# Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

## Abstract

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partitionmore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22447592

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics; Journal Volume: 353; Other Information: Copyright (c) 2014 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; ANNIHILATION OPERATORS; EIGENSTATES; ELECTRON-PHONON COUPLING; HARMONIC OSCILLATORS; OSCILLATORS; PARTIAL DIFFERENTIAL EQUATIONS; PARTITION FUNCTIONS

### Citation Formats

```
Toutounji, Mohamad, E-mail: Mtoutounji@uaeu.ac.ae.
```*Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach*. United States: N. p., 2015.
Web. doi:10.1016/J.AOP.2014.10.010.

```
Toutounji, Mohamad, E-mail: Mtoutounji@uaeu.ac.ae.
```*Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach*. United States. doi:10.1016/J.AOP.2014.10.010.

```
Toutounji, Mohamad, E-mail: Mtoutounji@uaeu.ac.ae. Sun .
"Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach". United States.
doi:10.1016/J.AOP.2014.10.010.
```

```
@article{osti_22447592,
```

title = {Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach},

author = {Toutounji, Mohamad, E-mail: Mtoutounji@uaeu.ac.ae},

abstractNote = {This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.},

doi = {10.1016/J.AOP.2014.10.010},

journal = {Annals of Physics},

number = ,

volume = 353,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}