Matrix elements of the radial-angular factorized harmonic oscillator

1972 ◽  
Vol 12 (2) ◽  
pp. 185-196 ◽  
Author(s):  
T. G. Haskell ◽  
B. G. Wyboturne
Keyword(s):  
Harmonic Oscillator ◽  
Matrix Elements

Matrix Elements of the Lorentzian Term

1976 ◽  
Vol 31 (6) ◽  
pp. 553-556 ◽  
Author(s):  
Ch. V. S. Ramachandrá Rao
Keyword(s):  
Harmonic Oscillator ◽  
Matrix Elements ◽  
Practical Application ◽  
The Matrix

Recursion formulae for the matrix elements of the Lorentzian term 1/(C2 + q2) as well as 1/(C2 + q2)2, on the basis of harmonic oscillator eigenfunctions, are obtained. A practical application where these formulae would be useful is discussed

Quantum algebras and parity-dependent spectra

2007 ◽  
Vol 463 (2086) ◽  
pp. 2415-2427 ◽  
Author(s):  
C.V Sukumar ◽  
Andrew Hodges
Keyword(s):  
Quantum Optics ◽  
Harmonic Oscillator ◽  
Combinatorial Theory ◽  
Squeezed States ◽  
Quantum Algebras ◽  
Matrix Elements ◽  
Euler Numbers ◽  
Simple Harmonic Oscillator ◽  
Non Gaussian ◽  
Special Case

We study the structure of a quantum algebra in which a parity-violating term modifies the standard commutation relation between the creation and annihilation operators of the simple harmonic oscillator. We discuss several useful applications of the modified algebra. We show that the Bernoulli and Euler numbers arise naturally in a special case. We also show a connection with Gaussian and non-Gaussian squeezed states of the simple harmonic oscillator. Such states have been considered in quantum optics. The combinatorial theory of Bernoulli and Euler numbers is developed and used to calculate matrix elements for squeezed states.

Properties of the Quantum Double Oscillator

1975 ◽  
Vol 30 (12) ◽  
pp. 1730-1741 ◽  
Author(s):  
Jürgen Brickmann
Keyword(s):  
Harmonic Oscillator ◽  
Large Class ◽  
Electronic Transitions ◽  
Matrix Elements ◽  
Quantum Double ◽  
The Matrix ◽  
Analytic Expressions ◽  
Condon Factors ◽  
Finite Sums ◽  
Franck Condon

Abstract A formalism is presented to obtain approximate analytic expressions for the eigenstates and eigenvalues of a quantum double oscillator (QDO). The matrix elements of a large class of operators with respect to states of different double oscillators result as finite sums of explicit functions of the respective parameters. Matrix elements between states of a harmonic oscillator and a double oscillator are also determined. The analytic expressions were used to calculate Franck-Condon factors for electronic transitions including double oscillator anharmonicities.

On a Flexible Double Minimum Potential

1979 ◽  
Vol 34 (9) ◽  
pp. 1106-1112 ◽  
Author(s):  
J. Bohmann ◽  
W. Witschel
Keyword(s):  
Harmonic Oscillator ◽  
Molecular Spectroscopy ◽  
Additional Term ◽  
Numerical Control ◽  
Upper And Lower Bounds ◽  
Parabolic Cylinder ◽  
Matrix Elements ◽  
The Matrix ◽  
Minimum Potential ◽  
Parabolic Cylinder Functions

Abstract The Gauss-perturbed harmonic oscillator, a customary double minimum potential of molecular spectroscopy, is made more flexible by addition of a term α4 · exp(-γ4X4). The matrix elements of the additional term are calculated in the harmonic oscillator basis in terms of parabolic cylinder functions. A sum rule for matrix elements serves as an independent numerical control. The eigenvalues can be given by straightforward diagonalization of the Hamilton-matrix. In addition, upper and lower bounds are given for the partition function.

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