Editing Stern–Gerlach experiment

{{Short description|1922 physical experiment demonstrating that atomic spin is quantized}}
{{Use American English|date=January 2019}}
[[File:Stern-Gerlach experiment.svg|thumb|upright=1.35|'''Stern–Gerlach experiment''': Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result]]
{{Quantum mechanics|Topic=Experiments}}

In [[quantum physics]], the '''Stern–Gerlach experiment''' demonstrated that the spatial orientation of [[angular momentum]] is [[quantization (physics)|quantized]]. Thus an [[Atomic spacing|atomic-scale]] system was shown to have intrinsically quantum properties. In the original experiment, silver [[atom]]s were sent through a spatially-varying [[magnetic field]], which [[Deflection (physics)|deflected]] them before they struck a detector screen, such as a [[glass slide]]. Particles with non-zero [[magnetic moment]] were deflected, owing to the magnetic field [[spatial gradient|gradient]], from a straight path. The screen revealed discrete points of accumulation, rather than a continuous distribution,<ref name=SG/> owing to their quantized [[Spin (physics)|spin]]. Historically, this experiment was decisive in convincing physicists of the reality of angular-momentum quantization in all atomic-scale systems.<ref>{{cite book|first1=Allan |last1=Franklin |first2=Slobodan |last2=Perovic |chapter= Experiment in Physics, Appendix 5 |title=The Stanford Encyclopedia of Philosophy |title-link=Stanford Encyclopedia of Philosophy |edition= Winter 2016 |editor-first=Edward N. |editor-last=Zalta |editor-link=Edward N. Zalta |chapter-url=https://plato.stanford.edu/archives/win2016/entries/physics-experiment/app5.html |access-date=2018-08-14}}</ref><ref name="FH2003">
{{cite journal
 |last1=Friedrich |first1=B. |last2=Herschbach |first2=D.
 |title=Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics
 |journal=[[Physics Today]]
 |volume=56 |page=53
 |year=2003
 |doi=10.1063/1.1650229
 |issue=12
 |bibcode = 2003PhT....56l..53F |s2cid=17572089 |doi-access=free
}}</ref><ref>{{cite journal |last1=Castelvecchi |first1=D. |date=2022 |title=The Stern–Gerlach experiment at 100 |url=https://doi.org/10.1038/s42254-022-00436-4 |journal=Nature Reviews Physics |volume=4 |issue=3 |pages=140–142 | doi=10.1038/s42254-022-00436-4|bibcode=2022NatRP...4..140C |s2cid=247184330 |url-access=subscription }}</ref>

After its conception by [[Otto Stern]] in 1921, the experiment was first successfully conducted with [[Walther Gerlach]] in early 1922.<ref name=SG>
{{cite journal
 |last1=Gerlach |first1=W.
 |last2=Stern |first2=O.
 |title=Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld
 |trans-title=The experimental proof of directional quantization in the magnetic field
 |journal=[[Zeitschrift für Physik]]
 |volume=9 |issue=1
 |pages=349–352
 |year=1922
 |doi=10.1007/BF01326983
 |bibcode = 1922ZPhy....9..349G |s2cid=186228677
}}</ref><ref>
{{cite journal
 |last1=Gerlach |first1=W.
 |last2=Stern |first2=O.
 |title=Das magnetische Moment des Silberatoms
 |trans-title=The magnetic moment of the silver atom
 |journal=[[Zeitschrift für Physik]]
 |volume=9 |issue=1
 |pages=353–355
 |year=1922
 |doi=10.1007/BF01326984
 |bibcode = 1922ZPhy....9..353G |s2cid=126109346
}}</ref><ref>
{{cite journal
 |last1=Gerlach |first1=W.
 |last2=Stern |first2=O.
 |title=Der experimentelle Nachweis des magnetischen Moments des Silberatoms
 |trans-title=The experimental proof of the magnetic moment of the silver atom
 |journal=[[Zeitschrift für Physik]]
 |volume=8 |pages=110–111
 |year=1922
 |issue=1
 |doi=10.1007/BF01329580
 |bibcode = 1922ZPhy....8..110G |s2cid=122648402
 |url=https://zenodo.org/record/1525119
}}</ref>

== Description ==
{{See also|Spin quantum number}}
[[File:Quantum spin and the Stern-Gerlach experiment.ogv|thumb|upright=1.5|Video explaining quantum spin versus classical magnet in the Stern–Gerlach experiment]]

