Part of a series on |

Quantum mechanics |
---|

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,^{ [1] } 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.^{ [2] }^{ [3] }

- Description
- Experiment using particles with +1⁄2 or −1⁄2 spin
- Sequential experiments
- Experiment 1
- Experiment 2
- Experiment 3
- History
- Importance
- See also
- References
- Further reading
- External links

After its conception by Otto Stern in 1921, the experiment was first successfully conducted by Walther Gerlach in early 1922.^{ [1] }^{ [4] }^{ [5] }

The Stern–Gerlach experiment involves sending a beam of silver atoms through an inhomogeneous magnetic field and observing their deflection.

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 particles 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.^{ [6] }^{ [7] }

If the particle is treated as a classical spinning magnetic dipole, it will precess in a magnetic field because of the torque that the magnetic field exerts on the dipole (see torque-induced precession).^{[ vague ]} 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 random and continuous. Each particle would be deflected by an amount proportional to its magnetic moment, 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 angular momentum, which demonstrated possible outcomes of a measurement where the observable has a discrete set of values or point spectrum.

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 in the history of science.

Theoretically, quantum angular momentum *of any kind* has a discrete spectrum, which is sometimes briefly expressed as "angular momentum is quantized".

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 .^{[ specify ]} 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.

This section may be confusing or unclear to readers. In particular, a detailed explanation of the picture may be beneficial to many readers.(February 2018) (Learn how and when to remove this template message) |

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 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.^{ [8] }

The top illustration shows that when a second, identical, S-G apparatus is placed at the exit of the z+ beam resulting of the first apparatus, **only z+ is seen** in the output of the second apparatus. This result is expected since all neutrons at this point are expected to have z+ spin, as only the z+ beam from the first apparatus entered the second apparatus.^{ [9] }

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 +.^{ [9] }

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.^{ [9] }

The Stern–Gerlach experiment was conceived by Otto Stern in 1921 and performed by him and Walther Gerlach in Frankfurt in 1922.^{ [8] } At the time, Stern was an assistant to Max Born at the University of Frankfurt's Institute for Theoretical Physics,^{[ citation needed ]} and Gerlach was an assistant at the same university's Institute for Experimental Physics.^{[ citation needed ]}

At the time of the experiment, the most prevalent model for describing the atom was the Bohr model,^{[ citation needed ]} which described electrons as going around the positively charged nucleus only in certain discrete atomic orbitals or energy levels. Since the electron was quantized to be only in certain positions in space, the separation into distinct orbits was referred to as space quantization. The Stern–Gerlach experiment was meant to test the Bohr–Sommerfeld hypothesis that the direction of the angular momentum of a silver atom is quantized.^{ [10] }

Note that the experiment was performed several years before Uhlenbeck and Goudsmit formulated their hypothesis of the existence of the electron spin.^{[ citation needed ]} 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 experiment should be seen as a corroboration of the Bohr–Sommerfeld theory.^{ [11] }

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.^{ [12] } However, in 1926 the non-relativistic Schrödinger equation had incorrectly predicted the magnetic moment of hydrogen to be zero in its ground state. To correct this problem Wolfgang Pauli introduced "by hand", so to speak, the 3 Pauli matrices which now bear his name, but which were later shown by Paul Dirac in 1928 to be intrinsic in his relativistic equation.^{ [13] }^{[ self-published source? ]}

The experiment was first performed with an electromagnet that allowed the non-uniform magnetic field to be turned on gradually from a null value.^{ [1] } 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.^{ [14] } 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.

