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American Journal of Physics

Visualizing the phonon wave function

Scott C. Johnson, T. D. Gutierrez

Published Mar 1, 2002• 10 citations
Publisher

Abstract

A phonon often is described as “a quantum of lattice vibration,” but this description can be difficult to reconcile with the wave functions explored in a typical undergraduate quantum mechanics class. A phonon wave function is similar to the harmonic oscillator wave functions studied in introductory quantum mechanics, except that it is many-dimensional. We suggest a way to visualize the probability density for this very high-dimensional wave function. The resulting pictures are especially clear and intuitive for a coherent state, which is both a good approximation to a sound wave and a discrete analog to laser light. These pictures can also provide a qualitative introduction to quantum field theory.

DOI
10.1119/1.1446858

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A phonon often is described as “a quantum of lattice vibration,” but this description can be difficult to reconcile with the wave functions explored in a typical undergraduate quantum mechanics class. A phonon wave function is similar to the harmonic oscillator wave functions studied in introductory quantum mechanics, except that it is many-dimensional. We suggest a way to visualize the probability density for this very high-dimensional wave function. The resulting pictures are especially clear and intuitive for a coherent state, which is both a good approximation to a sound wave and a discrete analog to laser light. These pictures can also provide a qualitative introduction to quantum field theory.
Phonon is undoubtedly one of the most important concepts in the physics of materials. Phonon is the result when we quantize vibrational field. Whereas Schrodinger’s wave function method is the most popular and intuitive method in doing the first quantization, one usually uses Heisenberg’s operator method in the second quantization. We feel that there is a methodological and pedagogical discontinuity here. So in this paper we will use Schrodinger’s wave functional method to quantize the lattice vibration to produce phonons. Wave functional is difficult due to the nondenumerably infinite number of dimensions of its domain. In this paper we will approximate this infinity through the discrete nature of crystalline lattice so that phonon can be represented by wave function with many variables
Mechanical motions of atoms near their equilibrium geometry are approximated by a universal mapping on a collection of independent harmonic oscillators. The respective frequencies are characteristic to the material, for example, lattice phonons or molecular vibrations. The quantum description of atomic motions involves the solution of the Schrödinger equation for the harmonic oscillator in the space of proper wave functions. The energy levels obtained are shown to be equally spaced where the level spacing is proportional to the oscillator frequency. The stationary solutions are identified in terms of Hermite polynomials, demonstrating remarkable differences from the classical harmonic oscillator. The importance of the zero-point energy is emphasized in the context of the stability of chemical bonds, where the Harmonic approximation is shown to be reasonably valid for typical interatomic bonds in standard thermal conditions. This explains the relevance of the harmonic approximation for analyzing the absorption spectrum of infrared radiation by molecules.
Abstract Quantum field theory (QFT) is the basis of some of the most fundamental theories in modern physics, but it is not an easy subject to learn. In the present article we intend to pave the way from quantum mechanics to QFT for students at early graduate or advanced undergraduate level. More specifically, we propose a new way of visualising the wave function Ψ of a linear chain of interacting quantum harmonic oscillators, which can be seen as a model for a simple one-dimensional bosonic quantum field. The main idea is to draw randomly chosen classical states of the chain superimposed upon each other and use a grey scale to represent the value of Ψ at the corresponding coordinates of the quantised system. Our goal is to establish a better intuitive understanding of the mathematical objects underlying quantum field theories and solid state physics.
Quantum mechanics holds a very important position as the theoretical foundation of the micro world. Harmonic oscillator system is a very classical model in quantum mechanics. In this paper, the wave function of a three-dimensional linear harmonic oscillator is derived based on the linear harmonic oscillator model in quantum mechanics. According to the probabilistic interpretation of quantum mechanics, we obtain the probability density function of the harmonic oscillator electron cloud. In this paper, Monte Carlo selection method is used for sampling, and Mathematica software is used to draw electron cloud images with different quantum number, so that people can intuitively see the distribution of electron cloud in three-dimensional space.
This research paper presents a simulation of a particle in a quantum mechanical oscillator, modeled as a multidimensional wave function. The simulation was created by solving the time-dependent Schrodinger equation, which was discretized in both the spatial and time domains, and then approximated using a finite difference method. The objective of this research was to gain a deeper understanding of the behavior of a particle in a quantum mechanical oscillator and to provide visualizations and calculations of relevant physical observables. Initial conditions for the wave function, such as a Gaussian wave packet, were chosen and parameters for the simulation, such as the spatial step size and total simulation time, were set. The results of the simulation showed the evolution of the wave function over time, providing a clear picture of the particles behavior within the potential well of the oscillator. Additionally, physical observables, such as the average position, average kinetic energy, and average potential energy, were calculated and used to validate the simulation and the predictions of quantum mechanics. In conclusion, this research provides a comprehensive understanding of the behavior of a particle in a quantum mechanical oscillator and highlights the importance of simulations in advancing our knowledge of quantum mechanics. The results of this research can contribute to various fields, such as quantum computation and quantum control, and have the potential to drive further breakthroughs in these fields. KEYWORDS: Quantum mechanical oscillator, particle behavior, multidimensional wave function, Schrödinger equation, finite difference method, physical observables, simulation, quantum mechanics, quantum computation, quantum control.