The Stern–Gerlach experiment involves sending [[silver]] atoms through an [[Homogeneity and heterogeneity|inhomogeneous]] [[magnetic field]] and observing their deflection. Silver atoms were evaporated using an electric furnace in a vacuum. Using thin slits, the atoms were guided into a flat beam and the beam sent through an inhomogeneous magnetic field before colliding with a metallic plate. The laws of classical  physics predict that the collection of condensed silver atoms on the plate should form a thin solid line in the same shape as the original beam. However, the inhomogeneous magnetic field caused the beam to split in two separate directions, creating two lines on the metallic plate.<ref name="FH2003"/>

The results show that particles possess an intrinsic [[angular momentum]] that is closely analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values. Another important result is that only one component of a particle's spin can be measured at one time, meaning that the measurement of the spin along the z-axis destroys information about a particle's spin along the x and y axis.

The experiment is normally conducted using electrically [[neutral particle]]s such as silver atoms. This avoids the large deflection in the path of a charged particle moving through a magnetic field and allows spin-dependent effects to dominate.<ref>[[Nevill Francis Mott|Mott, N.F.]], [[Harrie Massey|Massey, H.S.W.]] (1965/1971). ''The Theory of Atomic Collisions'', third edition, Oxford University Press, Oxford UK, pp. 214–219, §2, Ch. IX, reprinted in {{cite book|first1=J.A.|last1=Wheeler|author1-link=John Archibald Wheeler|first2=W.H.|last2=Zurek|author2-link=Wojciech H. Zurek|year=1983|title=Quantum Theory and Measurement|publisher=Princeton University Press|location=Princeton NJ|pages= 701–706}}</ref><ref>{{cite journal|title=Comment on "Stern-Gerlach Effect for Electron Beams"|journal=Phys. Rev. Lett.|volume=81|issue=4772|pages=4772|first1=George H. |last1=Rutherford |first2=Rainer |last2=Grobe|doi=10.1103/PhysRevLett.81.4772|year=1997|bibcode = 1998PhRvL..81.4772R }}</ref>

If the particle is treated as a classical spinning [[magnetic dipole]], it will [[Larmor precession|precess]] in a magnetic field because of the torque that the magnetic field exerts on the dipole (see [[Gyroscopic precession|torque-induced precession]]). If it moves through a homogeneous magnetic field, the forces exerted on opposite ends of the dipole cancel each other out and the trajectory of the particle is unaffected. However, if the magnetic field is inhomogeneous then the force on one end of the dipole will be slightly greater than the opposing force on the other end, so that there is a net force which deflects the particle's trajectory. If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be [[Boltzmann distribution|random]] and [[random variable|continuous]]. Each particle would be deflected by an amount proportional to the [[dot product]] of its magnetic moment with the external field gradient, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount. This was a measurement of the quantum [[observable]] now known as [[spin operator|spin]] angular momentum, which demonstrated possible outcomes of a measurement where the observable has a discrete set of values or [[point spectrum]].<ref>{{cite book|first=Asher |last=Peres |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |year=1993 |publisher=[[Kluwer]] |isbn=0-7923-2549-4 |pages=14–18}}</ref>

Although some discrete quantum phenomena, such as [[atomic spectra]], were observed much earlier, the Stern–Gerlach experiment allowed scientists to directly observe separation between discrete quantum states for the first time.

Theoretically, [[angular momentum operator#Quantization|quantum angular momentum ''of any kind'' has a discrete spectrum]], which is sometimes briefly expressed as "angular momentum is quantized".

=== Experiment using particles with +1/2 or −1/2 spin ===
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.

[[File:Quantum projection of S onto z for spin half particles.svg|left|thumb|upright=0.6|Spin values for fermions]]