The Stern–Gerlach experiment 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.
^{[ example needed ]}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.^{ [15] } - In the 1930s, using an extended version of the Stern–Gerlach apparatus, Isidor Rabi and colleagues showed that by using a varying magnetic field, one can force the magnetic moment to go from one state to the other.
^{[ citation needed ]}The series of experiments culminated in 1937 when they discovered that state transitions could be induced using time varying fields or RF fields. The so-called Rabi oscillation is the working mechanism for the Magnetic Resonance Imaging equipment found in hospitals.^{[ citation needed ]} - Norman F. Ramsey later modified the Rabi apparatus to increase the interaction time with the field. The extreme sensitivity due to the frequency of the radiation makes this very useful for keeping accurate time, and it is still used today in atomic clocks.
^{[ citation needed ]} - In the early sixties, Ramsey and Daniel Kleppner used a Stern–Gerlach system to produce a beam of polarized hydrogen as the source of energy for the hydrogen maser, which is still one of the most popular frequency standards.
- The direct observation of the spin is the most direct evidence of quantization in quantum mechanics.
^{[ why? ]}^{[ citation needed ]} - The Stern–Gerlach experiment has become a prototype
^{ [16] }^{ [17] }^{ [18] }for*quantum measurement*, demonstrating the observation of a single, real value (*eigenvalue)*of an initially unknown physical property. Entering the Stern-Gerlach magnet, the direction of the silver atom’s magnetic moment is indefinite, but it is observed to be either parallel, or anti-parallel to the direction of the magnetic field,**B**, at the exit of the magnet. Atoms with a magnetic moment parallel to**B**have been accelerated in that direction by the magnetic field gradient; those with anti-parallel moments were accelerated the opposite way. So, each atom traversing the magnet will strike the detector ((5) in the diagram) at just one of the two spots. According to*quantum measurement theory,*the wavefunction representing the atom’s magnetic moment is in a*superposition*of those two directions entering the magnet. A single, spin-direction eigenvalue is recorded when a momentum quantum is transferred, from the magnetic field, to the atom, initiating acceleration, and displacement, in that momentum direction.^{ [19] }

In atomic physics, the **Bohr model** or **Rutherford–Bohr model**, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar System, but with attraction provided by electrostatic forces in place of gravity. After the cubical model (1902), the plum pudding model (1904), the Saturnian model (1904), and the Rutherford model (1911) came the *Rutherford–Bohr model* or just *Bohr model* for short (1913). The improvement over the 1911 Rutherford model mainly concerned the new quantum physical interpretation.

The **electron** is a subatomic particle, symbol ^{}e^{−}_{} or ^{}β^{−}_{}, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, *ħ*. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

A **hydrogen atom** is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. **Atomic hydrogen** constitutes about 75% of the baryonic mass of the universe.

The **quantum Hall effect** is a quantized version of the Hall effect and which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance *R*_{xy} exhibits steps that take on the quantized values at certain level

In quantum mechanics, the **principal quantum number** is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers making it a discrete variable.

The **azimuthal quantum number** is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron. It is also known as the **orbital angular momentum** quantum number, **orbital quantum number** or **second quantum number**, and is symbolized as **ℓ**.

The **magnetic quantum number** is one of four quantum numbers in atomic physics. The set is: principal quantum number, azimuthal quantum number, magnetic quantum number, and spin quantum number. Together, they describe the unique quantum state of an electron. The magnetic quantum number distinguishes the orbitals available within a subshell, and is used to calculate the azimuthal component of the orientation of orbital in space. Electrons in a particular subshell are defined by values of *ℓ*. The value of *m _{l}* can range from -

In atomic physics, the **spin quantum number** is a quantum number that describes the intrinsic angular momentum of a given particle. The spin quantum number is designated by the letter s, and is the fourth of a set of quantum numbers, which completely describe the quantum state of an electron. The name comes from a physical spinning of the electron about an axis that was proposed by Uhlenbeck and Goudsmit. However this simplistic picture was quickly realized to be physically impossible, and replaced by a more abstract quantum-mechanical description.

The **old quantum theory** is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics.

In atomic physics, the **electron magnetic moment**, or more specifically the **electron magnetic dipole moment**, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is approximately −9.284764×10^{−24} J/T. The electron magnetic moment has been measured to an accuracy of 7.6 parts in 10^{13}.

In quantum physics, the **spin–orbit interaction** is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is one cause of magnetocrystalline anisotropy and the spin Hall effect.

**Quantum mechanics** is the study of very small things. It explains the behavior of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the Moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original scientific paradigm: the *theory of relativity* and the development of *quantum mechanics*. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century. It describes these concepts in roughly the order in which they were first discovered. For a more complete history of the subject, see *History of quantum mechanics*.

In quantum mechanics, spin is an intrinsic property of all elementary particles. All known fermions, the particles that constitute ordinary matter, have a spin of ½. The spin number describes how many symmetrical facets a particle has in one full rotation; a spin of ½ means that the particle must be fully rotated twice before it has the same configuration as when it started.

In quantum mechanics, the **angular momentum operator** is one of several related operators analogous to classical angular momentum. The angular momentum operator plays a central role in the theory of atomic and molecular physics and other quantum problems involving rotational symmetry. Such an operator is applied to a mathematical representation of the physical state of a system and yields an angular momentum value if the state has a definite value for it. In both classical and quantum mechanical systems, angular momentum is one of the three fundamental properties of motion.

