Calculation of Matrix Elements for One‐Dimensional Quantum‐Mechanical Problems and the Application to Anharmonic Oscillators

1965 ◽  
Vol 43 (5) ◽  
pp. 1515-1517 ◽  
Author(s):  
David O. Harris ◽  
Gail G. Engerholm ◽  
William D. Gwinn
Keyword(s):  
Quantum Mechanical ◽  
Matrix Elements ◽  
Anharmonic Oscillators ◽  
One Dimensional

QUANTIZATION CONDITION FOR HIGHLY EXCITED STATES

1999 ◽  
Vol 14 (19) ◽  
pp. 1237-1242 ◽  
Author(s):  
FRANCISCO M. FERNÁNDEZ ◽  
RAFAEL GUARDIOLA
Keyword(s):  
Perturbation Theory ◽  
Excited States ◽  
Rational Approximation ◽  
Logarithmic Derivative ◽  
Quantization Condition ◽  
Quantum Mechanical ◽  
Anharmonic Oscillators ◽  
One Dimensional ◽  
Simple Quantum ◽  
Mechanical Models

We develop a quantization condition for the excited states of simple quantum-mechanical models. The approach combines perturbation theory for the oscillatory part of the eigenfunction with a rational approximation to the logarithmic derivative of the nodeless part of it. We choose one-dimensional anharmonic oscillators as illustrative examples.

The Riccati–Padé quantization method for one-dimensional quantum-mechanical models

1996 ◽  
Vol 74 (9-10) ◽  
pp. 697-700 ◽  
Author(s):  
Francisco M. Fernández ◽  
R. H. Tipping
Keyword(s):  
Ground State ◽  
Logarithmic Derivative ◽  
Quantum Mechanical ◽  
Quantization Method ◽  
Anharmonic Oscillators ◽  
Eigenvalues And Eigenfunctions ◽  
One Dimensional ◽  
Continuum States ◽  
Separable Models ◽  
Mechanical Models

We improve on a previously developed method for the calculation of accurate eigenvalues and eigenfunctions of separable models in quantum mechanics. It consists of the approximation of the logarithmic derivative of the eigenfunction by means of a rational function or Padé approximant. Here we modify the approach by the separation of the function just mentioned into its odd and even parts, thus making the procedure more efficient for treating asymmetric one-dimensional potentials. We obtain the ground-state eigenvalue of anharmonic oscillators with one and two wells and the lowest resonances of anharmonic oscillators that support only continuum states.

SYMMETRY DECOUPLING IN LINEAR ALGEBRAIC VARIATIONAL SCATTERING PROBLEMS

1995 ◽  
Vol 06 (01) ◽  
pp. 105-121
Author(s):  
MEISHAN ZHAO
Keyword(s):  
Angular Momentum ◽  
Variational Principle ◽  
Total Angular Momentum ◽  
Numerical Calculations ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Scattering Problems ◽  
Identical Particles ◽  
Generalized Newton

This paper discusses the symmetry decoupling in quantum mechanical algebraic variational scattering calculations by the generalized Newton variational principle. Symmetry decoupling for collisions involving identical particles is briefly discussed. Detailed discussion is given to decoupling from evaluation of matrix elements with nonzero total angular momentum. Example numerical calculations are presented for BrH2 and DH2 systems to illustrate accuracy and efficiency.

scholarly journals Contact Electrification by Quantum-Mechanical Tunneling

Research ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Morten Willatzen ◽  
Zhong Lin Wang
Keyword(s):  
Potential Versus ◽  
Coulomb Repulsion ◽  
Quantum Mechanical ◽  
Quantum Mechanical Tunneling ◽  
One Dimensional ◽  
Contact Electrification ◽  
Order Of Magnitude ◽  
Left And Right ◽  
Tunneling Dynamics ◽  
Vacuum Gap

A simple model of charge transfer by loss-less quantum-mechanical tunneling between two solids is proposed. The model is applicable to electron transport and contact electrification between e.g. a metal and a dielectric solid. Based on a one-dimensional effective-mass Hamiltonian, the tunneling transmission coefficient of electrons through a barrier from one solid to another solid is calculated analytically. The transport rate (current) of electrons is found using the Tsu-Esaki equation and accounting for different Fermi functions of the two solids. We show that the tunneling dynamics is very sensitive to the vacuum potential versus the two solids conduction-band edges and the thickness of the vacuum gap. The relevant time constants for tunneling and contact electrification, relevant for triboelectricity, can vary over several orders of magnitude when the vacuum gap changes by one order of magnitude, say, 1 Å to 10 Å. Coulomb repulsion between electrons on the left and right material surfaces is accounted for in the tunneling dynamics.

Efficient computation of one-electron two-centre exchange integrals in ion–atom collisions

1976 ◽  
Vol 54 (9) ◽  
pp. 944-949 ◽  
Author(s):  
Alfred Msezane
Keyword(s):  
Principal Quantum Number ◽  
Computing Time ◽  
Translational Motion ◽  
Matrix Elements ◽  
Efficient Computation ◽  
Close Coupling ◽  
One Dimensional ◽  
State Approximation ◽  
The Matrix ◽  
Exchange Matrix

A scheme is presented for the reduction to one-dimensional integrals of any one-electron two-centre exchange matrix elements which incorporate the momentum associated with the translational motion of the electron. These elements are of the types occurring in close coupling-based treatments of ion–atom collisions. It is shown in a six state approximation, by coupling both eigenstates and pseudostates for the asymmetric He2+–H collision process, that computing time for the evaluation of the matrix elements is determined mainly by the number of different exponents in the matrix elements. The coupling of additional states with the same principal quantum number as the already coupled ones alters computing time insignificantly.

scholarly journals One-loop non-planar anomalous dimensions in super Yang-Mills theory

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Tristan McLoughlin ◽  
Raul Pereira ◽  
Anne Spiering
Keyword(s):  
Perturbation Theory ◽  
Integrable Systems ◽  
Chaotic Systems ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Mechanical Perturbation ◽  
Yang Mills ◽  
Anomalous Dimensions ◽  
Dilatation Operator ◽  
Scalar Products

Abstract We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues. Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use non-degenerate quantum-mechanical perturbation theory to compute the leading 1/N2 corrections to operator dimensions and as an example compute the large R-charge limit for two-excitation states through subleading order in the R-charge. Finally, we numerically study the distribution of level spacings for these theories and show that they transition from the Poisson distribution for integrable systems at infinite N to the GOE Wigner-Dyson distribution for quantum chaotic systems at finite N.

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