Electrons are [[spin-1/2]] particles. These have only two possible spin angular momentum values measured along any axis, <math>+\frac{\hbar}{2}</math> or <math>-\frac{\hbar}{2}</math>, 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-1/2 particles, it is easiest to use [[Paul Adrien Maurice Dirac|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 <math>j</math>, which can take on one of the two possible allowed values, either +1/2 or -1/2. The act of observing (measuring) the momentum along the <math>z</math> axis corresponds to the <math>z</math>-axis [[angular momentum operator]], often denoted <math>J_z</math>. In mathematical terms, the initial state of the particles is
:<math>|\psi\rangle = c_1\left|\psi_{j = +\frac{1}{2}}\right\rangle + c_2\left|\psi_{j = -\frac{1}{2}}\right\rangle</math>
where constants <math>c_1</math> and <math>c_2</math> are complex numbers. This initial state spin can point in any direction. The squares of the [[absolute value]]s 
<math>|c_1|^2</math> and <math>|c_2|^2</math> are respectively the probabilities for a system in the state <math>|\psi\rangle</math> to be found in <math>\left|\psi_{j = +\frac{1}{2}}\right\rangle</math> and <math>\left|\psi_{j = -\frac{1}{2}}\right\rangle</math> after the measurement along <math>z</math> axis is made. The constants <math>c_1</math> and <math>c_2</math> must also be normalized in order that the probability of finding either one of the values be unity, that is we must ensure that <math>|c_1|^2 + |c_2|^2 = 1</math>. However, this information is not sufficient to determine the values of <math>c_1</math> and <math>c_2</math>, because they are complex numbers. Therefore, the measurement yields only the squared magnitudes of the constants, which are interpreted as probabilities.

== Sequential experiments ==

If we link multiple Stern–Gerlach apparatuses (the rectangles containing ''S-G''), we can clearly see that they do not act as simple selectors, i.e. filtering out particles with one of the states (pre-existing to the measurement) and blocking the others. Instead they alter the state by observing it (as in [[photon polarization|light polarization]]). In the figure below, x and z name the directions of the (inhomogenous) magnetic field, with the x-z-plane being orthogonal to the particle beam. In the three S-G systems shown below, the cross-hatched squares denote the blocking of a given output, i.e. each of the S-G systems with a blocker allows only particles with one of two states to enter the next S-G apparatus in the sequence.<ref name="Sakurai1985">
{{cite book |last=Sakurai |first=J.-J. |author-link=J. J. Sakurai |title=Modern Quantum Mechanics |title-link=Modern Quantum Mechanics |publisher=[[Addison-Wesley]] |year=1985 |isbn=0-201-53929-2}}</ref>
[[File:Sg-seq.svg|left|640px]]

{{clear}}
[[File:Stern-Gerlach Analyzer Sequential Series E1.png|alt=3D model of 2 S-G analyzers in sequence, showing the path of neutrons. Both analyzers measure the z-axis|thumb|Exp. 1 - Notice that no z- neutrons are detected at the second S-G analyzer]]

=== Experiment 1 ===
The top illustration shows that when a second, identical, S-G apparatus is placed at the exit of the first apparatus, '''only z+ is seen''' in the output of the second apparatus. This result is expected since all particles at this point are expected to have z+ spin, as only the z+ beam from the first apparatus entered the second apparatus.<ref name=":0">{{cite journal |last1=Qinxun |first1=Li |date=June 8, 2020 |title=Stern Gerlach Experiment:Descriptions and Developments |url=https://dokumen.tips/documents/stern-gerlach-experiment-descriptions-and-liqinxunintrotosgepdf-2020-6.html?page=1 |journal=University of Science and Technology of China |pages=2–5 |access-date=24 November 2020}}</ref>
[[File:Stern-Gerlach Analyzer Sequential Series E2.png|alt=3D model of 2 S-G analyzers in sequence, showing the path of neutrons. The first one measures the z-axis spin, and the second one the x-axis spin.|thumb|Exp. 2 - The z-spin is known, now measuring the x-spin.]]

=== Experiment 2 ===
The middle system shows what happens when a different S-G apparatus is placed at the exit of the z+ beam resulting of the first apparatus, the second apparatus measuring the deflection of the beams on the x axis instead of the z axis. The second apparatus produces x+ and x- outputs. Now classically we would expect to have one beam with the x characteristic oriented + and the z characteristic oriented +, and another with the x characteristic oriented - and the z characteristic oriented +.<ref name=":0" />
[[File:Stern-Gerlach Analyzer Sequential Series E3.png|alt=3D model of 3 S-G analyzers in sequence, showing the path of neutrons through them. The first one measures the z-axis spin, and the second one the x-axis spin, and the third one the z-spin again.|thumb|Exp. 3 - Neutrons thought to have only z+ spin are measured again, finding that the z-spin has been 'reset'.]]

