# Quantum Nonlocality and Quantum Correlations in the Stern–Gerlach Experiment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Quantum Nonlocality

## 3. The SGE in A Complete Quantum Treatment

## 4. Quantum Correlations and Nonlocality in the Stern–Gerlach Experiment

#### 4.1. A Pure State

#### 4.2. Violation of Bell’s Inequalities

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Friedrich, B.; Herschbach, D. Space Quantization: Otto Stern’s Lucky Star. Daedalus
**1998**, 127, 165–191. [Google Scholar] - Friedrich, B.; Herschbach, D. Stern and Gerlach: How a bad cigar helped reorient atomic physics. Phys. Today
**2003**, 56, 53–59. [Google Scholar] [CrossRef] - Schmidt-Böcking, H.; Schmidt, L.; Lüdde, H.J.; Trageser, W.; Templeton, A.; Sauer, T. The Stern–Gerlach experiment revisited. Eur. Phys. J. H
**2016**, 41, 327–364. [Google Scholar] [CrossRef] - Weinert, F. Wrong theory-Right experiment: The significance of the Stern–Gerlach experiments. Stud. Hist. Phil. Mod. Phys.
**1995**, 26, 75–86. [Google Scholar] [CrossRef] - Rodríguez, E.B.; Aguilar, L.A.; Martínez, E.P. A full quantum analysis of the Stern–Gerlach experiment using the evolution operator method: Analyzing current issues in teaching quantum mechanics. Eur. J. Phys.
**2017**, 38, 025403. [Google Scholar] [CrossRef] - Rodríguez, E.B.; Aguilar, L.A.; Martínez, E.P. Corrigendum: A full quantum analysis of the Stern–Gerlach experiment using the evolution operator method: Analysing current issues in teaching quantum mechanics. Eur. J. Phys.
**2017**, 38, 069501. [Google Scholar] [CrossRef] - Home, D.; Pan, A.K.; Ali, M.M.; Majumdar, A.S. Aspects of nonideal Stern–Gerlach experiment and testable ramifications. J. Phys. A: Math. Theor.
**2007**, 40, 13975. [Google Scholar] [CrossRef] - Roston, G.B.; Casas, M.; Plastino, A.; Plastino, A.R. Quantum entanglement, spin-1/2 and the Stern–Gerlach experiment. Eur. J. Phys.
**2005**, 26, 657–672. [Google Scholar] [CrossRef] - Scully, M.O.; Lamb, W.E.; Barut, A. On the theory of the Stern–Gerlach apparatus. Found. Phys.
**1987**, 17, 575–583. [Google Scholar] [CrossRef] - Platt, D.E. A modern analysis of the Stern–Gerlach experiment. Am. J. Phys.
**1992**, 60, 306–308. [Google Scholar] [CrossRef] - Hsu, B.C.; Berrondo, M.; Van Huele, J.F.S. Stern–Gerlach dynamics with quantum propagators. Phys. Rev. A
**2011**, 83, 012109. [Google Scholar] [CrossRef] - Sparaciari, C.; Paris, M.G. Canonical Naimark extension for generalized measurements involving sets of Pauli quantum observables chosen at random. Phys. Rev. A
**2013**, 87, 012106. [Google Scholar] [CrossRef] - Potel, G.; Barranco, F.; Cruz-Barrios, S.; Gómez-Camacho, J. Quantum mechanical description of Stern–Gerlach experiments. Phys. Rev. A
**2005**, 71, 052106. [Google Scholar] [CrossRef] - Sparaciari, C.; Paris, M.G. Probing qubit by qubit: Properties of the POVM and the information/disturbance tradeoff. Int. J. Quantum Inf.
**2014**, 12, 1461012. [Google Scholar] [CrossRef] - Fratini, F.; Safari, L. Quantum mechanical evolution operator in the presence of a scalar linear potential: Discussion on the evolved state, phase shift generator and tunneling. Phys. Scr.
**2014**, 89, 085004. [Google Scholar] [CrossRef] [Green Version] - Wennerström, H.; Westlund, P.O. A Quantum Description of the Stern–Gerlach Experiment. Entropy
**2017**, 19, 186. [Google Scholar] [CrossRef] - Rossi, M.A.; Benedetti, C.; Paris, M.G. Engineering decoherence for two-qubit systems interacting with a classical environment. Int. J. Quantum Inf.
**2014**, 12, 1560003. [Google Scholar] [CrossRef] - Boustimi, M.; Bocvarski, V.; de Lesegno, B.V.; Brodsky, K.; Perales, F.; Baudon, J.; Robert, J. Atomic interference patterns in the transverse plane. Phys. Rev. A
**2000**, 61, 033602. [Google Scholar] [CrossRef] - Larson, J.; Garraway, B.M.; Stenholm, S. Transient effects on electron spin observation. Phys. Rev. A
**2004**, 69, 032103. [Google Scholar] [CrossRef] - Machluf, S.; Japha, Y.; Folman, R. Coherent Stern–Gerlach momentum splitting on an atom chip. Nat. Commun.
**2013**, 4, 2424. [Google Scholar] [CrossRef] [PubMed] - Quijas, P.G.; Aguilar, L.A. Factorizing the time evolution operator. Phys. Scr.
