerlach apparatus measures the spin component along its field-gradient axis and can also prepare a state: blocking one output beam leaves a pure \( (+z) \) or \( (-z) \) beam. Passing that filtered beam through a second \( z \)-oriented apparatus sends everything out the same port, because the \( (+z) \) and \( (-z) \) states are orthogonal.<sup>[6](https://mitocw.ups.edu.ec/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_02.pdf)</sup> Passing the \( (+z) \) beam through an \( x \)-oriented apparatus instead gives a 50/50 split between \( (+x) \) and \( (-x) \), and for any given atom the output port cannot be predicted.<sup>[7](https://qm.utasphys.cloud.edu.au/01-stern-gerlach/1-1-exp/)</sup> Feynman's calculation gives the amplitudes explicitly: \( C_+ = 1 \) and \( C_- = 0 \) for the \( (+z) \) state yield \( C_+' = 1/\sqrt{2} \) for \( (+x) \) and \( -1/\sqrt{2} \) for \( (-x) \), equal probabilities with opposite phase.<sup>[3](https://www.feynmanlectures.caltech.edu/III_06.html)</sup>

## The Stern-Gerlach experiment

Stern and Gerlach generated a beam of neutral silver atoms by evaporating silver from an oven in a vacuum, collimated it with slits, and sent it through a region with a large non-uniform magnetic field. A field gradient \( \partial B_z/\partial z \) exerts a force \( (\partial B_z/\partial z)\,\mu_z \) on a magnetic moment, so the deflection on a cold metallic plate measures the \( z \)-component of the atoms' magnetic moment. After about 8 to 10 hours of deposition, the plate showed two marks, meaning the silver atoms had only two possible values of \( \mu_z \).<sup>[1](https://pages.hep.wisc.edu/~prepost/407/stern-gerlach/stern.pdf)</sup>

[Classical physics](https://www.edgepedia.org/classical-physics) predicted something different. If the atomic moment could point at any angle \( \theta \) to the field axis, the component \( \mu_z = \mu\cos\theta \) would take every value from \( -\mu \) to \( +\mu \), producing a continuous band on the detector. The original Schrödinger theory, applied without spin, would predict an odd number of states, for example three values \( -\mu, 0, +\mu \). The observed two states are the evidence that electrons carry an intrinsic spin that cannot be removed, with \( \vec{S} = \tfrac{1}{2}\hbar\vec{\sigma} \) and component eigenvalues \( \pm\tfrac{1}{2}\hbar \).<sup>[1](https://pages.hep.wisc.edu/~prepost/407/stern-gerlach/stern.pdf)</sup> The deflections measured in the experiment are consistent with \( S_z = \pm\hbar/2 \), where \( \hbar = h/2\pi \) and \( h = 6.62607015 \times 10^{-34} \ \mathrm{J \cdot Hz^{-1}} \).<sup>[7](https://qm.utasphys.cloud.edu.au/01-stern-gerlach/1-1-exp/)</sup>

The result was interpreted correctly only in 1925, when Goudsmit and Uhlenbeck introduced the spin hypothesis.<sup>[8](https://www.lp.uni-goettingen.de/get/text/8217)</sup> The experiment confirmed two consequences of quantum mechanics: space quantization can occur even in non-periodic systems, and some particles have intrinsic angular momentum and therefore magnetic moment.<sup>[2](https://www.physics.wisc.edu/courses/home/spring2018/407/experiments/stern-gerlach/stern_html.html)</sup>

**Why silver atoms and not electrons.** The silver atom's total angular momentum is half-valued because of its unpaired electron's spin, so the atomic beam splits in two.<sup>[9](https://batelaan.unl.edu/electron-stern-gerlach-magnets/)</sup> A beam of free electrons would not work in the same apparatus: Mott, Bohr, and Pauli argued that Lorentz forces on a charged particle in the magnet's field blur the splitting, and a beam narrow enough to avoid spatially varying Lorentz forces would be so badly diffraction limited that it would no longer be a beam. A longitudinal version, with the field gradient along the beam so that spin-forward and spin-backward electrons separate along the propagation direction, was proposed by Brillouin and later reinstated experimentally by the Batelaan group at the University of Nebraska.<sup>[9](https://batelaan.unl.edu/electron-stern-gerlach-magnets/)</sup>

