Harmonic Oscillator Integrals Research Articles

Harmonic oscillator integrals by operator methods, a simple unified approach

One- and two-centre matrix elements for a wide class of operators in an harmonic oscillator basis are derived by operator methods. The advantage of the method is that all calculations are reduced to the well known Hausdorff transform. The elementary operator steps in the harmonic oscillator occupation number representation make this technique suitable for introductory quantum mechanics classes.Zusammenfassung. Für eine Vielzahl von Operatoren wurden Ein- und Zweizentren-Matrixelemente des harmonischen Oszillators mit Operatormethoden abgeleitet. Die elementaren Operatorschritte in der Besetzungszahldarstellung wurden auf die bekannte Hausdorff-Formel zurückgeführt.

Recurrence relations for two-center harmonic oscillator integrals and their solutions using generating functions

Using the algebraic second quantization formalism, the general recurrence relations for some two-center harmonic oscillator integrals are obtained. The various known special cases for one-center integrals are evident in the formalism. Generating functions for these integrals are obtained and most of them are solved to find explicit expressions for polynomial, exponential, and Gaussian operator integrals. For these cases, all the generating integrals needed for utilizing the recurrence relations are also calculated in their simplest forms.

Generating functions and recurrence relations for harmonic oscillator matrix elements

A simple and straightforward way of obtaining generating functions and recurrence relations for harmonic oscillator integrals is presented. The method, which is based on the properties of unnormalized coherent states and canonical transformations, allows a unified treatment of several different physical problems. Matrix elements of Gaussian and exponential functions, Franck-Condon overlaps, and transition probabilities for time-dependent quadratic Hamiltonians are discussed as illustrative examples.

Closed formulas for one- and two-center harmonic oscillator integrals

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.

Recurrence relations for two-center harmonic oscillator integrals

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.

Hypervirial theorem and ladder operators. Recurrence relations for harmonic oscillator integrals

AbstractA 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.

General harmonic oscillator integrals by the operator method

AbstractThe operator technique with a minimum of commutator algebra is employed to calculate matrix elements of any number of operators between distorted, displaced harmonic oscillator wavefunctions. The results are valid for multidimensional integrals, and regardless of the extent of the Duschinsky effect. General recursion relations useful in machine calculations are given. The formalism is illustrated for the well‐known one‐dimensional Franck–Condon integrals.

Algebraic approach to one- and two-center harmonic oscillator integrals

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.

Graphical method for computing the determinant and inverse of a matrix. Generating functions for harmonic oscillator integrals

A graph G with n vertices is associated with the n*n matrix x. Det x and xij-1 are expressed in terms of sums over sets of subgraphs of G. The method is used to compute generating functions for products of overlaps involving harmonic oscillator wavefunctions.

Second quantization and the general two-centre harmonic oscillator integrals

Second quantization is used to derive the overlap integral between two harmonic-oscillator wave functions on different centres and with different force constants. The appropriate dipole moment and kinetic-energy integrals are also obtained.

A remark on the calculation of two-centre harmonic oscillator integrals

An improved method for the calculation of two-centre oscillator integrals is given for harmonic oscillators of equal frequencies. Formulae for a transition integral of arbitrary order are developed necessary for the calculation of optical transitions in colour centres.

On the use of second quantization for the calculation of two-centre harmonic oscillator integrals

Second quantization of the harmonic oscillator is used to derive two-centre overlap, dipole moment and kinetic energy integrals.