=== Experiment 3 ===
The bottom system contradicts that expectation. The output of the third apparatus which measures the deflection on the z axis again shows an '''output of''' '''z-''' as well as z+. Given that the input to the second S-G apparatus consisted '''only of z+''', it can be inferred that a S-G apparatus must be altering the states of the particles that pass through it. This experiment can be interpreted to exhibit the [[uncertainty principle]]: since the angular momentum cannot be measured on two perpendicular directions at the same time, the measurement of the angular momentum on the x direction destroys the previous determination of the angular momentum in the z direction. That's why the third apparatus measures renewed z+ and z- beams like the x measurement really made a clean slate of the z+ output.<ref name=":0" />

== History ==
[[File:SternGerlach2.jpg|thumb|A plaque at the Frankfurt institute commemorating the experiment]]
The Stern–Gerlach experiment was conceived by [[Otto Stern]] in 1921 and performed by him and [[Walther Gerlach]] in [[Frankfurt]] in 1922.<ref name=Sakurai1985/> 
At the time of the experiment, the most prevalent model for describing the [[atom]] was the [[Bohr-Sommerfeld model]],<ref>{{cite journal |last1=Castelvecchi |first1=Davide |title=Just a moment |journal=Nature Physics |date=2020 |volume=16 |issue=9 |page=994 |doi=10.1038/s41567-020-1022-6|bibcode=2020NatPh..16..994C |s2cid=225215678 }}</ref><ref>{{cite book |last1=Kragh |first1=Helge |title=Niels Bohr and the quantum atom: The Bohr model of atomic structure 1913–1925 |date=2012 |publisher=Oxford University Press |url=https://doi.org/10.1093/acprof:oso/9780199654987.003.0004 |chapter=The Bohr–Sommerfeld theory|pages=140–181 |doi=10.1093/acprof:oso/9780199654987.003.0004 |isbn=978-0-19-965498-7 }}</ref> which described [[electrons]] as going around the positively charged [[Atomic nucleus|nucleus]] only in certain discrete [[atomic orbital]]s or [[energy levels]]. Since the electron was [[Quantization (physics)|quantized]] to be only in certain positions in space, the separation into distinct orbits was referred to as [[Old quantum theory#Rotator|space quantization]]. The Stern–Gerlach experiment was meant to test the [[Old quantum theory|Bohr–Sommerfeld hypothesis]] that the direction of the angular momentum of a silver atom is quantized.<ref>
{{cite journal
 |first=O. |last=Stern
 |title=Ein Weg zur experimentellen Pruefung der Richtungsquantelung im Magnetfeld
 |trans-title=A way to experimentally test directional quantization in a magnetic field
 |journal=[[Zeitschrift für Physik]]
 |volume=7 |issue=1
 |pages=249–253
 |year=1921
 |doi=10.1007/BF01332793
 |bibcode = 1921ZPhy....7..249S |s2cid=186234469
}}</ref>
 
The experiment was first performed with an electromagnet that allowed the non-uniform magnetic field to be turned on gradually from a null value.<ref name=SG/> When the field was null, the silver atoms were deposited as a single band on the detecting glass slide. When the field was made stronger, the middle of the band began to widen and eventually to split into two, so that the glass-slide image looked like a lip-print, with an opening in the middle, and closure at either end.<ref>[[Anthony French|French, A.P.]], Taylor, E.F. (1979). ''An Introduction to Quantum Physics'', Van Nostrand Reinhold, London, {{ISBN|0-442-30770-5}}, pp. 428–442.</ref> In the middle, where the magnetic field was strong enough to split the beam into two, statistically half of the silver atoms had been deflected by the non-uniformity of the field.

Note that the experiment was performed several years before [[George Uhlenbeck]] and [[Samuel Goudsmit]] formulated their hypothesis about the existence of [[Spin (physics)|electron spin]] in 1925.<ref>{{cite book |last1=Eisberg |first1=Robert |last2=Resnick |first2=Robert |author-link2=Robert Resnick |title=Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles|publisher=Wiley|url=https://archive.org/details/quantumphysicsat00eisb |url-access=limited |edition=2nd |year=1985  |page=277|isbn=9780471873730}}</ref> Even though the result of the Stern−Gerlach experiment has later turned out to be in agreement with the predictions of quantum mechanics for a spin-1/2 particle, the experimental result was also consistent with the [[Old quantum theory|Bohr–Sommerfeld theory]].<ref>
{{cite journal
 |last=Weinert |first=F.
 |title=Wrong theory—right experiment: The significance of the Stern–Gerlach experiments
 |journal=[[Studies in History and Philosophy of Modern Physics]]
 |volume=26B |issue=1
 |pages=75–86
 |year=1995
 |doi=10.1016/1355-2198(95)00002-B
 |bibcode=1995SHPMP..26...75W
}}</ref>