Modern teaching replications simplify the original setup by using potassium instead of silver, because potassium evaporates at 63.6 °C rather than the 961.9 °C needed for silver.<sup>[1](https://pages.hep.wisc.edu/~prepost/407/stern-gerlach/stern.pdf)</sup>

## Rotation and the transformation of amplitudes

Because the two-state space is small, spin-1/2 is the standard setting for working out how quantum amplitudes transform between coordinate systems. Consider two Stern-Gerlach apparatuses \( S \) and \( T \) with different orientations. There are four amplitudes: from \( (+S) \) or \( (-S) \) into \( (+T) \) or \( (-T) \), which are the four coefficients of the transformation matrix \( R_{ji} \) between the two representations.<sup>[3](https://www.feynmanlectures.caltech.edu/III_06.html)</sup>

The physics does not depend on how the whole apparatus is oriented in space, so \( R_{ji} \) depends only on the rotation carrying \( S \) to \( T \). Knowing the amplitudes of any state with respect to one frame's \( x \), \( y \), and \( z \) axes, one can calculate the fraction entering either beam of an apparatus with any other orientation.<sup>[3](https://www.feynmanlectures.caltech.edu/III_06.html)</sup> A state polarized up along an axis \( A \) defined by angles \( \theta \) and \( \phi \) has computable amplitudes \( C_+ \) and \( C_- \) to be up or down along \( z \).<sup>[3](https://www.feynmanlectures.caltech.edu/III_06.html)</sup>

## Why spin-1/2 matters

The half-integer possibility in the angular momentum algebra has no analog in orbital motion, where no single-valued spherical harmonic represents a half-integer state; it instead describes the internal angular momentum of particles.<sup>[5](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)</sup> This split organizes all particles into two families: integer-spin bosons such as the photon (\( s = 1 \)) and half-integer-spin fermions such as the electron (\( s = \tfrac{1}{2} \)).<sup>[4](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/content/angularmomentum.html)</sup> The two-dimensional state space also makes spin-1/2 the natural physical realization of a quantum bit, or qubit, whose two basis states play the role of the classical 0 and 1 in quantum computing.<sup>[10](https://www.sjsu.edu/people/kenneth.wharton/courses/263/s0/QuantumInformationv2.pdf)</sup>

## References

1. [Stern-Gerlach experiment notes, Physics 407, University of Wisconsin (revised 12/28/06)](https://pages.hep.wisc.edu/~prepost/407/stern-gerlach/stern.pdf)
2. [The Stern-Gerlach Experiment, Advanced Laboratory, Physics 407, University of Wisconsin](https://www.physics.wisc.edu/courses/home/spring2018/407/experiments/stern-gerlach/stern_html.html)
3. [The Feynman Lectures on Physics, Vol. III, Ch. 6: Spin One-Half](https://www.feynmanlectures.caltech.edu/III_06.html)
4. [Angular momentum, Introduction to Quantum Mechanics, TU Delft interactive textbook](https://interactivetextbooks.tudelft.nl/introduction-to-quantum-mechanics/content/angularmomentum.html)
5. [Spin 1/2 and other 2 State Systems, UCSD Physics 130 notes](https://quantummechanics.ucsd.edu/ph130a/130_notes/node28.html)
6. [Quantum Physics II, Lecture Notes 2, MIT OpenCourseWare 8.05](https://mitocw.ups.edu.ec/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_02.pdf)
7. [The Stern-Gerlach experiment, University of Tasmania quantum mechanics course](https://qm.utasphys.cloud.edu.au/01-stern-gerlach/1-1-exp/)
8. [Der Stern-Gerlach-Versuch, Lehrprogramm Physik, Universität Göttingen](https://www.lp.uni-goettingen.de/get/text/8217)
9. [Electron Stern-Gerlach Magnets, Herman Batelaan, University of Nebraska](https://batelaan.unl.edu/electron-stern-gerlach-magnets/)
10. [The Bloch Sphere, course notes, San José State University](https://www.sjsu.edu/people/kenneth.wharton/courses/263/s0/QuantumInformationv2.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum phenomena and measurement › Superposition and quantum interference › Stern–Gerlach experiment as superposition demonstration*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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