The Relaxation of the Vibrational Motion in an Isolated of Harmonic Oscillators

Elmar G. Petrov, V. I. Teslenko

L. L. Northup

Shabnam Siddiqui

In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.

Antonia Chimonidou

Matteo Bonfanti, Keith H. Hughes, Irène Burghardt +1 more

Vibrational relaxation is a key issue in chemical reaction dynamics in condensed phase and at the gas‐surface interface, where the environment is typically highly structured and cannot be expressed in terms of a simple friction coefficient. Rather, full knowledge of the coupling of the molecular oscillator to the environment is required, as typically subsumed in the spectral density of the environmental coupling. Here, we focus on harmonic Brownian motion and investigate the effectiveness of classical, canonical position autocorrelation functions to compute the spectral density of the coupling needed to describe vibrational relaxation in complex environments. Classical dynamics is performed on model systems, and several effects are investigated in detail, notably the presence of anharmonicity, the role of a high‐frequency “Debye” cutoff in the environment and the influence of the detailed structure of the latter. The spectral densities are then used in standard independent oscillator Hamiltonian models which are numerically solved at T = 0 K to investigate quantum relaxation and decoherence. image

Amita Dua, Chayannika Singh

Linear harmonic oscillator is the simplest system to understand the vibrational motion of microscopic particles, therefore, it is also called as simple harmonic oscillator. For the sake of simplicity, we treat here only the simple harmonic motion of a single particle which oscillates to and fro about its mean position. Thus, oscillations take place periodically in one dimension only, like ball of mass m , tied to a spring through a support shown in Fig. 6.1. Fig. 6.1 Schematic diagram of linear harmonic oscillator. https://www.w3.org/1999/xlink" content-type="black-white" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003490135/247a5333-15ea-46ba-8a55-ecb37a472373/content/fig6_1_B.tif "/>