ISSUE 2007 VOLUME 2 - The World of Mathematical Equations

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April, 2007 PROGRESS IN PHYSICS Volume 2 Since the duality relation is a mathematically true statement (theorem) then it cannot be disproved by experiment and certainly means that Afshar’s arguments, through which he violates the duality relation, are inconsistent. Indeed the calculation of V and D depends on the principle of complementarity and distinguishability of the states ψ1 and ψ2. The calculation of V and D in Unruh’s and Afshar’s setup is different for pure state and mixed state setups. 5.1 Mixed state setup In view of the foregoing explanation for Unruh’s claim with mixed density matrix, the calculation simply yields D = 1 and V = 0. This will be true if we label the paths by different polarization filters, or if we investigate a statistical mixture of two single path/slit experiments. D1: |ψ1| = 1 √ , |ψ2| = 0 , 2 D2: |ψ1| = 0, |ψ2| = 1 √ 2 . Thus the two paths 1 and 2 are distinguishable and they do not quantum mechanically cross-interfere. This cannot be said for the quantum coherent setup with both paths/slits unimpeded. 5.2 Pure state setup The correct analysis of Unruh’s and Afshar’s setup suggests a pure state density matrix, and amplitudes for each of the exit gates being |ψ1| = |ψ2| = 1 √ 8 . Thus one gets D = 0 and V = 1: D1:|ψ1| = 1 √ 8 , |ψ2| = 1 √ 8 , D2:|ψ1| = 1 √ 8 , |ψ2| = 1 √ 8 . The two paths 1 and 2 are indistinguishable, and they quantum mechanically cross-interfere. 6 Conclusions It is wrongly believed that the lens at the image plane always provides “which way” information (Afshar [1, 2]; Drezet [5]). However we have shown that Afshar’s analysis is inconsistent, and that the distinguishability and visibility in Afshar’s setup are erroneously calculated by Afshar and colleagues [3]. The two peak image at the image plane in Afshar’s setup, even without wire grid in the path of the photons, is an interference pattern and does not provide any “which way” information. Exact calculations for the lens setup have been performed by Qureshi [9] and Reitzner [10], showing that once the two paths interfere the interference cannot be undone, and the “which way” information cannot be regained. The probability distribution can look like the one in a mixed setup, but the retrospective reconstruction of the setup for times before the detector click must be done with interfering waves which do not carry the “which way” information. Afshar’s mathematics is inconsistent, hence Afshar’s setup does not disprove MWI, or any other rival interpretation of Quantum Mechanics that opposes the standard Copenhagen paradigm. References Submitted on March 01, 2007 Accepted on March 05, 2007 After correction: March 20, 2007 1. Afshar S. S. Violation of the principle of complementarity, and its implications. Proceedings of SPIE, 2005, v. 5866, 229–244; arXiv: quant-ph/0701027. 2. Afshar S. S. Violation of Bohr’s complementarity: one slit or both? AIP Conference Proceedings, 2006, v. 810, 294–299; arXiv: quant-ph/0701039. 3. Afshar S. S., Flores E., McDonald K. F. and Knoesel E. Paradox in wave-particle duality. Foundations of Physics, 2007, v. 37(2), 295–305; arXiv: quant-ph/0702188. 4. Cramer J. G. The transactional interpretation of quantum mechanics. Reviews of Modern Physics, 1986, v. 58, 647–688. 5. Drezet A. Complementarity and Afshar’s experiment. arXiv: quant-ph/0508091. 6. Elitzur A. C. and Vaidman L. Is it possible to know about something without ever interacting with it? Vistas in Astronomy, 1993, v. 37, 253–256. 7. Feynman R. P. and Hibbs A. R. Quantum physics and path integrals. McGraw-Hill, New York, 1965. 8. Penrose R. On gravity’s role in quantum state reduction. General Relativity and Gravitation, 1996, v. 28(5), 581–600. 9. Qureshi T. Complementarity and the Afshar experiment. arXiv: quant-ph/0701109. 10. Reitzner D. Comment on Afshar’s expriment. arXiv: quantph/0701152. 11. Tegmark M. and Wheeler J. A. 100 years of the quantum. Scientific American, 2001, v. 284, 68–75; arXiv: quant-ph/ 0101077. 12. Vedral V. Modern foundations of quantum optics. Imperial College Press, London (UK), 2001. 13. Unruh W. Quantum rebel. 2004, http://axion.physics.ubc.ca/ rebel.html. 14. Zeh H. D. The meaning of decoherence. Lecture Notes in Physics, 2000, v. 538, 19–42; arXiv: quant-ph/ 9905004. D. D. Georgiev. Single Photon Experiments and Quantum Complementarity 103
