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  • 1
    Electronic Resource
    Electronic Resource
    Generalization of the Darboux transform (2001)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 42 (2001), S. 966-975 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: This article presents a generalization of the standard Darboux transform applied to Sturm–Liouville differential equations. This is achieved with the aid of an ansatz as a particular solution for the Riccati relationship involved, which in turn led us to obtain its generalized Darboux solution that contains, as a particular case, the standard Darboux transform. The proposed generalized Darboux transform (GDT), applied to the quantum mechanical field, gives the opportunity to prove the existence of standard and generalized Darboux potentials that match with the so-called isospectral potentials. This is exemplified by obtaining, through the GDT, a set of standard and generalized Darboux potentials that form the partner of the one-dimensional harmonic oscillator model for any quantum principal number. The worked example indicates how the GDT can be used to obtain the isospectral potentials associated to any known specific potential. We consider also the application of our method as proposed to the theory of solitons in order to show why the GDT will be important in other fields of application where the standard Darboux transform is usually concerned. © 2001 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1334904
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  • 2
    Electronic Resource
    Electronic Resource
    Electron nuclear dynamics of proton collisions with methane at 30 eV (1997)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 107 (1997), S. 6146-6155 
    Details
    ISSN: 1089-7690
    Source: AIP Digital Archive
    Topics: Physics , Chemistry and Pharmacology
    Notes: The reactive collisions of protons with methane molecules at 30 eV in the laboratory frame are studied with the electron nuclear dynamics (END). The results from this theoretical approach, which does not invoke the Born–Oppenheimer approximation and does not impose any constraints on the nuclear dynamics, are compared to the results from time-of-flight measurements. Total differential cross sections and integral cross sections as well as fragmentation ratios and energy loss spectra are discussed.© 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.474343
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  • 3
    Electronic Resource
    Electronic Resource
    Hypervirial theorem and parameter differentiation: Closed formulation for harmonic oscillator integrals (1987)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 28 (1987), S. 1032-1035 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A simple method has been developed to generate a closed formula for the calculation of matrix elements of arbitrary functions f(x) in the representation of the harmonic oscillator. The proposed algebraic procedure is based on the combined use of the hypervirial theorem with and without the second quantization formalism along with the parameter differentiation technique. The closed formula thus obtained is given in terms of a sum involving the jth derivative of f(x) evaluated at zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527543
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  • 4
    Electronic Resource
    Electronic Resource
    Recurrence relations for two-center harmonic oscillator integrals (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 2966-2972 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: General recurrence relations for the calculation of two-center harmonic oscillator (HO) integrals are obtained by means of a hypervirial-like-theorem commutator algebra procedure, combined with a second quantization formalism. The method is based on a linear transformation between the creation and annihilation operators of two displaced HO with different frequencies. Ansbacher's recurrence relations for the calculation of Franck–Condon factors are obtained straightforwardly from the proposed general recurrence relations. The application to polynomial, exponential, and Gaussian operator integrals is shown and new recurrence relations are given. In all cases, the proposed recurrence relations reduce, as particular cases, to the corresponding formulas for the calculation of one-center integrals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527277
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  • 5
    Electronic Resource
    Electronic Resource
    The generalization of the binomial theorem (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 393-397 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: As is well known, the binomial theorem is a classical mathematical relation that can be straightforwardly proved by induction or through a Taylor expansion, albeit it remains valid as long as [A,B]=0. In order to generalize such an important equation to cases where [A,B]≠0, an algebraic approach based on Cauchy's integral theorem in conjunction with the Baker–Campbell–Hausdorff series is presented that allows a partial extension of the binomial theorem when the commutator [A,B]=c, where c is a constant. Some useful applications of the new proposed generalized binomial formula, such as energy eigenvalues and matrix elements of power, exponential, Gaussian, and arbitrary f(xˆ) functions in the one-dimensional harmonic oscillator representation are given. The results here obtained prove to be consistent in comparison to other analytical methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528457
