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Chinese Journal of Physics

Franck-Condon Factors for Harmonic Oscillators- An Operator Approach

Chen‐Shiung Hsue

Published Dec 1, 1994• 0 citations
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Abstract

We give an explicit derivation of the Franck-Condon factors for harmonic oscillators in the general case where both the origin and the frequencies are changed for one and two dimensions, using operator manipulations in our two-step method. We also discuss the case with rotated coordinates in the excited states of two dimensional transitions. The method is straightforward and can be easily generalized to higher dimensions.

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Two-step approach to the Morse-potential derivation of the energy spectrum and its application to Frank-Condon factors

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Physical Review A•Feb 1, 1986•9 citations
We give an explicit derivation of the Franck-Condon factors for harmonic oscillators in the general case where both the origin and the frequencies are changed for one and two dimensions, using operator manipulations in our two-step method. We also discuss the case with rotated coordinates in the excited states of two dimensional transitions. The method is straightforward and can be easily generalized to higher dimensions.
Multidimensional harmonic oscillator matrix elements of arbitrary products of exponentials for operators containing only linear and quadratic terms in the creation and annihilation operators are obtained. A simple and straightforward method is presented that leads to recurrence relations and generating functions for such matrix elements. Results are shown to be general enough to cover the cases of Franck–Condon overlaps, transition probabilities for time-dependent harmonic oscillators, and correlation functions for electronic spectra.
The purpose of this paper is two-fold. First, we would like to write down algebraic expression for the wave function of general excited state of harmonic oscillator which doesn't include derivative signs (this is to be contrasted with typical physics textbook which only gets rid of derivative signs for first few excited states, while leaving derivatives in when it comes to Hermite polynomial for general n). Secondly, we would like to write similar expression for two dimensional case as well. In the process of tackling two dimensions, we will highlight the interplay between Cartesian and polar coordinates in 2D in the context of an oscillator. All of the above mentioned results have probably been derived by others but unfortunately they are not easily available. The purpose of this paper is to make it easier for both students and general public to look up said results and their derivations, should the need arise. We also attempt to illustrate different angles from which one could look at the problem and this way encourage students to think more deeply about the material.
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.
A recently proposed two-step method is applied to the quantum-mechanical problem of one-dimensional Morse potential. The method is composed of two steps. In the first step a variational principle is applied to a generalized coherent-state ansatz which in the present case is nothing but a Gaussian with a shifted origin and modified frequency. Correlation corrections are then added in the second step by standard perburbation. The ``rigorous formula'' for the energy spectrum is derived by a lowest-order perturbation calculation. As a severe test of the theory, we have applied the method to calculate the Frank-Condon factors. The agreement with numerical calculation is excellent over eight orders of magnitude.