In 1927, T.E. Phipps and J.B. Taylor reproduced the effect using [[hydrogen]] atoms in their [[ground state]], thereby eliminating any doubts that may have been caused by the use of [[silver]] atoms.<ref>
{{cite journal
 |last1=Phipps |first1=T.E.
 |last2=Taylor |first2=J.B.
 |title=The Magnetic Moment of the Hydrogen Atom
 |journal=[[Physical Review]]
 |volume=29 | issue=2 | pages=309–320
 |year=1927
 |bibcode=1927PhRv...29..309P
 |doi=10.1103/PhysRev.29.309
}}</ref> However, in 1926 the non-relativistic scalar [[Schrödinger equation]] had incorrectly predicted the [[magnetic moment]] of hydrogen to be zero in its ground state. To correct this problem [[Wolfgang Pauli]] considered a spin-1/2 version of the Schrödinger equation using the 3 [[Pauli matrices]] which now bear his name, which was later shown by [[Paul Dirac]] in 1928 to be a consequence of his relativistic [[Dirac equation]].

In the early 1930s Stern, together with [[Otto Robert Frisch]] and [[Immanuel Estermann]] improved the [[molecular beam]] apparatus sufficiently to measure the magnetic moment of the [[proton]], a value nearly 2000 times smaller than the electron moment. In 1931, theoretical analysis by [[Gregory Breit]] and [[Isidor Isaac Rabi]] showed that this apparatus could be used to measure nuclear spin whenever the electronic configuration of the atom was known. The concept was applied by Rabi and Victor W. Cohen in 1934 to determine the <math>3/2</math> spin of [[sodium]] atoms.<ref name=KelloggMillman>{{Cite journal |last1=Kellogg |first1=J. B. M. |last2=Millman |first2=S. |date=1946-07-01 |title=The Molecular Beam Magnetic Resonance Method. The Radiofrequency Spectra of Atoms and Molecules |url=https://link.aps.org/doi/10.1103/RevModPhys.18.323 |journal=Reviews of Modern Physics |language=en |volume=18 |issue=3 |pages=323–352 |doi=10.1103/RevModPhys.18.323 |issn=0034-6861|url-access=subscription }}</ref>

In 1938 Rabi and coworkers inserted an oscillating magnetic field element into their apparatus, inventing [[nuclear magnetic resonance]] spectroscopy.<ref name=":1">
{{cite journal
 |author = Rabi, I.I.
 |author2 = Zacharias, J.R.
 |author3 = Millman, S.
 |author4 = Kusch, P.
 |name-list-style = amp
 |title = A New Method of Measuring Nuclear Magnetic Moment
 |journal = [[Physical Review]]
 |volume = 53
 |date = 1938
 |doi = 10.1103/PhysRev.53.318
 |issue = 4
 |bibcode = 1938PhRv...53..318R
 |pages = 318–327 |doi-access = free
}}</ref><ref>{{Cite journal |last=Hamilton |first=Donald R. |date=1941-12-01 |title=Molecular Beams and Nuclear Moments |url=https://pubs.aip.org/ajp/article/9/6/319/1037623/Molecular-Beams-and-Nuclear-Moments |journal=American Journal of Physics |language=en |volume=9 |issue=6 |pages=319–337 |doi=10.1119/1.1991712 |issn=0002-9505|url-access=subscription }}</ref> By tuning the frequency of the oscillator to the frequency of the nuclear precessions they could selectively tune into each quantum level of the material under study. Rabi was awarded the Nobel Prize in 1944 for this work.<ref>{{Cite web|url=https://www.nobelprize.org/prizes/physics/1944/rabi/biographical/|title=The Nobel Prize in Physics 1944|website=NobelPrize.org}}</ref>

==Importance==

The Stern–Gerlach experiment  was the first direct evidence of angular-momentum quantization in quantum mechanics,<ref>{{Cite journal |last=Johnston |first=Hamish |date=2022-12-01 |title=100 years of the Stern–Gerlach experiment |url=https://iopscience.iop.org/article/10.1088/2058-7058/35/12/30 |journal=Physics World |volume=35 |issue=12 |pages=27–30 |doi=10.1088/2058-7058/35/12/30 |issn=0953-8585|url-access=subscription }}</ref> and it strongly influenced later developments in [[modern physics]]:

