page_title: Stern–Gerlach experiment
section_title: Experiment using particles with +​1⁄2 or −​1⁄2 spin
hierarchical_section_title: Stern–Gerlach experiment / Description / Experiment using particles with +​1⁄2 or −​1⁄2 spin
caption_attribution_description: Diagram showing the possible spin angular momentum values for 1/2 spin particles (for example, electrons) Italiano: Diagramma che mostra i possibili valori del momento angolare di spin per una particella di spin 1/2 (per esempio, un elettrone)
context_page_description: The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent through a spatially varying magnetic field, which deflected them before they struck a detector screen, such as a glass slide. Particles with non-zero magnetic moment are deflected, due to the magnetic field gradient, from a straight path. The screen reveals discrete points of accumulation, rather than a continuous distribution, owing to their quantized spin. Historically, this experiment was decisive in convincing physicists of the reality of angular-momentum quantization in all atomic-scale systems.
  After its conception by Otto Stern in 1921, the experiment was first successfully conducted by Walther Gerlach in early 1922.
context_section_description: If the experiment is conducted using charged particles like electrons, there will be a Lorentz force that tends to bend the trajectory in a circle. This force can be cancelled by an electric field of appropriate magnitude oriented transverse to the charged particle's path.
  Electrons are spin-​¹⁄₂ particles. These have only two possible spin angular momentum values measured along any axis,  or , a purely quantum mechanical phenomenon. Because its value is always the same, it is regarded as an intrinsic property of electrons, and is sometimes known as "intrinsic angular momentum" (to distinguish it from orbital angular momentum, which can vary and depends on the presence of other particles). If one measures the spin along a vertical axis, electrons are described as "spin up" or "spin down", based on the magnetic moment pointing up or down, respectively.
  To mathematically describe the experiment with spin  particles, it is easiest to use Dirac's bra–ket notation. As the particles pass through the Stern–Gerlach device, they are deflected either up or down, and observed by the detector which resolves to either spin up or spin down. These are described by the angular momentum quantum number , which can take on one of the two possible allowed values, either  or . The act of observing (measuring) the momentum along the  axis corresponds to the operator . In mathematical terms, the initial state of the particles is
  where constants  and  are complex numbers. This initial state spin can point in any direction. The squares of the absolute values 
   and  determine the probabilities that for a system in the initial state  one of the two possible values of  is found after the measurement is made. The constants  and  must also be normalized in order that the probability of finding either one of the values be unity, that is we must ensure that . However, this information is not sufficient to determine the values of  and , because they are complex numbers. Therefore, the measurement yields only the squared magnitudes of the constants, which are interpreted as probabilities.
caption: Spin values for fermions
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