**2007**, 75, 185–194. [Google Scholar] [CrossRef] - Quijas, P.G.; Aguilar, L.A. Overcoming misconceptions in quantum mechanics with the time evolution operator. Eur. J. Phys.
**2007**, 28, 147–159. [Google Scholar] [CrossRef] - Aguilar, L.A.; Quijas, P.G. Reply to Comment on “Overcoming misconceptions in quantum mechanics with the time evolution operator”. Eu. J. Phys.
**2013**, 34, L77. [Google Scholar] [CrossRef] - Aguilar, L.A.; Luna, F.V.; Robledo-Sánchez, C.; Arroyo-Carrasco, M.L. The infinite square well potential and the evolution operator method for the purpose of overcoming misconceptions in quantum mechanics. Eur. J. Phys.
**2014**, 35, 025001. [Google Scholar] [CrossRef] - Quijas, P.C.; Aguilar, L.M. A quantum coupler and the harmonic oscillator interacting with a reservoir: Defining the relative phase gate. Quantum Inf. Comput.
**2010**, 10, 190–200. [Google Scholar] - Toyama, F.M.; Nogami, Y. Comment on ‘Overcoming misconceptions in quantum mechanics with the time evolution operator’. Eur. J. Phys.
**2013**, 34, L73. [Google Scholar] [CrossRef] - Amaku, M.; Coutinho, F.A.; Masafumi Toyama, F. On the definition of the time evolution operator for time-independent Hamiltonians in non-relativistic quantum mechanics. Am. J. Phys.
**2017**, 85, 692–697. [Google Scholar] [CrossRef] - Singh, C.; Belloni, M.; Christian, W. Improving students’ understanding of quantum mechanics. Phys. Today
**2006**, 59, 43–49. [Google Scholar] [CrossRef] - Chhabra, M.; Das, R. Quantum mechanical wavefunction: Visualization at undergraduate level. Eur. J. Phys.
**2017**, 38, 015404. [Google Scholar] [CrossRef] - Cataloglu, E.; Robinett, R.W. Testing the development of student conceptual and visualization understanding in quantum mechanics through the undergraduate career. Am. J. Phys.
**2002**, 70, 238–251. [Google Scholar] [CrossRef] - Emigh, P.J.; Passante, G.; Shaffer, P.S. Student understanding of time dependence in quantum mechanics. Phys. Rev. ST Phys. Educ. Res.
**2015**, 11, 020112. [Google Scholar] [CrossRef] - Dini, V.; Hammer, D. Case study of a successful learner’s epistemological framings of quantum mechanics. Phys. Rev. Phys. Educ. Res.
**2017**, 13, 010124. [Google Scholar] [CrossRef] - Zhu, G.; Singh, C. Improving students understanding of quantum mechanics via the Stern–Gerlach experiment. Am. J. Phys.
**2011**, 79, 499–507. [Google Scholar] [CrossRef] - Carr, L.D.; McKagan, S.B. Graduate quantum mechanics reform. Am. J. Phys.
**2009**, 77, 308–319. [Google Scholar] [CrossRef] - Passante, G.; Emigh, P.J.; Shaffer, P.S. Examining student ideas about energy measurements on quantum states across undergraduate and graduate levels. Phys. Rev. Spec. Top. Phys. Educ. Res.
**2015**, 11, 020111. [Google Scholar] [CrossRef] - Passante, G.; Emigh, P.J.; Shaffer, P.S. Student ability to distinguish between superposition states and mixed states in quantum mechanics. Phys. Rev. Spec. Top. Phys. Educ. Res.
**2015**, 11, 020135. [Google Scholar] [CrossRef] - Greca, I.M.; Freire, O. Meeting the Challenge: Quantum Physics in Introductory Physics Courses. In International Handbook of Research in History, Philosophy and Science Teaching; Springer: Dordrecht, The Netherlands, 2014; pp. 183–209. [Google Scholar]
- Kohnle, A.; Bozhinova, I.; Browne, D.; Everitt, M.; Fomins, A.; Kok, P.; Kulaitis, G.; Prokopas, M.; Raine, D.; Swinbank, E. A new introductory quantum mechanics curriculum. Eur. J. Phys.
**2014**, 35, 015001. [Google Scholar] [CrossRef] - Singh, C. Students understanding of quantum mechanics at the beginning of graduate instruction. Am. J. Phys.
**2008**, 76, 277–287. [Google Scholar] [CrossRef] - Singh, C.; Marshman, E. Review of student difficulties in upper-level quantum mechanics. Phys. Rev. Spec. Top. Phys. Educ. Res.
**2015**, 11, 020117. [Google Scholar] [CrossRef] - Johansson, A.; Andersson, S.; Salminen-Karlsson, M.; Elmgren, M. “Shut up and calculate”: The available discursive positions in quantum physics courses. Cult. Stud. Sci. Educ.
