Simple Harmonic Oscillator: One Dimension

Claude N. Cohen-Tannoudji, Bernard Diu, Franck Laloë

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 "/>

James K. Freericks

Until this point, we have been solving problems that involve just two states or are reducible to two-state problems. While we could examine a plethora of complex quantum phenomena, it is not sufficient to describe the behavior of electrons in atoms or how quantum particles interact with each other. That all changes now as we work to set-up the quantum mechanics of the atom. There is much for us to do before we get to the hydrogen atom. We start, in this chapter, with the simple harmonic oscillator in one dimension. We show how to determine the canonical commutation relation between position and momentum from the fact that atomic spectra have sharp lines and then move into describing free particles and free particles on a circle. We introduce two of our operator identities, the Hadamard lemma and the Baker-Campbell-Hausdorff formula and we revisit exponential disentangling, now for the symplectic group Sp(2). Then we show how to solve the harmonic oscillator energy eigenstates using operator algebra, discuss Heisenberg uncertainty, determine harmonic oscillator wavefunctions, and examine coherent and squeezed states. We end with a discussion of applications of the simple harmonic oscillator.

John P. Lowe, Kirk A. Peterson

John P. Lowe

S. Balasubramanian

A simple derivation based on the generalized Hellmann–Feynman theorem is given for the three-term recursion relation for 〈rN〉 in the energy eigenstates of a hydrogenic atom and the harmonic oscillator in one, two, and three dimensions.