Unusual Bound or Localized States

Zeitschrift fur Naturforschung - Section A Journal of Physical Sciences

Volumn 56, Issue 1-2, 2001, Pages 48-60

  Cirone, M A ^a   Metikas, G ^a   Schleich, W P ^a,b  

^a UNIVERSITY OF ULM   (Germany)

^b UNIVERSITY OF NORTH TEXAS   (United States)

Author keywords

Bound States; Parametric Oscillator; Periodic Potential; Quantum Mechanics

Indexed keywords

EID: 0347242156     PISSN: 09320784     EISSN: None     Source Type: Journal    
DOI: 10.1515/zna-2001-0109     Document Type: Article

References (48)

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2. P. Ehrenfest, Naturwiss. 11, 543 (1923); for an excellent introduction into and comprehensive summary of the early quantum mechanics we refer to D. ter Haar, The Old Quantum Theory, Pergamon Press, Oxford 1967, p. 44. A classic in this field is M. Born, Atom- mechanik, Springer, Heidelberg 1925; see also J. Duck and E. C. G. Sudarshan, 100 Years of Planck's Quantum, World Scientific, Singapore 2000.
3. P. Ehrenfest, Naturwiss. 11, 543 (1923); for an excellent introduction into and comprehensive summary of the early quantum mechanics we refer to D. ter Haar, The Old Quantum Theory, Pergamon Press, Oxford 1967, p. 44. A classic in this field is M. Born, Atom-mechanik, Springer, Heidelberg 1925; see also J. Duck and E. C. G. Sudarshan, 100 Years of Planck's Quantum, World Scientific, Singapore 2000.
4. P. Ehrenfest, Naturwiss. 11, 543 (1923); for an excellent introduction into and comprehensive summary of the early quantum mechanics we refer to D. ter Haar, The Old Quantum Theory, Pergamon Press, Oxford 1967, p. 44. A classic in this field is M. Born, Atom- mechanik, Springer, Heidelberg 1925; see also J. Duck and E. C. G. Sudarshan, 100 Years of Planck's Quantum, World Scientific, Singapore 2000.
5. J. A. Wheeler, in C. and B. S. De Witt (eds), Relativity, Groups and Topology, Gordon and Breach, New York 1964.
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7. For a summary of the problem of acceleration of charged particles to high energies see the article by J. P. Blewett, in E. U. Condon and H. Odishaw (eds.), Handbook of Physics, McGraw-Hill, New York 1958, Chapt. 9, p. 153.
8. J. A. Wheeler in many discussions with one of us (WPS) repeatedly credited H. Bethe with the discovery of the problems leading to the principle of strong focusing. Unfortunately, we have not been able to locate the appropriate reference. We have talked to Prof. Bethe, but also he was not able to point out the article.
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16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York 1965; for a particularly illuminating method to motivate the stability chart of the Mathieu equation using WKB wave functions, see M. J. Richardson, Amer. J. Phys. 39, 560 (1971).
17. W. Paul, Rev. Mod. Phys. 62, 531 (1990). A pedagogical introduction into the physics of Paul traps emphasizing the concept of the effective potential is given by P. E. Toschek, in G. Grynberg and R. Stora (eds), New trends in atomic physics, North-Holland, Amsterdam 1984, p. 390. For a summary of current activities in Paul traps see H. Walther, in B. Bederson and H. Walther (eds.), Advances in Atomic, Molecular and Optical Physics, Academic Press, Boston 1995.
18. W. Paul, Rev. Mod. Phys. 62, 531 (1990). A pedagogical introduction into the physics of Paul traps emphasizing the concept of the effective potential is given by P. E. Toschek, in G. Grynberg and R. Stora (eds), New trends in atomic physics, North-Holland, Amsterdam 1984, p. 390. For a summary of current activities in Paul traps see H. Walther, in B. Bederson and H. Walther (eds.), Advances in Atomic, Molecular and Optical Physics, Academic Press, Boston 1995.
19. W. Paul, Rev. Mod. Phys. 62, 531 (1990). A pedagogical introduction into the physics of Paul traps emphasizing the concept of the effective potential is given by P. E. Toschek, in G. Grynberg and R. Stora (eds), New trends in atomic physics, North-Holland, Amsterdam 1984, p. 390. For a summary of current activities in Paul traps see H. Walther, in B. Bederson and H. Walther (eds.), Advances in Atomic, Molecular and Optical Physics, Academic Press, Boston 1995.
20. K. Richter and D. Wintgen, Phys. Rev. Lett. 65, 1965 (1990); J. Phys. B 24, L565 (1991): for an overview of the full classical and quantum dynamics of two-electron systems see K. Richter, G. Tanner, and D. Wintgen, Phys. Rev. A 48, 4182 (1993).
21. K. Richter and D. Wintgen, Phys. Rev. Lett. 65, 1965 (1990); J. Phys. B 24, L565 (1991): for an overview of the full classical and quantum dynamics of two-electron systems see K. Richter, G. Tanner, and D. Wintgen, Phys. Rev. A 48, 4182 (1993).
22. K. Richter and D. Wintgen, Phys. Rev. Lett. 65, 1965 (1990); J. Phys. B 24, L565 (1991): for an overview of the full classical and quantum dynamics of two-electron systems see K. Richter, G. Tanner, and D. Wintgen, Phys. Rev. A 48, 4182 (1993).
23. We can also interpret the system of the two electrons in the helium atom as one realization of the Fermi accelerator. In the most elementary version of this device an oscillating wall confines the unbounded motion of a particle in a linear potential. For a review of the Fermi accelerator see F. Saif, I. Białynicki-Birula, M. Fortunato, and W. P. Schleich, Physics Reports, to be published.
24. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Pergamon Press, Oxford 1965, p. 267; There exists a large amount of literature on the physics of negative ions. See for example the classic book H. Massey, Negative Ions, Cambridge University Press, Cambridge 1976. For the most recent activities in this field see the Springer series, Production and Neutralization of Negative Ions and Beams. A similar effect appears also for an electron in the field of a super-heavy nuclei, see for example F. G. Werner and J. A. Wheeler, Phys. Rev. 109, 126 (1958).
25. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Pergamon Press, Oxford 1965, p. 267; There exists a large amount of literature on the physics of negative ions. See for example the classic book H. Massey, Negative Ions, Cambridge University Press, Cambridge 1976. For the most recent activities in this field see the Springer series, Production and Neutralization of Negative Ions and Beams. A similar effect appears also for an electron in the field of a super-heavy nuclei, see for example F. G. Werner and J. A. Wheeler, Phys. Rev. 109, 126 (1958).
26. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Pergamon Press, Oxford 1965, p. 267; There exists a large amount of literature on the physics of negative ions. See for example the classic book H. Massey, Negative Ions, Cambridge University Press, Cambridge 1976. For the most recent activities in this field see the Springer series, Production and Neutralization of Negative Ions and Beams. A similar effect appears also for an electron in the field of a super-heavy nuclei, see for example F. G. Werner and J. A. Wheeler, Phys. Rev. 109, 126 (1958).
27. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, Pergamon Press, Oxford 1965, p. 267; There exists a large amount of literature on the physics of negative ions. See for example the classic book H. Massey, Negative Ions, Cambridge University Press, Cambridge 1976. For the most recent activities in this field see the Springer series, Production and Neutralization of Negative Ions and Beams. A similar effect appears also for an electron in the field of a super-heavy nuclei, see for example F. G. Werner and J. A. Wheeler, Phys. Rev. 109, 126 (1958).
28. For summary see C. N. Cohen-Tannoudji, in J. Dalibard et al. (eds), Fundamental Systems in Quantum Optics, Elsevier Science, Amsterdam, 1992; see also the Nobel Lectures by S. Chu, C.N. Cohen-Tannoudji, and W. D. Phillips, Rev. Mod. Phys. 70, 685 (1998).
29. For summary see C. N. Cohen-Tannoudji, in J. Dalibard et al. (eds), Fundamental Systems in Quantum Optics, Elsevier Science, Amsterdam, 1992; see also the Nobel Lectures by S. Chu, C.N. Cohen-Tannoudji, and W. D. Phillips, Rev. Mod. Phys. 70, 685 (1998).
30. For a most striking and counter-intuitive bound state resulting from the peculiar behavior of the radial wave function at the origin of three-dimensional space we refer to J. Rauch and M. Reed, Comm. Math. Phys. 29, 105 (1973); and M. Reed, and B. Simon, Methods of Modern Mathematical Analysis II, Academic Press, New York 1975. In this example the potential is a sequence of appropriately constructed steps that lead continuously downwards as the radial variable increases. Classically a particle of given energy has to fall down the steps. However, the reflections of the quantum wave at the individual steps interfere in a way as to localize the particle. Likewise, the same authors discuss a potential consisting of an infinite sequence of potential spikes that classically would keep a particle trapped, however, due to the tunneling effect the quantum particle escapes.
31. For a most striking and counter-intuitive bound state resulting from the peculiar behavior of the radial wave function at the origin of three-dimensional space we refer to J. Rauch and M. Reed, Comm. Math. Phys. 29, 105 (1973); and M. Reed, and B. Simon, Methods of Modern Mathematical Analysis II, Academic Press, New York 1975. In this example the potential is a sequence of appropriately constructed steps that lead continuously downwards as the radial variable increases. Classically a particle of given energy has to fall down the steps. However, the reflections of the quantum wave at the individual steps interfere in a way as to localize the particle. Likewise, the same authors discuss a potential consisting of an infinite sequence of potential spikes that classically would keep a particle trapped, however, due to the tunneling effect the quantum particle escapes.
32. A similar reasoning appears [2] in general relativity, that is in geometrodynamics, when we determine the metric coefficients of a system. They follow from Brill's equation [2], which is similar to the time independent Schrödinger equation for zero energy. In contrast to quantum mechanics where the Schrödinger equation determines the energy eigenvalues we now have to solve the equation under the constraints that the wave is not allowed to have nodes and corresponds to zero energy.
33. S. Flügge, Practical Quantum Mechanics, Springer, Heidelberg 1971.
34. This observation is in complete accordance with a recent paper arguing that the time dependent Schrödinger equation is an approximation of the time independent Schrödinger equation resulting from the elimination of degrees of freedom. See for example J. S. Briggs and J. M. Rost, EPJD 10, 311 (2000).
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38. Two examples illustrate the unusual bound states that originate from the mutual interaction of the electrons in a heavy atom: The Thomas-Fermi potential together with the centrifugal potential can form a second potential minimum, which is very deep and located close to the nucleus. This effect occurs provided the atomic number is larger than 57 and we are dealing with an energy eigenstate corresponding to the angular momentum quantum number l = 3, see for example M. Goeppert-Mayer, Phys. Rev. 60, 184 (1941). Since A. Sommerfeld we associate the motion of an electron in an atom with an ellipse or a circle. However, the electron at the top of the sea of filled atomic states moves in an effective screened potential giving rise to a necklace orbit as pointed out by J. A. Wheeler, in E. H. Lieb et al. (eds), Studies in Mathematical Physics, Princeton University Press, Princeton 1976, p. 383.
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