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CRC Press eBooks

Linear Harmonic Oscillator: Quantisation of Vibrational Energy

Amita Dua, Chayannika Singh

Published Mar 15, 2024• 0 citations
Publisher

Abstract

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 "/>

DOI
10.1201/9781003490135-6

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A linear harmonic oscillator is a system having 1 degree of freedom and executing in its direction a harmonic oscillation . We devote this heading to a study of an important physical system called the one-dimensional harmonic oscillator. We are concerned with a one-dimensional motion of a point mass that is attracted to a fixed center by a restoring force https://www.w3.org/1998/Math/MathML"> F x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , which is proportional to the displacement https://www.w3.org/1998/Math/MathML"> x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> from that center: 6.1 https://www.w3.org/1998/Math/MathML"> F x = - m ω 2 x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math6_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>
All the properties and motion of a particle of mass ‘ m ’ which moves with potential energy, V ( x, y, z ) can be obtained from solution of well behaved Schrodinger equation https://www.w3.org/1998/Math/MathML" display="block"> − ħ 2 2 m ∇ 2 ψ + V ( x , y , z ) ψ = E ψ 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/math4-1_B.tif "/>
A simple harmonic oscillator, or oscillator in short, is one of the most important systems in physics as well as in quantum mechanics. The path integral and Hamiltonian of a simple harmonic oscillator are leading exemplars for the description of a wide class of physical systems. An oscillator is a theoretical model of great utility, primarily due to its simplicity that allows for the exact derivation of many results. The simple harmonic oscillator can be generalized to the case of infinitely many oscillators, and is also the starting point for analysis of many nonlinear quantum systems. Path integrals for the simple harmonic oscillator are all based on Gaussian integration, which has been discussed in Section 7.2. The harmonic oscillator is described by Gaussian path integrals , and is a bedrock of the general theory of path integration. Moreover, all perturbation expansions of nonlinear path integrals about the oscillator path integral, which includes the semi-classical expansion, are based on results of Gaussian integration. The properties of the harmonic oscillator are analyzed in the coordinate space representation, and its path integral as well as its correlation functions are analyzed. All formulas and derivations are for Euclidean time. In Sections 11.1 and 11.2 the Hamiltonian and state space of the oscillator are studied and the correlator is derived using state space methods. In Section 11.2 the infinite time oscillator is introduced and in Sections 11.3–11.5 the path integral for the oscillator is evaluated.
In physics, harmonic motion is among the most representative types of motion. A simple harmonic oscillator is often the source of any vibration with a restoring force proportional to Hooke’s law. Every minimum potential has a solution in the form of the harmonic oscillator potential. Little oscillations at the minimum are characteristic of almost all natural potentials and of many quanta mechanical systems. Harmonic motion is an essential building block for these more complex uses. The Schrödinger equation is a defining feature of the harmonic oscillator. Here, we demonstrate that the time-frequency plane is a useful tool for analyzing their dynamics. We numerically integrate several examples involving different input forces and demonstrate that the oscillations are clearly displayed and easily interpretable in the time-frequency plane.
The n -puzzle is a classical problem for modeling algorithms. For n -puzzle there are https://www.w3.org/1998/Math/MathML" display="inline"> n + 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_1.tif "/> different objects: n cells and one empty cell. Each object can be coded by https://www.w3.org/1998/Math/MathML" display="inline"> ρ = ⌈ log 2 ⁡ n + 1 ⌉ https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_2.tif "/> qubits and a configuration of https://www.w3.org/1998/Math/MathML" display="inline"> n + 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_3.tif "/> objects can be represented by a register of https://www.w3.org/1998/Math/MathML" display="inline"> z := ρ ⋅ ( n + 1 ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_4.tif "/> qubits https://www.w3.org/1998/Math/MathML" display="inline"> | x ⟩ https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_5.tif "/> . The function https://www.w3.org/1998/Math/MathML" display="inline"> g ( x ) https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_6.tif "/> is represented a unitary operator T . T acts on the https://www.w3.org/1998/Math/MathML" display="inline"> z + 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_7.tif "/> qubits with https://www.w3.org/1998/Math/MathML" display="inline"> x ∈ B z https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_8.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> c ∈ B 1 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_9.tif "/> https://www.w3.org/1998/Math/MathML" display="block"> T ⋅ | x ⟩ | c ⟩ = | x ⟩ | f ( x ) ⊕ c ⟩ . https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003374404/4745344c-4074-4386-85e6-ca8a3849de1f/content/math10_10.tif "/>
One of the most commonly used quantum mechanical systems is that of the simple harmonic oscillator . The prototype system is the simple pendulum. Classically, the motion of the system exhibits harmonic, or sinusoidal oscillations. The sinusoidal motion of classical mechanics is the simplest system that produces oscillatory behavior, and therefore is found in almost an infinity of physical systems. Thus, the properties of the harmonic oscillator become important for their use in describing quite diverse physical systems. In a solid, the motion of the atoms is much like the motion of free particles, with the important exception that the atoms are forced on average to retain positions within the solid that specify a particular crystal structure. The motion remains that of a harmonic oscillator, but now the shift of the wave function and the energy levels are hybrids of the confinement harmonic oscillator and the magnetic harmonic oscillator.