In this short note we discuss how to solve the quantum harmonic oscillator. There is a well known traditional method of solving the quantum harmonic oscillator. Here we present a matrix-eigenvalue method. We also present another equivalent matrix-eigenvalue method, which has pedagogical value.

In recent years, the Legendre and other Sturm-Liouville polynomials have been computed using matrix-eigenvalue algorithms. In the present work, we extend the Legendre matrix-eigenvalue algorithm to compute the associated Legendre polynomials, which have broad applications in a large variety of fields including quantum mechanics and computer science.

Recently the Legendre and other Sturm-Liouville (SL) polynomials were found as eigenvectors of certain matrices [2, 3, 4, 5]. However, the proposed algorithms are computationally incomplete and do not lead to general formulas to calculate the coefficients of SL polynomials of any order. In this paper, we complete the algorithms based on a matrix-eigenvector method, which can be used to compute SL polynomials of any order. This includes Legendre, Hermite, Laguerre, and Chebyshev polynomials.

Structural and magnetic properties of self-assembled Co nanorods on a Cu(110)-(2x3)N surface have been investigated by low-energy electron diffraction (LEED), Auger-electron spectroscopy (AES), magneto-optical Kerr effect (MOKE), and x-ray magnetic circular dichroism (XMCD). The LEED observation confirms that the Co nanorod grows epitaxially along the [1 (1) over bar0] axis and its interval exhibits the (1x6) periodicity. The AES clarifies that the N atom locates always at the surface even after 5 monolayer (ML) Co deposition. Angle dependent magnetization curves of the Co nanorods recorded by MOKE and XMCD show that the magnetic easy axis is perpendicular to the rod within the substrate plane, irrespective of the Co thickness down to 0.8 ML. This implies that the magnetic anisotropy is not dominated by the shape anisotropy but by the magneto-crystalline anisotropy. The XMCD sum-rule analysis reveals significant enhancement of the orbital magnetic moment along the easy axis compared to the hard axes. The magnetocrystalline anisotropy is found to be directly related to the anisotropy of the orbital magnetic moment.

Scanning tunneling microscopy (STM) images show that adsorbed formate has a profound affect on the step edges of Cu(110) surfaces at room temperature. For low exposures, the presence of formate enhances step fluctuations as confirmed by a correlation function analysis. For formate coverages approaching 0.5 monolayers, drastic restructuring of step edges is observed. Quantum chemical calculations help to explain this behavior.

We have performed quantum chemical calculations for coupled oscillations examining CO adsorbed on Cu(110) and atomic oxygen on the (100) diamond face. For CO, we find that our calculations show coupling consistent with a dynamic dipole–dipole interaction. In the case of the adsorbed oxygen atoms the dominant coupling seems to be mechanical. The limits of the semiempirical method SAM1 are tested for consistency and precision. Our model gives accuracy within 10% when compared to experiment and works well with near neighbor adsorbates.

Recent scanning tunneling microscopy (STM) and photoelectron diffraction studies of the Cu(1 0 0)-c(2×2)N system have led to the proposal of a “new type of adsorbate-induced surface reconstruction” involving the movement of Cu atoms perpendicular to the surface. Claims were made that images from prior STM studies were misinterpreted. We present STM images that clearly contradict these claims. We discuss the role of tip structure in the imaging process and demonstrate how tips with asymmetric apexes can result in STM images that can easily be misinterpreted.

We calculate the preferred adsorption sites, molecular structure, infrared spectra, and energies relative to stable gas phase molecules for carbon monoxide, the formate ion, and the acetate ion when adsorbed on the (110) face of copper. This substrate is modeled using small clusters of (8–46) copper atoms. Our semiempirical calculations compare well with those from ab initio, density functional theory, and with experiments. We offer a method of estimating the adsorption energy.

Recent scanning tunneling microscopy (STM) and photoelectron diffraction studies of the Cu(1 0 0)-c(2 x 2)N system have led to the proposal of a "new type of adsorbate-induced surface reconstruction" involving the movement of Cu atoms perpendicular to the surface. Claims were made that images from prior STM studies were misinterpreted. We present STM images that clearly contradict these claims. We discuss the role of tip structure in the imaging process and demonstrate how tips with asymmetric apexes can result in STM images that can easily be misinterpreted. (C) 2002 Elsevier Science B.V. All rights reserved.

Self-assembled nanostructures can be formed on Cu(100), Cu(110) and Cu(111) surfaces by bombarding these surfaces with quantities of activated nitrogen followed by annealing. Coupled with epitaxial deposition, these surfaces can be used to create arrays of metallic nanostructures. On Cu(111) surfaces, N deposition can result in the mass production of quantum confinement structures. The use of scanning tunneling microscopy to selectively modify these nanostructures is also demonstrated.

High-quality vicinal Si(111) surfaces are used as templates to create single domains of the Si(111)3×1-Ca reconstruction which exhibit atomic chains parallel to Si steps. Scanning tunneling microscope images support the formation of honeycomb chains of Si atoms, rather than zigzag chains proposed in earlier models. Angle-resolved photoemission is used to map out the dispersion of valence band states parallel and perpendicular to the chains. A gap of ≈0.9 eV is found below the Fermi level with both in-plane and out-of-plane polarization of the synchrotron light. The observed semiconducting behavior suggests that the honeycomb chain channel model proposed for the alkali-induced 3×1 reconstruction be modified for divalent alkaline earths, e.g. a 3×2 structure with 1/6 monolayer coverage.

We have exposed clean and oxygen-precovered Cu(110) surfaces to acetic acid to create adsorbed acetate. Our studies show the formation of a variety of ordered acetate structures depending on the initial oxygen exposure. We have observed the restructuring of step edges and the formation of ordered arrays of up–down steps. Using a purposeful coadsorption strategy, we have obtained images that assist us in developing models for the acetate structures. Fast Fourier transform infrared spectra have been obtained and Hartree–Fock calculations are included for comparison.

We have used scanning tunneling microscopy to examine N-terminated Cu(111) surfaces. For substaturation N coverages, elongated rectangular islands are formed, the internal structures of which are in good agreement with a model proposed in a previous LEED study. These islands sufficiently perturb the surface so as to reflect surface state electrons in the adjacent clean surface regions. This results in the observation of electron standing wave patterns. We have simultaneously observed both the electron wave patterns and atomic resolution. We have also shown that under conditions of high electric fields, the N atoms at the surface can be forced down into the second layer with Cu atoms liberated from the first layer to form single layer high islands.

We demonstrate how we can use scanning tunneling microscopy to create nanometer-scale pits several layers deep on Ag(1 1 1) films grown on a Cu(1 1 1) substrate. The creation of these pits is accompanied by the formation of multilayer-high islands. We also demonstrate the ability to manipulate small one-layer deep pits. In addition, the evolution of the islands and pits is also studied. Images show island and pit coalescence and instances of rapid and gradual decay.

We demonstrate the ability to use scanning tunneling microscopy to create nanometer-scale pits on Ag films grown on Cu(100) substrates. Atomic resolution images show that the Ag structures are intact within these pits. We also demonstrate how we can selectively modify segments of Ag nanowire arrays grown on atomic nitrogen modified Cu(100) surfaces.

We demonstrate, using scanning tunneling microscopy, that the growth of Co onto Cu(110) surfaces can be dramatically altered by first terminating the Cu(110) surface with an atomic nitrogen-induced (2 x 3) structure. Co growth onto such surfaces results in the formation of ordered arrays of Co nanowires. These results are contrasted with Co growth on clean Cu(110) surfaces and suggest that the N termination also prevents the diffusion of Cu atoms into the Co layers.