*In the decade that followed, scientists showed using similar techniques, that the nuclei of some atoms also have quantized angular momentum.<ref name=":1" /> It is the interaction of this nuclear angular momentum with the spin of the electron that is responsible for the [[hyperfine structure]] of the spectroscopic lines.<ref>
{{cite book
 |first=David | last=Griffiths |author-link=David J. Griffiths
 |title=Introduction to Quantum Mechanics
 |title-link=Introduction to Quantum Mechanics (book)
 |edition=2nd
 |publisher=[[Pearson Prentice Hall]]
 |page=267
 |year=2005
 |isbn=0-13-111892-7
}}
</ref>
* [[Norman F. Ramsey]] later modified the Rabi apparatus to improve its sensitivity (using the [[Ramsey interferometry|separated oscillatory field method]]).<ref name="autobiog">{{cite web |title=Norman F. Ramsey – Autobiography |url=https://www.nobelprize.org/prizes/physics/1989/ramsey/auto-biography/ |access-date=June 13, 2013 |publisher=The Nobel Foundation}}</ref> In the early sixties, Ramsey, H. Mark Goldenberg, and [[Daniel Kleppner]] used a Stern–Gerlach system to produce a beam of polarized hydrogen as the source of energy for the [[hydrogen maser]].<ref>{{Cite journal |last=Ramsey |first=Norman F. |author-link=Norman F. Ramsey |date=1968 |title=The Atomic Hydrogen Maser |url=https://www.jstor.org/stable/27828332 |journal=American Scientist |volume=56 |issue=4 |pages=420–438 |jstor=27828332 |issn=0003-0996}}</ref> This led to developing an extremely stable clock based on a hydrogen maser. From 1967 until 2019, [[2019 revision of the SI|the second was defined]] based on 9,192,631,770&nbsp;Hz hyperfine transition of a [[cesium-133]] atom; the [[atomic clock]] which is used to set this standard is an application of Ramsey's work.<ref>{{cite web |title=Nobel Prize press release |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1989/press.html |access-date=June 13, 2013 |publisher=The Nobel Foundation}}</ref>
* The Stern–Gerlach experiment has become a prototype for ''[[quantum measurement]]'', demonstrating the observation of a discrete value (''[[Eigenvalues and eigenvectors|eigenvalue]])'' of a physical property, previously assumed to be continuous.<ref>{{Cite book|last=Bohm|first=David |author-link=David Bohm |title=Quantum Theory|publisher=Prentice-Hall|year=1951|location=New York|pages=326–330}}</ref><ref>{{Cite book|last=Gottfried|first=Kurt |author-link=Kurt Gottfried |title=Quantum Mechanics|publisher=W. A. Benjamin, Inc.|year=1966|location=New York|pages=170–174}}</ref><ref>{{Cite book|last=Eisberg|first=Robert|title=Fundamentals of Modern Physics|publisher=John Wiley & Sons|year=1961|isbn=0-471-23463-X|location=New York|pages=334–338}}</ref> Entering the Stern–Gerlach magnet, the direction of the silver atom's magnetic moment is indefinite, but when the atom is registered at the screen, it is observed to be at either one spot or the other, and this outcome cannot be predicted in advance. Because the experiment illustrates the character of quantum measurements, ''[[The Feynman Lectures on Physics]]'' use idealized Stern–Gerlach apparatuses to explain the basic mathematics of quantum theory.<ref>{{Cite book |last1=Feynman |first1=Richard P. |author-link=Richard Feynman |title=The Feynman Lectures on Physics |title-link=The Feynman Lectures on Physics |last2=Leighton |first2=Robert B. |author-link2=Robert B. Leighton |last3=Sands |first3=Matthew L. |author-link3=Matthew Sands |date=2006 |publisher=Pearson Addison Wesley |isbn=978-0-8053-9045-2 |edition=Definitive |volume=3 |location=San Francisco, Calif. |chapter=5. Spin One |orig-date=1966 |chapter-url=https://www.feynmanlectures.caltech.edu/III_05.html}}</ref><ref>{{Cite journal |last1=Gale |first1=W. |last2=Guth |first2=E. |last3=Trammell |first3=G. T. |date=1968-01-25 |title=Determination of the Quantum State by Measurements |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.165.1434 |journal=[[Physical Review]] |language=en |volume=165 |issue=5 |pages=1434–1436 |doi=10.1103/PhysRev.165.1434|url-access=subscription }}</ref><ref>{{Cite journal |last1=Ududec |first1=Cozmin |last2=Barnum |first2=Howard |last3=Emerson |first3=Joseph |date=March 2011 |title=Three Slit Experiments and the Structure of Quantum Theory |url=http://link.springer.com/10.1007/s10701-010-9429-z |journal=[[Foundations of Physics]] |language=en |volume=41 |issue=3 |pages=396–405 |arxiv=0909.4787 |bibcode=2011FoPh...41..396U |doi=10.1007/s10701-010-9429-z}}</ref>

