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Wave-Particle Duality

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Quantum Mechanics: Genesis and Achievements

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Abstract

The wave-particle duality was one of main issues for the Schrödinger theory. Now we will dwell upon this question in framework of the Schrödinger formalism calculating the corresponding density of particles. However, particles cannot be assigned any fixed position and momentum, as there are restrictions due to the Heisenberg Uncertainty Principle.

The wave nature of electron beam was confirmed experimentally and is perfectly explained by the Schrödinger wave theory.

However, the problem of wave-particle duality acquires new appearance as ‘reduction of wave packets’ in diffraction of a week electron beam. This key phenomenon was discovered experimentally and suggests the Born probabilistic interpretation of the wave function.

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Bibliography

  1. V.M. Babich, V.S. Buldirev, Short-Wavelength Diffraction Theory. Asymptotic Methods (Springer, Berlin, 1991)

    Book  Google Scholar 

  2. A. Bensoussan, C. Iliine, A. Komech, Breathers for a relativistic nonlinear wave equation. Arch. Ration. Mech. Anal. 165, 317–345 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Berestycki, P.L. Lions, Arch. Ration. Mech. Anal. 82(4), 313–375 (1983)

    MathSciNet  MATH  Google Scholar 

  4. A.B. Bhatia, P.C. Clemmow, D. Gabor, A.R. Stokes, A.M. Taylor, P.A. Wayman, W.L. Wilcock, in Principles of Optics, Electromagnetic Theory of Propagation, Interference and Diffraction of Light, ed. by M. Born, E. Wolf (Pergamon, London, 1959)

    Google Scholar 

  5. N. Bournaveas, Local existence for the Maxwell–Dirac equations in three space dimensions. Commun. Partial Differ. Equ. 21(5–6), 693–720 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. V.S. Buslaev, G.S. Perelman, Scattering for the nonlinear Schrödinger equation: states close to a soliton. St. Petersburg Math. J. 4, 1111–1142 (1993)

    MathSciNet  Google Scholar 

  7. V.S. Buslaev, G.S. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations. Am. Math. Soc. Trans. (2) 164, 75–98 (1995)

    MathSciNet  Google Scholar 

  8. V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20(3), 419–475 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. R.G. Chambers, Shift of an electron interference pattern by enclosed magnetic flux. Phys. Rev. Lett. 5, 3–5 (1960)

    Article  ADS  Google Scholar 

  10. T. Dudnikova, A. Komech, H. Spohn, Energy-momentum relation for solitary waves of relativistic wave equation. Russ. J. Math. Phys. 9(2), 153–160 (2002)

    MathSciNet  MATH  Google Scholar 

  11. W. Eckhaus, A. van Harten, The Inverse Scattering Transformation and the Theory of Solitons. An Introduction (North-Holland, Amsterdam, 1981)

    MATH  Google Scholar 

  12. M. Esteban, V. Georgiev, E. Sere, Stationary solutions of the Maxwell–Dirac and the Klein–Gordon–Dirac equations. Calc. Var. Partial Differ. Equ. 4(3), 265–281 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Frabboni, G.C. Gazzadi, G. Pozzi, Young’s double-slit interference experiment with electrons. Am. J. Phys. 75(11), 1053–1055 (2007)

    Article  ADS  Google Scholar 

  14. J. Fröhlich, T.P. Tsai, H.T. Yau, On the point-particle (Newtonian) limit of the nonlinear Hartree equation. Commun. Math. Phys. 225(2), 223–274 (2002)

    Article  ADS  MATH  Google Scholar 

  15. Y. Guo, K. Nakamitsu, W. Strauss, Global finite-energy solutions of the Maxwell–Schrödinger system. Commun. Math. Phys. 170(1), 181–196 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. K. Hannabus, An Introduction to Quantum Theory (Clarendon, Oxford, 1997)

    Google Scholar 

  17. V. Imaikin, A. Komech, N. Mauser, Soliton-type asymptotics for the coupled Maxwell–Lorentz equations. Ann. Inst. Poincaré, Phys. Theor. 5, 1117–1135 (2004)

    MathSciNet  MATH  Google Scholar 

  18. C. Jönsson, Electron diffraction at multiple slits. Am. J. Phys. 42, 4–11 (1974)

    Article  ADS  Google Scholar 

  19. A. Komech, H. Spohn, Soliton-like asymptotics for a classical particle interacting with a scalar wave field. Nonlinear Anal. 33(1), 13–24 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Komech, N. Mauser, A. Vinnichenko, On attraction to solitons in relativistic nonlinear wave equations. Russ. J. Math. Phys. 11(3), 289–307 (2004)

    MathSciNet  MATH  Google Scholar 

  21. A. Komech, E. Kopylova, On asymptotic stability of moving kink for relativistic Ginzburg–Landau equation. Commun. Math. Phys. 302(1), 225–252 (2011). arXiv:0910.5538

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. A. Komech, E. Kopylova, On asymptotic stability of kink for relativistic Ginzburg–Landau equation. Arch. Rat. Mech. Anal. 202, 213–245 (2011). arXiv:0910.5539

    Article  MathSciNet  Google Scholar 

  23. A. Komech, M. Kunze, H. Spohn, Effective dynamics for a mechanical particle coupled to a wave field. Commun. Math. Phys. 203, 1–19 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. E. Long, D. Stuart, Effective dynamics for solitons in the nonlinear Klein–Gordon–Maxwell system and the Lorentz force law. Rev. Math. Phys. 21(4), 459–510 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Miller, M. Weinstein, Asymptotic stability of solitary waves for the regularized long-wave equation. Commun. Pure Appl. Math. 49(4), 399–441 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  26. R. Newton, Quantum Physics (Springer, New York, 2002)

    Google Scholar 

  27. R.L. Pego, M.I. Weinstein, On asymptotic stability of solitary waves. Phys. Lett. A 162, 263–268 (1992)

    Article  ADS  Google Scholar 

  28. R.L. Pego, M.I. Weinstein, Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. M. Peshkin, A. Tonomura, The Aharonov–Bohm’s Effect (Springer, New York, 1989)

    Google Scholar 

  30. A. Soffer, M.I. Weinstein, Multichannel nonlinear scattering in nonintegrable systems. Commun. Math. Phys. 133, 119–146 (1990)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. A. Soffer, M.I. Weinstein, Multichannel nonlinear scattering and stability, II: the case of anisotropic and potential and data. J. Differ. Equ. 98, 376–390 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Soffer, M.I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136(1), 9–74 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  33. A. Soffer, M.I. Weinstein, Selection of the ground state for nonlinear Schrödinger equations. Rev. Math. Phys. 16(8), 977–1071 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Soffer, M.I. Weinstein, Theory of nonlinear dispersive waves and selection of the ground state. Phys. Rev. Lett. 95, 213905 (2005)

    Article  ADS  Google Scholar 

  35. A. Sommerfeld, Optics (Academic Press, New York, 1954)

    MATH  Google Scholar 

  36. G.P. Thomson, The diffraction of cathode rays by thin films of platinum. Nature 120, 802 (1927)

    Article  ADS  Google Scholar 

  37. A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki, H. Ezawa, Demonstration of single-electron buildup of an interference pattern. Am. J. Phys. 57(2), 117–120 (1989)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Alexander Komech

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  1. Alexander Komech
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Komech, A. (2013). Wave-Particle Duality. In: Quantum Mechanics: Genesis and Achievements. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5542-0_5

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