**2016**, 13, 205–226. [Google Scholar] [CrossRef] - Greca, I.M.; Freire, O. Teaching introductory quantum physics and chemistry: Caveats from the history of science and science teaching to the training of modern chemists. Chem. Educ. Res. Pract.
**2014**, 15, 286–296. [Google Scholar] [CrossRef] - Coto, B.; Arencibia, A.; Suárez, I. Monte Carlo method to explain the probabilistic interpretation of atomic quantum mechanics. Comput. Appl. Eng. Educ.
**2016**, 24, 765–774. [Google Scholar] [CrossRef] - Marshman, E.; Singh, C. Investigating and improving student understanding of the expectation values of observables in quantum mechanics. Eur. J. Phys.
**2017**, 38, 045701. [Google Scholar] [CrossRef] - Siddiqui, S.; Singh, C. How diverse are physics instructors’ attitudes and approaches to teaching undergraduate level quantum mechanics? Eur. J. Phys.
**2017**, 38, 035703. [Google Scholar] [CrossRef] - Marshman, E.; Singh, C. Investigating and improving student understanding of quantum mechanical observables and their corresponding operators in Dirac notation. Eur. J. Phys.
**2018**, 39, 015707. [Google Scholar] [CrossRef] - Kohnle, A.; Baily, C.; Campbell, A.; Korolkova, N.; Paetkau, M.J. Enhancing student learning of two-level quantum systems with interactive simulations. Am. J. Phys.
**2015**, 83, 560–566. [Google Scholar] [CrossRef] [Green Version] - Baily, C.; Finkelstein, N.D. Teaching quantum interpretations: Revisiting the goals and practices of introductory quantum physics courses. Phys. Rev. Spec. Top. Phys. Educ. Res.
**2015**, 11, 020124. [Google Scholar] [CrossRef] - McKagan, S.B.; Perkins, K.K.; Wieman, C.E. Design and validation of the Quantum Mechanics Conceptual Survey. Phys. Rev. Spec. Top. Phys. Educ. Res.
**2010**, 6, 020121. [Google Scholar] [CrossRef] - Sadaghiani, H.R.; Pollock, S.J. Quantum mechanics concept assessment: Development and validation study. Phys. Rev. Spec. Top. Phys. Educ. Res.
**2015**, 11, 010110. [Google Scholar] [CrossRef] - Wuttiprom, S.; Sharma, M.D.; Johnston, I.D.; Chitaree, R.; Soankwan, C. Development and Use of a Conceptual Survey in Introductory Quantum Physics. Int. J. Sci. Educ.
**2009**, 31, 631–654. [Google Scholar] [CrossRef] - Bao, L.; Redish, E.F. Understanding probabilistic interpretations of physical systems: A prerequisite to learning quantum physics. Am. J. Phys.
**2002**, 70, 210–217. [Google Scholar] [CrossRef] - Archer, R.; Bates, S. Asking the right questions: Developing diagnostic tests in undergraduate physics. New Dir. Teach. Phys. Sci.
**2009**, 5, 22–25. [Google Scholar] - Clauser, J.F.; Shimony, A. Bell’s theorem. Experimental tests and implications. Rep. Prog. Phys.
**1978**, 41, 1881–1927. [Google Scholar] [CrossRef] - Gisin, N. Quantum Chance: Nonlocality, Teleportation and Other Quantum Marvels; Springer International Publishing: Cham, Switzerland, 2014. [Google Scholar]
- Augusiak, R.; Demianowicz, M.; Acín, A. Local hidden variable models for entangled quantum states. J. Phys. A Math. Theor.
**2014**, 47, 424002. [Google Scholar] [CrossRef] - Gisin, N. Bell’s inequality holds for all non-product states. Phys. Lett. A
**1991**, 154, 201–202. [Google Scholar] [CrossRef] - Popescu, S.; Rohrlich, D. Generic quantum nonlocality. Phys. Lett. A
**1992**, 166, 293–297. [Google Scholar] [CrossRef] - Popescu, S. Bell’s inequalities versus teleportation: What is nonlocality? Phys. Rev. Lett.
**1994**, 72, 797–799. [Google Scholar] [CrossRef] [PubMed] - Brunner, N.; Gisin, N.; Scarani, V. Entanglement and non-locality are different resources. New J. Phys.
**2005**, 7, 88. [Google Scholar] [CrossRef] - Bennett, C.H.; DiVincenzo, D.P.; Fuchs, C.A.; Mor, T.; Rains, E.; Shor, P.W.; Smolin, J.A.; Wootters, W.K. Quantum nonlocality without entanglement. Phys Rev A
**1999**, 59, 1070–1091. [Google Scholar] [CrossRef] - Jammer, M. The Philosophy of Quantum Mechanics; John Wiley & Sons: New York, NY, USA, 1974. [Google Scholar]
- Fine, A. The Shaky Game; The University of Chicago Press: London, UK, 1986. [Google Scholar]
- Norsen, T. Einstein’s boxes. Am. J. Phys.
