Abstract
Chapters 1–12 form what could be the basis for a one semester course in quantum mechanics. Before moving on to advanced topics such as time-independent perturbation theory, the variational method, the WKB approximation and irreducible tensor operators, it can’t hurt to review the basic concepts of quantum mechanics that I have covered up to this point. I also use this chapter to discuss an alternative formulation of quantum mechanics given by Feynman based on path-integrals. Finally I present a proof of Bell’s theorem and see how it can be tested.
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Notes
- 1.
R. P. Feynman, Reviews of Modern Physics, Vol. 20, pp. 367–387 (1948).
- 2.
A general discussion of the path-integral approach can be found in L. S. Schulman, Techniques and Applications of Path Integration (Dover Publications, Mineola, N.Y., 2005).
- 3.
John Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 195–200 (1964).
- 4.
Some authors object to Bell using the same value of λ for different measurements. See, for example, Karl Hess, Einstein Was Right! (CRC Press, Boca Raton, FL, 2015).
- 5.
J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt (1969), Proposed experiment to test local hidden-variable theories, Physical Review Letters 23, 880–4 (1969).
- 6.
Although the singlet state is written for a quantization axis in the z direction, the same state is realized for any quantization axis.
- 7.
B. Hensen, H. Bernien, A. E. Dréau, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abellán, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau and R. Hanson, Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature 526, 682–686 (2015).
- 8.
S. J. Freedman and J. F. Clauser, Experimental Test of Local Hidden-Variable Theories, Physical Review Letters 28, 938–941 (1972); E. S. Fry and R. C. Thompson, R. C. (1976), Experimental Test of Local Hidden Variables Theories, Physical Review Letters 37, 465-468 (1976); A. Aspect, P. Grangier, and G. Roger, Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedankenexperiment: A New Violation of Bell’s Inequalities, Physical Review Letters 49, 91–94 (1982); W. Tittel, J. Brendel, H. Zbinden, N. Gisin, Violation of Bell inequalities by photons more than 10 km apart, Physical Review Letters 81, 3563–3566 (1998); J-A. Larsson, M. Giustina, J. Kofler, B. Wittmann, R. Ursin, and S. Ramelow, Bell violation with entangled photons, free of the coincidence-time loophole. Physical. Review A 90, 032107 (2014).
- 9.
This argument follows that given by Robert Adair in The Great Design, Particles, Fields, and Creation (Oxford University Press, New York, 1987), pp. 185–187.
- 10.
See, for example, The Physics of Quantum Information, edited by Dirk Bouwmeester, Artur Ekert, and Anton Zeilinger (Springer-Verlag, Berlin, 2000), and references therein.
- 11.
See, for example, Xiao-Song Ma, Thomas Herbst, Thomas Scheidl, Daqing Wang, Sebastian Kropatschek, William Naylor, Bernhard Wittmann, Alexandra Mech, Johannes Kofler, Elena Anisimova, Vadim Makarov, Thomas Jennewein, Rupert Ursin and Anton Zeilinger, Quantum teleportation over 143 kilometres using active feed-forward, Nature 489, 269–273 (2012); Raju Valivarthi, Marcel.li Grimau Puigibert, Qiang Zhou, Gabriel Aguilar, Varun Verma, Francesco Marsili, Matthew D. Shaw, Sae Woo Nam, Daniel Oblak and Wolfgang Tittel, Quantum teleportation across a metropolitan fibre network, Nature Photonics 10, 676–680 (2016); J. Yin et al., Satellite-based entanglement distribution over 1200 kilometers, Science 356, 1140-1144 (2017).
- 12.
For example, she can make such a Bell state measurement by entangling her single photon states using beam splitters [Dik Bouwmeester, Jian-Wei Pan, Klaus Mattle, Manfred Eibl, Harald Weinfurter and Anton Zeilinger, Experimental quantum teleportation, Nature 390, 575–579 (1997)] or nonlinear crystals [Yoon-Ho Kim, Sergei Kulik, and Yanhua Shih, Quantum Teleportation of a Polarization State with a Complete Bell State Measurement, Physical Review Letters 86, 1370–1374 (2001)].
- 13.
See, for example, S. Huerfano, S. Sahu, and M. Socolovsky, Quantum Mechanics and the Weak Equivalence Principle, International Journal of Pure and Applied Mathematics 49, 153–166 (2008).
- 14.
See, for example, K. Hira, Derivation of the harmonic oscillator propagator using the Feynman path integral and recursive relations, European Journal of Physics 34, 777–785 (2013).
- 15.
J. R. Schaibley, A. P. Burgers, G. A. McCracken, L.-M. Duan, P. R. Berman, A. S. Bracker, D. Gammon, L. Sham, and D. G. Steel, Entanglement between a Single Electron Spin Confined to an InAs Quantum Dot and a Photon, Physical Review Letters 110, 167401 pp. 1–5 (2013).
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Berman, P.R. (2018). (A) Review of Basic Concepts (B) Feynman Path Integral Approach (C) Bell’s Inequalities Revisited. In: Introductory Quantum Mechanics. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-68598-4_13
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