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American Journal of Physics

An acoustic analog for a quantum mechanical level-splitting route to band formation

Marissa D’Onofrio, Mitchell Crum, Shawn A. Hilbert, Herman Batelaan, Timothy Canalichio, Tyler Bull

Published Oct 20, 2016• 10 citations
Publisher

Abstract

This paper explores band structure in a simple acoustic apparatus that acts as an analog to the quantum infinite square well with multiple delta-function perturbations. The apparatus can be used to visualize abstract quantum phenomena in a concrete and easily understandable way. It consists of regular sections of PVC pipes connected by variable aluminum diaphragms to allow coupling between the pipe sections. The equivalence between standing waves in the acoustic system and stationary states in the quantum system is examined for multiple-cavity situations. We show that the experimental results from the acoustic system and the analytic solutions of the quantum system demonstrate the same resonance structure. We also experimentally show that the acoustic system supports band structure and that the band width is dependent on the hole size of the diaphragms.

DOI
10.1119/1.4962227

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This paper explores band structure in a simple acoustic apparatus that acts as an analog to the quantum infinite square well with multiple delta-function perturbations. The apparatus can be used to visualize abstract quantum phenomena in a concrete and easily understandable way. It consists of regular sections of PVC pipes connected by variable aluminum diaphragms to allow coupling between the pipe sections. The equivalence between standing waves in the acoustic system and stationary states in the quantum system is examined for multiple-cavity situations. We show that the experimental results from the acoustic system and the analytic solutions of the quantum system demonstrate the same resonance structure. We also experimentally show that the acoustic system supports band structure and that the band width is dependent on the hole size of the diaphragms.
In this paper we study an acoustic system comprised of two identical closed tubes connected with a short one, and a quantum-mechanical square well with a delta function potential. Both systems can be thought of as mathematically equivalents to the homogeneous string with a concentrated mass at its middle point. We describe also a simple experimental device to measure the resonances of the acoustic system as a function of the connecting tube section. The experimental setup and the theoretical approach we have employed can be used in both undergraduate and graduate level courses. Moreover, this work may be useful as a teaching aid in other experimental research projects such as for instance the study of the emergence of the band structure in solid state physics.
Abstract We present an educational approach aimed at introducing the fundamental concepts of quantum mechanics (QM). By exploiting the formal analogy between sound wave behavior in an acoustic pipe (a drinking straw) and the quantum infinite square potential well, we provide an intuitive framework that explains the origin of energy quantization in bound quantum systems (such as atoms), introduces the concept of wavefunction, and lays the groundwork for discussing the Heisenberg uncertainty relations. The proposed method enables low-cost experimental activities that actively involve students, facilitating a meaningful connection between theoretical principles and empirical observations. Moreover, it encourages critical reflection on the Copenhagen interpretation of the wavefunction, promoting a deeper conceptual understanding.
When many individual atoms come together to form a solid, their interaction splits their electronic energy levels to form continuous bands separated by forbidden energy ranges known as band gaps. Introducing defects in a solid results in new electron energy levels that may lie inside the bandgaps. The presence of these defect levels is the heart of the semiconductor-based devices that play a significant role in the modern world. Quantum mechanics provides the best description of interacting atoms. However, band formation is not unique to small-scale atomic interactions but rather is a result of the wave-nature of Schrödinger's equation, which governs quantum mechanics. Using oscillations in a mass-spring system, we present a table-top, classical analog to the quantum system illustrating how defects in a one-dimensional lattice produce changes to the band structure. A pair of masses connected by a spring plays the role of a single atom. Interactions between “atoms” are introduced with weak coupling springs producing two distinct frequency bands from the translational and fundamental modes. Defects are introduced by altering an oscillator pair's total mass or internal spring constant. We provide the theoretical groundwork and experimental verification of the model along with a discussion of the value and limitations of the model as a macroscopic tool to visualize the microscopic world.
Quantum mechanics is a notoriously difficult subject to learn, due to a lack of real-world analogies that might help provide an intuitive grasp of the underlying ideas. Discrete energy levels and absorption and emission wavelengths in atoms are sometimes described as uniquely quantum phenomena, but are actually general to spatially confined waves of any sort. Here, we provide an experiment demonstrating the acoustic spectroscopy of a drum. We show that a struck drum emits sounds of discrete frequencies (tones), and that the lifetime of each emitted tone is directly related by Heisenberg’s uncertainty principle to its line width in the drum’s frequency spectrum. We also show that a still drum absorbs only those same frequencies when exposed to monochromatic sound from an audio speaker. The resonant motion of the drum membrane is too fast to see by eye (>60 Hz), but can be observed with the aid of a strobe light. The observed resonant modes of the drum are the eigenfunctions of a particle trapped in an infinite circular well, and analogous to the shapes of atomic orbitals (1s, 2p, 3d, 2s, etc.). This experiment can be built for under $50, and is an excellent demo or lab exercise for general and physical chemistry courses to provide a visual example of abstract quantum mechanical ideas.
Abstract The experimental investigation of quantum devices incorporating mechanical resonators has opened up new frontiers in the study of quantum mechanics at a macroscopic level. It has recently been shown that surface acoustic waves (SAWs) can be piezoelectrically coupled to superconducting qubits, and confined in high-quality Fabry–Perot cavities in the quantum regime. Here we present measurements of a device in which a superconducting qubit is coupled to a SAW cavity, realising a surface acoustic version of cavity quantum electrodynamics. We use measurements of the AC Stark shift between the two systems to determine the coupling strength, which is in agreement with a theoretical model. This quantum acoustodynamics architecture may be used to develop new quantum acoustic devices in which quantum information is stored in trapped on-chip acoustic wavepackets, and manipulated in ways that are impossible with purely electromagnetic signals, due to the 10 5 times slower mechanical waves.