The **mass-to-charge ratio** (*m*/*Q*) is a physical quantity that is most widely used in the electrodynamics of charged particles, e.g. in electron optics and ion optics. It appears in the scientific fields of electron microscopy, cathode ray tubes, accelerator physics, nuclear physics, Auger electron spectroscopy, cosmology and mass spectrometry. The importance of the mass-to-charge ratio, according to classical electrodynamics, is that two particles with the same mass-to-charge ratio move in the same path in a vacuum, when subjected to the same electric and magnetic fields. Its SI units are kg/C. In rare occasions the thomson has been used as its unit in the field of mass spectrometry.

A ** g-factor** is a dimensionless quantity that characterizes the magnetic moment and angular momentum of an atom, a particle or the nucleus. It is essentially a proportionality constant that relates the observed magnetic moment

**Spin** is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.

The **Planck constant**, or **Planck's constant**, is the quantum of electromagnetic action that relates a photon's energy to its frequency. The Planck constant multiplied by a photon's frequency is equal to a photon's energy. The Planck constant is a fundamental physical constant denoted as , and of fundamental importance in quantum mechanics. In metrology it is used to define the kilogram in SI units.

**Magnetic resonance** is a quantum mechanical resonant effect that can appear when a magnetic dipole is exposed to a static magnetic field and perturbed with another, oscillating electromagnetic field. Due to the static field, the dipole can assume a number of discrete energy eigenstates, depending on the value of its angular momentum quantum number. The oscillating field can then make the dipole transit between its energy states with a certain probability and at a certain rate. The overall transition probability will depend on the field's frequency and the rate will depend on its amplitude. When the frequency of that field leads to the maximum possible transition probability between two states, a magnetic resonance has been achieved. In that case, the energy of the photons composing the oscillating field matches the energy difference between said states. If the dipole is tickled with a field oscillating far from resonance, it is unlikely to transition. That is analogous to other resonant effects, such as with the forced harmonic oscillator. The periodic transition between the different states is called Rabi cycle and the rate at which that happens is called Rabi frequency. The Rabi frequency should not be confused with the field's own frequency. Since many atomic nuclei species can behave as a magnetic dipole, this resonance technique is the basis of nuclear magnetic resonance, including nuclear magnetic resonance imaging and nuclear magnetic resonance spectroscopy.

Electrons in free space can carry quantized orbital angular momentum (OAM) projected along the direction of propagation. This orbital angular momentum corresponds to helical wavefronts, or, equivalently, a phase proportional to the azimuthal angle. Electron beams with quantized orbital angular momentum are also called **electron vortex beams**.

