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INFRARED SPECTROSCOPY OF DIATOMIC MOLECULES — A FRACTIONAL CALCULUS APPROACH

RICHARD HERRMANN

Berliner Ring 80, D-63303 Dreieich, Germany

https://doi.org/10.1142/S0217979213500197Cited by:9 (Source: Crossref)

Abstract

The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically, solving the fractional Schrödinger equation based on the Riemann and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.

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References

C. Huygens , Horologium Oscillatorium Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae ( F. Muguet , Paris, France , 1673 ) . Crossref, Google Scholar
G. Herzberg , Molecular Spectra and Molecular Structure I: Structure of Diatomic Molecules , 2nd edn. ( D. Van Nostrand Company , New York, USA , 1951 ) . Google Scholar
K. W. Whittenet al., Chemistry, 9th edn. (Cengage Learning, Stamford, CT, 2009). Google Scholar
K. B. Oldham and J. Spanier , The Fractional Calculus ( Dover Publications , Mineola, New York , 1976 ) . Google Scholar
B. Rubin , Fractional Integrals and Potentials , Pitman Monographs and Surveys in Pure and Applied Mathematics 82 ( Longman , Harlow , 1996 ) . Google Scholar
I. Podlubny , Fractional Differential Equations ( Academic Press , New York , 1999 ) . Google Scholar
F. Mainardi , Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models ( World Scientific Publishing , Singapore , 2010 ) . Link, Google Scholar
M. D. Ortigueira , Fractional Calculus for Scientists and Engineers ( Springer , Berlin , 2011 ) . Crossref, Google Scholar
R. Herrmann , Fractional Calculus — An Introduction for Physicists ( World Scientific Publishing , Singapore , 2011 ) . Link, Google Scholar
B. Riemann, Versuch einer allgemeinen Auffassung der integration und differentiation, in Bernhard Riemann's Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, eds. H. Weber and R. Dedekind (Teubner, Leipzig, reprinted in Collected Works of Bernhard Riemann (Dover Publications, 1953), pp. 353–366 . Google Scholar
M. Caputo, Geophys. J. R. Astr. Soc. 13, 529 (1967). Crossref, Web of Science, ADS, Google Scholar
P. A. M. Dirac , The Principles of Quantum Mechanics ( The Clarendon Press , Oxford , 1930 ) . Google Scholar
A. Messiah , Quantum Mechanics ( John Wiley & Sons, North-Holland Publishing , New York , 1968 ) . Google Scholar
R. Herrmann, Properties of a fractional derivative Schrödinger type wave equation and a new interpretation of the charmonium spectrum, (2005) , arXiv:math-ph/0510099v4 . Google Scholar
R. P. Feynman, Phys. Rev. 76, 749 (1949), DOI: 10.1103/PhysRev.76.749. Crossref, Web of Science, ADS, Google Scholar
N. Laskin, Phys. Rev. E 66, 056108 (2002), DOI: 10.1103/PhysRevE.66.056108. Crossref, Web of Science, Google Scholar
H. Weyl, Z. Phys. A 46, 1 (1927), reprinted in Quantum Mechanics in Phase Space, eds. C. K. Zachos, D. B. Fairlie and T. L. Curtright (World Scientific Publishing, Singapore, 2006) . Google Scholar
V. E. Tarasov, J. Math. Phys. 49(10), 102112 (2008), arXiv:0907.2699 DOI: 10.1063/1.3009533. Crossref, Web of Science, Google Scholar
G. Parisi and Y. Wu, Scientia Sinica 24(4), 483 (1981). Google Scholar
S. C. Lim and S. V. Muniandy, Phys. Lett. A 324, 396 (2004), DOI: 10.1016/j.physleta.2004.02.073. Crossref, Web of Science, ADS, Google Scholar
S. C. Lim and L. P. Tao, J. Phys. A: Math. Theor. 41, 145403 (2008), DOI: 10.1088/1751-8113/41/14/145403. Crossref, Web of Science, ADS, Google Scholar
W. Greiner, Quantum Mechanics — An Introduction, 4th edn. (Springer, Berlin, 2009). Google Scholar