== See also ==
* [[Photon polarization]]
* [[Stern–Gerlach Medal]]
* [[German inventors and discoverers]]

== References ==
{{reflist}}

== Further reading ==
* {{cite journal
 |last1=Hsu |first1=B. |last2=Berrondo |first2=M. |last3=Van Huele |first3=J.-F. 
 |title=Stern-Gerlach dynamics with quantum propagators
 |journal=[[Physical Review A]]
 |volume=83 |pages=012109–1–12
 |year=2011
 |doi=10.1103/PhysRevA.83.012109
 |issue=1
 |bibcode = 2011PhRvA..83a2109H |url=https://zenodo.org/record/894868
}}
* {{cite journal |last1=Margalit |first1=Yair |last2=Zhou |first2=Zhifan |last3=Machluf |first3=Shimon |last4=Japha |first4=Yonathan |last5=Moukouri |first5=Samuel |last6=Folman |first6=Ron |date=2019-07-23 |title=Analysis of a high-stability Stern–Gerlach spatial fringe interferometer |journal=[[New Journal of Physics]] |volume=21 |issue=7 |pages=073040 |doi=10.1088/1367-2630/ab2fdc |bibcode=2019NJPh...21g3040M |doi-access=free }}
* {{cite journal
 |last=Reinisch |first=G.
 |title=Stern–Gerlach experiment as the pioneer—and probably the simplest—quantum entanglement test? 
 |journal=[[Physics Letters A]]
 |volume=259 |issue=6 |pages=427–430
 |year=1999
 |doi=10.1016/S0375-9601(99)00472-7
 |bibcode = 1999PhLA..259..427R
}}
* {{cite web|url=https://www.quantamagazine.org/the-often-overlooked-experiment-that-revealed-the-quantum-world-20231205/ |title=The (Often) Overlooked Experiment That Revealed the Quantum World |first=Zack |last=Savitsky |website=[[Quanta Magazine]] |date=2023-12-05}}
* {{cite journal
 |last=Venugopalan |first=A.
 |title=Decoherence and Schrödinger-cat states in a Stern−Gerlach-type experiment
 |journal=[[Physical Review A]]
 |volume=56 |pages=4307–4310
 |year=1997
 |doi=10.1103/PhysRevA.56.4307
 |issue=5
|bibcode = 1997PhRvA..56.4307V
}}
* {{cite book| last=Zwiebach |first=Barton |author-link=Barton Zwiebach |publisher=MIT Press |title=Mastering Quantum Mechanics: Essentials, Theory, and Applications |year=2022 |isbn=978-0-262-04613-8 |chapter=12.2: The Stern–Gerlach experiment }}

== External links ==
{{Commons category}}
* [http://www.if.ufrgs.br/~betz/quantum/SGPeng.htm Stern–Gerlach Experiment Java Applet Animation]
* [http://phet.colorado.edu/simulations/sims.php?sim=SternGerlach_Experiment Stern–Gerlach Experiment Flash Model]
* [http://galileo.phys.virginia.edu/classes/252/Angular_Momentum/Angular_Momentum.html Detailed explanation of the Stern–Gerlach Experiment]
* [http://www.toutestquantique.fr/en/spin Animation, applications and research linked to the spin] (Université Paris Sud)
* [https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/lecture-3-wave-mechanics-cont/ Wave Mechanics and Stern–Gerlach experiment] at [[MIT OpenCourseWare]]

{{Quantum mechanics topics}}
{{Authority control}}

{{DEFAULTSORT:Stern-Gerlach experiment}}
[[Category:Quantum measurement]]
[[Category:Foundational quantum physics]]
[[Category:Physics experiments]]
[[Category:Spintronics]]
[[Category:1922 in science]]
[[Category:Articles containing video clips]]