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  • 6
    Electronic Resource
    Electronic Resource
    Two new potentials for the free particle model (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 5555-5564 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this work, is proposed a very simple method for obtaining the generalized potential associated with a known standard potential. The procedure is straightforward because it only uses two Ricatti-type relationships as enough condition to find a generalized potential; one particular equation is needed to identify the specific potential under study and one general Ricatti relationship is used to find the corresponding generalized potential. Moreover, the method is completely general due to the fact that an arbitrary potential has been considered in its development for which the procedure can also be used to find new potentials which could be needed in the modeling of important quantum interactions. The usefulness of the proposed approach, is shown with the treatment of the three- and one-dimensional potential for the free particle model. This work example leads to two new potentials whose Hamiltonians are isospectral when they are compared with the former Hamiltonian. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533045
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  • 7
    Electronic Resource
    Electronic Resource
    Algebraic approach to one- and two-center harmonic oscillator integralsThis work was partially supported by CONACYT-MEXICO PVT/EN/NAL81/1557. (1984)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 26 (1984), S. 57-60 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: A second-quantization formalism is used to derive closed formulas for one- and two-center matrix elements of two harmonic oscillators with different force constants. The method is based on a linear transformation between the creation and annihilation operators of the two harmonic oscillators, the use of the Baker-Campbell-Hausdorff theorem, and Cauchy's integral formula. As presented, the proposed procedure is considerably simpler than the ones hitherto employed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560260809
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  • 8
    Electronic Resource
    Electronic Resource
    Hypervirial theorem and ladder operators. Recurrence relations for harmonic oscillator integrals (1986)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 29 (1986), S. 211-219 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: A second-quantization formalism combined with a hypervirial theorem is used to derive new recurrence relations for one-dimensional harmonic oscillator matrix elements. The most general case of 〈m|f(â, â+)|n〉 is considered, and the recurrence relations forf(â, â) = Xk, exp(-βX), and exp(-X2) are given as examples. The relations obtained are considerably simpler than those derived by using only the hypervirial theorem; comparatively, the recurrence relations presented here have the advantage of avoiding the use of the quantum mechanical sum-rules when determining initial matrix elements. The proposed procedure can be used to determine the recurrence relations for other potentials as well as to evaluate the two-center integrals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560290211
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  • 9
    Electronic Resource
    Electronic Resource
    Closed formulas for one- and two-center harmonic oscillator integrals (1987)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 32 (1987), S. 729-735 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Closed formulas for one- and two-center matrix elements are derived for an analytical arbitrary operator function in the harmonic oscillator basis. The method is based on the Baker-Campbell-Hausdorff (BCH) theorem and Cauchy's integral formula. A table is displayed showing some examples for the two-center case. A pertinent relationship with other works is discussed.
    Additional Material: 1 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560320771
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  • 10
    Electronic Resource
    Electronic Resource
    Binding energies and peak intensities of light hydrocarbon molecules through TOM I. Methane and acetylene (1986)
    New York, NY : Wiley-Blackwell
    International Journal of Quantum Chemistry 30 (1986), S. 555-561 
    Details
    ISSN: 0020-7608
    Keywords: Computational Chemistry and Molecular Modeling ; Atomic, Molecular and Optical Physics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Chemistry and Pharmacology
    Notes: Binding energies and peak intensities of methane and acetylene molecules are calculated using the transition operator method (TOM) adapted to ab-initio Hartree-Fock programs. The calculations were done for both systems with and without symmetry restrictions, and the results are analyzed within the framework of TOM. It is found that the symmetry constraints have an important role in the binding energies of the acetylene molecule but none in the methane molecule. The intensities were calculated using the generalized sudden approximation (GSA) and the dipolar approximation; those resulting intensities are compared and discussed for the core and valence orbitals. The results let us conclude that the GSA method is only applicable when comparing photoionization intensities of neighboring ionization energy orbitals.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/qua.560300749
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