**2005**, 73, 164–176. [Google Scholar] [CrossRef] - Einstein, A.; Podolsky, B.; Rosen, N. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev.
**1935**, 47, 777–780. [Google Scholar] [CrossRef] - Gallego, R.; Würflinger, L.E.; Acín, A.; Navascués, M. Operational Framework for Nonlocality. Phys. Rev. Lett.
**2012**, 109, 070401. [Google Scholar] [CrossRef] [PubMed] - Forster, M.; Winkler, S.; Wolf, S. Distilling Nonlocality. Phys. Rev. Lett.
**2009**, 102, 120401. [Google Scholar] [CrossRef] [PubMed] - Wódkiewicz, K. Nonlocality of the Schrödinger cat. New J. Phys.
**2000**, 2, 21. [Google Scholar] [CrossRef] - Banaszek, K.; Wódkiewicz, K. Testing Quantum Nonlocality in Phase Space. Phys. Rev. Lett.
**1999**, 82, 2009–2013. [Google Scholar] [CrossRef] - Haug, F.; Freyberger, M.; Wódkiewicz, K. Nonlocality of a free atomic wave packet. Phys. Lett. A
**2004**, 321, 6–13. [Google Scholar] [CrossRef] - Agarwal, G.; Home, D.; Schleich, W. Einstein-Podolsky-Rosen correlation—Parallelism between the Wigner function and the local hidden variable approaches. Phys. Lett. A
**1992**, 170, 359–362. [Google Scholar] [CrossRef] - Ben-Benjamin, J.S.; Kim, M.B.; Schleich, W.P.; Case, W.B.; Cohen, L. Working in phase-space with Wigner and Weyl. Fortschr. Phys.
**2017**, 65, 1600092. [Google Scholar] [CrossRef] - Case, W.B. Wigner functions and Weyl transforms for pedestrians. Am. J. Phys.
**2008**, 76, 937–946. [Google Scholar] [CrossRef] - Royer, A. Wigner function as the expectation value of a parity operator. Phys. Rev. A
**1977**, 15, 449–450. [Google Scholar] [CrossRef] - Hillery, M.O.S.M.; O’Connell, R.F.; Scully, M.O.; Wigner, E.P. Distribution functions in physics: Fundamentals. Phys. Rep.
**1984**, 106, 121–167. [Google Scholar] [CrossRef] - Zurek, W.H. Decoherence and the Transition from Quantum to Classical. Phys. Today
**1991**, 44, 36. [Google Scholar] [CrossRef] - Gerry, C.C.; Knight, P.L. Quantum superpositions and Schrödinger cat states in quantum optics. Am. J. Phys.
**1997**, 65, 964–974. [Google Scholar] [CrossRef] - Ballentine, L.E. Quantum Mechanics: A Modern Development; World Scientific Publishing: Singapore, 1998. [Google Scholar]
- Jeong, H.; Son, W.; Kim, M.S.; Ahn, D.; Brukner, Č. Quantum nonlocality test for continuous-variable states with dichotomic observable. Phys. Rev. A
**2003**, 67, 012106. [Google Scholar] [CrossRef] - Chen, Z.B.; Pan, J.W.; Hou, G.; Zhang, Y.D. Maximal Violation of Bell’s Inequalities for Continuous Variable Systems. Phys. Rev. Lett.
**2002**, 88, 040406. [Google Scholar] [CrossRef] [PubMed] - Zukowski, M. Bell’s Theorem Tells Us Not What Quantum Mechanics Is, but What Quantum Mechanics Is Not. In Quantum [Un]Speakables II; Bertlmann, R., Zeilinger, A., Eds.; Springer: Cham, Switzerland, 2017; pp. 175–185. [Google Scholar] [CrossRef]
- Ferraro, A.; Paris, M.G.A. Nonlocality of two- and three-mode continuous variable systems. J. Opt. B Quantum Semiclassical Opt.
**2005**, 7, 174–182. [Google Scholar] [CrossRef] - Ferrie, C. Quasi-probability representations of quantum theory with applications to quantum information science. Rep. Prog. Phys.
**2011**, 74, 116001. [Google Scholar] [CrossRef] - Vourdas, A. Quantum systems with finite Hilbert space. Rep. Prog. Phys.
**2004**, 67, 267–320. [Google Scholar] [CrossRef] - Hinarejos, M.; Bañuls, M.C.; Pérez, A. Wigner formalism for a particle on an infinite lattice: dynamics and spin. New J. Phys.
**2015**, 17, 013037. [Google Scholar] [CrossRef] - Gomis, P.; Pérez, A. Decoherence effects in the Stern–Gerlach experiment using matrix Wigner Functions. Phys. Rev. A
**2016**, 94, 012103. [Google Scholar] [CrossRef] - Clauser, J.F.; Horne, M.A. Shimony A and Holt R A Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett.
**1969**, 23, 880–884. [Google Scholar] [CrossRef] - Rodríguez, E.B.; Aguilar, L.A. Disturbance-disturbance uncertainty relation: The statistical distinguishability of quantum states determines disturbance. Sci. Rep.
**2018**, 8, 4010. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**A plot of the correlation function between z and $\theta $ given by Equation (12). To obtain this plot, we have employed the following values: first, we set $\pi \hslash =1$ and $m=1$; then, we set ${\sigma}_{0}=0.005$, ${\mu}_{c}b/2=2.2$, ${p}_{z}=0.01$ and time $t=0.2$.