Volume 2 PROGRESS IN PHYSICS April, 2007 Upper Limit of the Periodic Table and Synthesis of Superheavy Elements Albert Khazan E-mail: albkhazan@list.ru For the first time, using the heaviest possible element, the diagram for known nuclides and stable isotopes is constructed. The direction of search of superheavy elements is indicated. The Periodic Table with an eighth period is tabulated. 1 Shell construction of a nucleus, magic numbers The nucleus of an atom is the central part of the atom, consisting of positively charged protons (Z) and electrically neutral neutrons (N). They interact by means of the strong interaction. If a nucleus of an atom is consider as a particle with a certain number of protons and neutrons it is called a nuclide. A nuclide is that version of an atom defined by its mass number (A = Z + N), its atomic number (Z) and a power condition of its nucleus. Nuclei with identical numbers of protons but different numbers of neutrons are isotopes. The majority of isotopes are unstable. They can turn into other isotopes or elements due to radioactive disintegration of the nucleus by one of the following means: β-decay (emission of electron or positron), α-decay (emission of particles consisting of two protons and two neutrons) or spontaneous nuclear fission of an isotope. If the product of disintegration is also unstable, it too breaks up in due course, and so on, until a stable product is formed. It has been shown experimentally that a set of these particles becomes particularly stable when the nuclei contain “magic” number of protons or neutrons. The stable structure can be considered as shells or spherical orbits which are completely filled by the particles of a nucleus, by analogy with the filled electronic shells of the noble gases. The numbers of particles forming such a shell are called “magic” numbers. Nuclei with magic number of neutrons or protons are unusually stable and in nuclei with one proton or other than a magic number, the neutron poorly binds the superfluous particle. The relevant values of these numbers are 2, 8, 20, 28, 50, 82, and 126, for which there exists more stable nuclei than for other numbers. Calculations indicate existence of a nucleus with filled shell at Z = 114 and N = 184 ( 298 114) which would be rather stable in relation to spontaneous division. There is experimental data for the connexion of magic numbers to a nucleus with Z = 164 [1, 2]. J. Oganesyan [3] has alluded to a Rutherford-model atom which assumes existence of heavy nuclei with atomic numbers within the limits of Z ∼ 170. At the same time there is a point of view holding that superheavy elements (SHEs) cannot have Z > 125 [4]. In October 2006 in was reported that element 118 had been synthesized in Dubna (Russia), with atomic weight 293 [5]. (It is known however, that this weight is understated, owing to technical difficulties associated with the experiments.) 2 The N-Z diagram of nuclei, islands of stability The search for superheavy nuclei, both in the Nature and by synthesis as products of nuclear reactions, has intensified. In the 1970’s 1200 artificially produced nuclei were known [6]. Currently the number is ∼3000, and it is estimated that this will increase to ∼6500 [7]. In Fig. 1 the neutron-proton diagram of nuclei of stable and artificial isotopes [8–10] is presented. Light stable or long-lived nuclei which arrangement can be arranged in a valley of stability as shown by small circles. The top set of border points represents a line of proton stability and bottom a line of neutron stability. Beyond these limits begins the so-called, “sea of instability”. There is apparently only a narrow strip of stability for which there exists a quite definite parity, N/Z. For nuclei with atomic weight below 40, the numbers of protons and neutrons are approximately identical. With increase in the quantity of neutrons the ratio increases, and in the field of A =(N+Z)= = 250 it reaches 1.6. The growth in the number of neutrons advances the quantity of protons in heavy nuclei, which in this case become energetically more stable. To the left of the stable nuclei are proton excess nuclei, and on the right neutron excess nuclei. These and others are called exotic nuclei. The diagram terminates in the last element from the table IUPAC [11] at No. 114, with mass number 289, while scientists suspect nucleus No. 114–298. Such isotopes should possess the increased stability and lifetime of superheavy elements. This diagram is specially constructed, only on the basis of tabulated data, but augmented by the theoretical upper limit of the Periodic Table [12]. Up to the Z∼60 the line of trend approaches the middle of a valley of stability, with N/Z ∼ 1.33. Furthermore, N/Z increases steadily to ∼1.5 up to Z ∼ 100. The equation of the line of trend represents a polynomial of the fourth degree. It is noteworthy that this implies rejection of the upper magic number for neutrons heretofore theoretically supposed. It is particularly evident from Fig. 2, in which small fragment of the N–Z diagram is amplified and augmented with 104 A. Khazan. Upper Limit of the Periodic Table and Synthesis of Superheavy Elements
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