A. Erdelyi et al. , Higher Transcendental Functions , Bateman Manuscript Project, California Institute of Technology II ( McGraw Hill , New York , 1953 ) . Google Scholar
R. Herrmannet al., Int. J. Mod. Phys. A 4(18), 4961 (1989), DOI: 10.1142/S0217751X89002107. Link, Web of Science, ADS, Google Scholar
R. Eidet al., Rom. J. Phys. 56(3–4), 323 (2011). Web of Science, Google Scholar
R. Herrmannet al., Physica A 389(4), 693 (2010), arXiv:0806.2300v1 DOI: 10.1016/j.physa.2009.11.016. Crossref, Web of Science, ADS, Google Scholar
R. Herrmann, Physica A 389(16), 3307 (2010), arXiv:0907.1953v1 DOI: 10.1016/j.physa.2010.03.033. Crossref, Web of Science, ADS, Google Scholar
J. A. Lissajous , Mèmoire sur l'ètude optique des mouvements vibratoires ( Mallet-Bachelier , Paris , 1857 ) . Google Scholar
H. Khunrath , Von hylealischen, das ist pri-materialischen catholischen oder allgemeinem natürlichen Chaos der naturgemessen Alchymiae und Alchymisten ( Franken , Magdeburg, Germany , 1597 ) . Google Scholar
R. Herrmann, Physica A 389(21), 4613 (2010), DOI: 10.1016/j.physa.2010.07.004. Crossref, Web of Science, ADS, Google Scholar
R. Lusztig , Introduction to Quantum Groups ( Birkhäuser , Boston , 1993 ) . Google Scholar
D. Bonatsos and C. Daskaloyannis, Prog. Part. Nucl. Phys. 43, 537 (1999), arXiv:nucl-th/9909003v1 DOI: 10.1016/S0146-6410(99)00100-3. Crossref, Web of Science, ADS, Google Scholar
D. Bonatsoset al., Phys. Rev. A 65, 033203 (2002), arXiv:cond-mat/0203125v1 DOI: 10.1103/PhysRevA.65.033203. Crossref, Web of Science, Google Scholar
J. T. Hartwig, D. Larsson and S. D. Silvestrov, J. Algebra 295, 314 (2006), DOI: 10.1016/j.jalgebra.2005.07.036. Crossref, Web of Science, Google Scholar
R. Hilfer and H. J. Seybold, Integral Transf. Special Funct. 17(9), 637 (2006), DOI: 10.1080/10652460600725341. Crossref, Web of Science, Google Scholar
H. J. Seybold and R. Hilfer, Fract. Calc. Appl. Anal. 8(2), 127 (2005). Google Scholar
H. J. Seybold and R. Hilfer, Siam J. Numer. Anal. 47(1), 69 (2008), DOI: 10.1137/070700280. Crossref, Web of Science, Google Scholar
P. Ehrenfest, In what way does it become manifest in the fundamental laws of physics that space has three dimensions?, Proc. Amsterdam Acad. 20 (I), 200 (1917), reprinted in Paul Ehrenfest — Collected Scientific Papers, ed. M. J. Klein (North Holland Publishing Co., Amsterdam, 1959) . Google Scholar
X. F. He, Phys. Rev. B 42(18), 11751 (1990), DOI: 10.1103/PhysRevB.42.11751. Crossref, Web of Science, ADS, Google Scholar
National Institute of Standards and Technology (NIST), Chemistry WebBook, NIST Standard Reference Database Number 69 (2012) , http://webbook.nist.gov/chemistry/ . Google Scholar
R. Herrmann, J. Phys. G: Nucl. Part. Phys. 34, 607 (2007), arXiv:nucl-th/0610091 DOI: 10.1088/0954-3899/34/4/001. Crossref, Web of Science, ADS, Google Scholar
R. Herrmann, Gam. Ori. Chron. Phys. 1(1), 13 (2013). Google Scholar
D. M. Villeneuveet al., Phys. Rev. Lett. 8, 542 (2000). Crossref, Web of Science, Google Scholar
L. Yuanet al., Proc. Nat. Acad. Sci. 108(17), 6872 (2011), DOI: 10.1073/pnas.1018669108. Crossref, Web of Science, ADS, Google Scholar
S. Abe, Phys. Lett. A 224, 326 (1997), DOI: 10.1016/S0375-9601(96)00832-8. Crossref, Web of Science, ADS, Google Scholar
R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific Publishing, Singapore, 2000) pp. 429–463. Link, Google Scholar
K. Krause , E. Wenger and K. Musgrave , Bryce — The First Name in 3D-Landscapes DAZ Productions ( Salt Lake City , UT, USA , 2012 ) , http://www.daz3d.com/products/bryce . Google Scholar