**Figure 2.**A plot of the function ${}_{z}$ given by Equation (15) considering ${z}^{\prime}=0.08$ and ${\theta}^{\prime}=\pi /2$. Once more, we set $\pi \hslash =1$, $m=1$, ${\sigma}_{0}=0.005$, ${\mu}_{c}b/2=2.2$, ${p}_{z}=0.01$ and time $t=0.2$.

**Figure 3.**A close up of the region of Figure 2 where the violation of Bell’s inequality is perceived.

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**MDPI and ACS Style**

Piceno Martínez, A.E.; Benítez Rodríguez, E.; Mendoza Fierro, J.A.; Méndez Otero, M.M.; Arévalo Aguilar, L.M.
Quantum Nonlocality and Quantum Correlations in the Stern–Gerlach Experiment. *Entropy* **2018**, *20*, 299.
https://doi.org/10.3390/e20040299

**AMA Style**

Piceno Martínez AE, Benítez Rodríguez E, Mendoza Fierro JA, Méndez Otero MM, Arévalo Aguilar LM.
Quantum Nonlocality and Quantum Correlations in the Stern–Gerlach Experiment. *Entropy*. 2018; 20(4):299.
https://doi.org/10.3390/e20040299

**Chicago/Turabian Style**

Piceno Martínez, Alma Elena, Ernesto Benítez Rodríguez, Julio Abraham Mendoza Fierro, Marcela Maribel Méndez Otero, and Luis Manuel Arévalo Aguilar.
2018. "Quantum Nonlocality and Quantum Correlations in the Stern–Gerlach Experiment" *Entropy* 20, no. 4: 299.
https://doi.org/10.3390/e20040299