Laboratory Experiment Cite This: J. Chem. Educ. XXXX, XXX, XXX-XXX
pubs.acs.org/jchemeduc
Musical Example To Visualize Abstract Quantum Mechanical Ideas Forrest W. Eagle, Kyser D. Seaney, and Michael P. Grubb* Department of Chemistry and Biochemistry, Fort Lewis College, Durango, Colorado 81301, United States S Supporting Information *
ABSTRACT: 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. KEYWORDS: Physical Chemistry, Hands-On Learning/Manipulatives, Upper-Division Undergraduate, Spectroscopy, Quantum Chemistry, Misconceptions/Discrepant Events
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INTRODUCTION Quantum chemistry is perceived by students as one of the most difficult topics to understand in the chemistry curriculum.1 The reasons for this perception include a lack of motivation/interest, a deficiency or fear of the required mathematics, and the abstract nature of the subject.1−4 According to one study of Turkish chemistry students by Sö zbilir, surveyed students most commonly identified “abstract concepts” as an obstacle to their understanding of physical chemistry.2 Although the use of computer simulations to visualize quantum behavior is increasingly popular,5−8 there are few real-world, macroscopically observable demonstrations of quantum mechanical ideas. The physics of musical instruments, however, provides a surprisingly deep analogy to many seemingly abstract ideas in quantum chemistry and spectroscopy. Musical instruments produce pleasing sounds because they generate discrete (quantized) frequencies of sound, instead of a continuum of frequencies which we would perceive as noise. The discrete frequencies arise from standing waves confined within the instrument, and indeed, guitar strings are commonly used to demonstrate standing waves in one-dimensional quantum mechanical systems such as a particle in a box.9 Vibrating Chladni plates covered in a thin layer of sand are often used to visualize two-dimensional standing waves,10 but the resulting wave patterns are not analogous to commonly encountered systems in quantum mechanics since the boundary conditions imposed by the plates do not fix the wave amplitude to zero at the plate edges (Dirichlet boundary conditions). Drums do impose Dirichlet boundary conditions, and thus, their standing wave© XXXX American Chemical Society and Division of Chemical Education, Inc.
forms represent the wave functions of quantum particles confined in a circular two-dimensional well, and are essentially two-dimensional projections of atomic orbitals.11 A drum’s interaction with acoustic waves is analogous to an atom’s interaction with electromagnetic waves, and similar spectroscopic experiments can be performed on both systems. Here we describe spectroscopy experiments that can be performed on a drum, providing macroscopic examples of discrete energy levels and wave functions, superpositions of states, resonant excitation, Heisenberg’s uncertainty principle, Zeeman splitting, and transition moments. The experiment has two parts: first, the acoustic emission spectrum of the struck drum is recorded, in order to identify the frequencies of the drum’s vibrational modes. A struck drum is excited indiscriminately, leading to a superposition of excited drum states. Second, each drum mode is excited with its resonant monochromatic acoustic frequency (analogous to the optical excitation of an atom to a single excited state by a laser), in order to observe the waveform associated with each drum state. These pure waveforms are analogous to the atomic orbital wave functions, and thus, the experiment provides an excellent demonstration of the origin of the atomic orbital shapes and the role of resonance in optical spectroscopy. Received: July 25, 2017 Revised: October 9, 2017
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DOI: 10.1021/acs.jchemed.7b00413 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
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Striking the drum with a hammer or drumstick excites a superposition of many drum modes simultaneously. Recording the resulting sound with a microphone and applying a Fourier transform to the raw sound data yields a frequency spectrum, where the relative contributions of each vibrational mode can be easily observed from the discrete peak intensities (Figure 2). Which modes are excited depends strongly upon where the drum is struck. For instance, hitting the drum dead-center will excite the 1s mode efficiently but will not excite the 2p mode at all, since the center is a node of the 2p waveform. The frequency of each mode is dependent upon the size of the drum and the tension of the membrane. These relationships can be qualitatively deduced from the relationship between the frequency of a wave f and its propagation speed s and wavelength λ: s f= (1) λ
HAZARDS Some individuals may have photosensitive conditions triggered by flashing lights, such as epilepsy or migraines, and thus should be informed beforehand that a strobe light will be used during the experiment so that they can take necessary precautions or withdraw from that portion of the lab. Amplified electrical current is always dangerous, so the instructor should completely insulate any exposed wiring coming from the audio amplifier, and students should be educated on the proper handling of electrical equipment.
Laboratory Experiment
PART 1: ACOUSTIC EMISSION
Mode Frequencies
EXPERIMENTAL SETUP The latex drum for the experiment was constructed by stretching 0.006 in. thick latex rubber across a 4.5 in. PVC pipe, and held in place by rubber bands. These parameters are not critical, and in fact it is instructive to have students make drums of different sizes with different latex thicknesses to demonstrate how the resonant frequencies of the drum are affected. The acoustic emission experiment requires only a microphone, connected to a computer with either a USB plug or 3.5 mm audio jack (Figure 1). It is not necessary to use a high-quality microphone. Our results were collected using the built-in microphone on a low-cost webcam (Logitech C720).
The discrete drum frequencies arise because only wavelengths that fulfill the radial and angular boundary conditions of the drum can lead to stable standing waves on the drum surface. Therefore, the λ of each mode is dependent only on the size and shape of the drum. The larger the drum is, the longer the standing wavelengths are, and thus the corresponding frequency is lowered. The size of the drum is analogous to the “size” of the Coulombic potential confining the electron to the nucleus in atoms. The other variable in this equation is the wave propagation speed s, which can be altered by changing the drum membrane tension. Increasing the tension of the drum increases the speed at which the acoustic wave propagates through the membrane, and thus, the frequency of each mode increases proportionally. This is how musical drums are tuned to the desired pitch. An interesting phenomenon is witnessed when the rubber drum membrane is stretched asymmetrically, tighter along one axis than the other: the p and d modes split into two frequencies (Figure 2, red spectrum). This removes the degeneracy of the frequency of different spatial orientations of the orbitals, and is analogous to the Zeeman splitting of degenerate atomic orbitals in an external magnetic field. This can be confirmed in the acoustic absorption part of the experiment: the split frequencies correspond to vibrational waveforms oriented in orthogonal directions.
Figure 1. Experimental setup diagram (left) and photograph (right). Materials include the following: a PC with a 192 kHz sound card (Creative Sound Blaster Audigy FX), an audio amplifier (Pyle PFA 200, 60 W), an LED light bulb (Bessky Outdoor Camping USB 5 W LED), a USB microphone (Logitech 720p webcam/mic), subwoofer speaker (Pyle GearX 300 W), and a latex drum (4.5 in. PVC pipe, 0.006 in. natural latex rubber sheet). The audio speaker should be connected to the “right” stereo output of the amplifier, and the strobe light should be connected to the “left” stereo output of the amplifier in order to be consistent with our control software.
Heisenberg’s Uncertainty Principle
The width of each peak in the drum frequency spectrum is not the same, and in fact depends upon the lifetime of acoustic emission from the corresponding mode, as determined by the frequency−time uncertainty principle:14
The acoustic absorption experiment requires an audio source to excite the drum, and a strobe light to observe the resulting motion of the drum membrane. Both can be controlled from the audio output of the computer’s sound card, amplified by a 60 W amplifier. The strobe light pulse is carried by the “left” stereo signal, and the audio frequency is carried by the “right” stereo signal. A subwoofer speaker should be used as the audio source, since the drum will likely have resonant frequencies in the 20− 200 Hz range. White USB LED light bulbs make excellent strobe lights, since LEDs can be flashed on and off quickly and only require a peak voltage of