- 1 2 3 Gerlach, W.; Stern, O. (1922). "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld".
*Zeitschrift für Physik*.**9**(1): 349–352. Bibcode:1922ZPhy....9..349G. doi:10.1007/BF01326983. S2CID 186228677. - ↑ Allan Franklin and Slobodan Perovic. "Experiment in Physics, Appendix 5". In Edward N. Zalta (ed.).
*The Stanford Encyclopedia of Philosophy*(Winter 2016 ed.). Retrieved 2018-08-14.CS1 maint: uses authors parameter (link) - ↑ Friedrich, B.; Herschbach, D. (2003). "Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics".
*Physics Today*.**56**(12): 53. Bibcode:2003PhT....56l..53F. doi:10.1063/1.1650229. S2CID 17572089. - ↑ Gerlach, W.; Stern, O. (1922). "Das magnetische Moment des Silberatoms".
*Zeitschrift für Physik*.**9**(1): 353–355. Bibcode:1922ZPhy....9..353G. doi:10.1007/BF01326984. S2CID 126109346. - ↑ Gerlach, W.; Stern, O. (1922). "Der experimentelle Nachweis des magnetischen Moments des Silberatoms".
*Zeitschrift für Physik*.**8**(1): 110–111. Bibcode:1922ZPhy....8..110G. doi:10.1007/BF01329580. S2CID 122648402. - ↑ Mott, N.F., 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 Wheeler, J.A.; Zurek, W.H. (1983).*Quantum Theory and Measurement*. Princeton NJ: Princeton University Press. pp. 701–706. - ↑ George H. Rutherford and Rainer Grobe (1997). "Comment on "Stern-Gerlach Effect for Electron Beams"".
*Phys. Rev. Lett*.**81**(4772): 4772. Bibcode:1998PhRvL..81.4772R. doi:10.1103/PhysRevLett.81.4772. - 1 2 Sakurai, J.-J. (1985).
*Modern quantum mechanics*. Addison-Wesley. ISBN 0-201-53929-2. - 1 2 3 Qinxun, Li (June 8, 2020). "Stern Gerlach Experiment:Descriptions and Developments" (PDF).
*University of Science and Technology of China*: 2–5. Retrieved 24 November 2020. - ↑ Stern, O. (1921). "Ein Weg zur experimentellen Pruefung der Richtungsquantelung im Magnetfeld".
*Zeitschrift für Physik*.**7**(1): 249–253. Bibcode:1921ZPhy....7..249S. doi:10.1007/BF01332793. S2CID 186234469. - ↑ Weinert, F. (1995). "Wrong theory—right experiment: The significance of the Stern–Gerlach experiments".
*Studies in History and Philosophy of Modern Physics*.**26B**(1): 75–86. Bibcode:1995SHPMP..26...75W. doi:10.1016/1355-2198(95)00002-B. - ↑ Phipps, T.E.; Taylor, J.B. (1927). "The Magnetic Moment of the Hydrogen Atom".
*Physical Review*.**29**(2): 309–320. Bibcode:1927PhRv...29..309P. doi:10.1103/PhysRev.29.309. - ↑ A., Henok (2002).
*Introduction to Applied Modern Physics*. Lulu.com. p. 76. ISBN 1-4357-0521-1.^{[ self-published source ]} - ↑ French, A.P., Taylor, E.F. (1979).
*An Introduction to Quantum Physics*, Van Nostrand Reinhold, London, ISBN 0-442-30770-5, pp. 428–442. - ↑ Griffiths, David (2005).
*Introduction to Quantum Mechanics, 2nd ed*. Pearson Prentice Hall. p. 267. ISBN 0-13-111892-7. - ↑ Bohm, David (1951).
*Quantum Theory*. New York: Prentice-Hall. pp. 326–330. - ↑ Gottfried, Kurt (1966).
*Quantum Mechanics*. New York: W. A. Benjamin, Inc. pp. 170–174. - ↑ Eisberg, Robert (1961).
*Fundamentals of Modern Physics*. New York: John Wiley & Sons. pp. 334–338. ISBN 0-471-23463-X. - ↑ Devereux, Michael (2015). "Reduction of the atomic wavefunction in the Stern–Gerlach magnetic field".
*Canadian Journal of Physics*.**93**(11): 1382–1390. Bibcode:2015CaJPh..93.1382D. doi:10.1139/cjp-2015-0031. hdl: 1807/69186 . ISSN 0008-4204.

- Devereux, M. (2015). "Reduction of the atomic wavefunction in the Stern-Gerlach magnetic field".
*Canadian Journal of Physics*.**93**(11): 1382–1390. Bibcode:2015CaJPh..93.1382D. doi:10.1139/cjp-2015-0031. hdl: 1807/69186 . - Friedrich, B.; Herschbach, D. (2003). "Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics".
*Physics Today*.**56**(12): 53. Bibcode:2003PhT....56l..53F. doi:10.1063/1.1650229. S2CID 17572089. - Reinisch, G. (1999). "Stern–Gerlach experiment as the pioneer—and probably the simplest—quantum entanglement test?".
*Physics Letters A*.**259**(6): 427–430. Bibcode:1999PhLA..259..427R. doi:10.1016/S0375-9601(99)00472-7. - Venugopalan, A. (1997). "Decoherence and Schrödinger-cat states in a Stern−Gerlach-type experiment".
*Physical Review A*.**56**(5): 4307–4310. Bibcode:1997PhRvA..56.4307V. doi:10.1103/PhysRevA.56.4307. - Hsu, B.; Berrondo, M.; Van Huele, J.-F. (2011). "Stern-Gerlach dynamics with quantum propagators".
*Physical Review A*.**83**(1): 012109–1–12. Bibcode:2011PhRvA..83a2109H. doi:10.1103/PhysRevA.83.012109. - Jeremy Bernstein (2010). "The Stern Gerlach Experiment". arXiv: 1007.2435v1 [physics.hist-ph].
- Use of ions

Wikimedia Commons has media related to . Stern-Gerlach experiment |

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.