Orbital Shaped Standing Waves Using Chladni Plates

doi.org/10.26434/chemrxiv.10255838.v1

Eric Janusson, Johanne Penafiel, Andrew Macdonald, Shaun MacLean, Irina Paci, J Scott McIndoe

Submitted date: 05/11/2019 • Posted date: 13/11/2019 Licence: CC BY-NC-ND 4.0 Citation information: Janusson, Eric; Penafiel, Johanne; Macdonald, Andrew; MacLean, Shaun; Paci, Irina; McIndoe, J Scott (2019): Orbital Shaped Standing Waves Using Chladni Plates. ChemRxiv. Preprint. https://doi.org/10.26434/chemrxiv.10255838.v1

Chemistry students are often introduced to the concept of atomic orbitals with a representation of a one-dimensional standing wave. The classic example is the harmonic frequencies which produce standing waves on a guitar string; a concept which is easily replicated in class with a length of rope. From here, students are typically exposed to a more realistic three-dimensional model, which can often be difficult to visualize. Extrapolation from a two-dimensional model, such as the vibrational modes of a drumhead, can be used to convey the standing wave concept to students more easily. We have opted to use Chladni plates which may be tuned to give a two-dimensional standing wave which serves as a cross-sectional representation of atomic orbitals. The demonstration, intended for first year chemistry students, facilitates the examination of nodal and anti-nodal regions of a Chladni figure which students can then connect to the concept of quantum mechanical parameters and their relationship to atomic orbital shape.

File list (4)

Chladni manuscript_20191030.docx (3.47 MiB) view on ChemRxiv download file

SUPPORTING INFORMATION Materials and Setup Phot... (6.79 MiB) view on ChemRxiv download file

MathSI.pdf (74.45 KiB) view on ChemRxiv download file

Chladni plates 2 - Petite.mov (1.56 MiB) view on ChemRxiv download file 1

1 Orbital shaped standing waves using Chladni plates

3 Eric Janusson, Johanne Penafiel, Shaun MacLean, Andrew Macdonald, Irina Paci*, J. Scot

4 McIndoe*

6 Department of Chemistry, University of Victoria, P.O. Box 3065 Victoria, BC V8W3V6, Canada.

7 Fax: +1 (250) 721-7147; Tel: +1 (250) 721-7181; E-mail: [email protected], [email protected]

8 2

12 3

13 Abstract

14Chemistry students are often introduced to the concept of atomic orbitals with a representation

15of a one-dimensional standing wave. The classic example is the harmonic frequencies which

16produce standing waves on a guitar string; a concept which is easily replicated in class with a

17length of rope. From here, students are typically exposed to a more realistic three-dimensional

18model, which can often be difficult to visualize. Extrapolation from a two-dimensional model,

19such as the vibrational modes of a drumhead, can be used to convey the standing wave concept

20to students more easily. We have opted to use Chladni plates which may be tuned to give a two-

21dimensional standing wave which serves as a cross-sectional representation of atomic orbitals.

22The demonstration, intended for first year chemistry students, facilitates the examination of

23nodal and anti-nodal regions of a Chladni figure which students can then connect to the concept

24of quantum mechanical parameters and their relationship to atomic orbital shape.

26 Graphical Abstract

28 Keywords 4

29First-Year Undergraduate / General, Demonstrations, Hands-On Learning / Manipulatives, Atomic

30Properties / Structure.

36 5

37 Introduction

39 Understanding that an electron in an orbital can be thought of as a three-dimensional

40standing wave is a tough concept to wrap one’s head around. A one dimensional standing wave

41is easy to visualize and indeed demonstrate in a lecture environment - a length of rope held at

42each end by volunteers can be induced into a standing wave, which if energetic enough, will

43readily display nodes. 1,2 However, making the jump to a three-dimensional wave is much

44tougher, perhaps because our minds run out of dimensions to consider. Length, breadth, and

45depth are conceivable; but where does the amplitude go? The meanings of the different regions

46of the wave can be even more confusing given the multi-dimensionality of the problem. If we

47back off one dimension, and consider a two-dimensional standing wave, it becomes easier - we

48can use the third dimension to convey the amplitude, and the standing waves, if chosen

49carefully, can illustrate the cross-sectional shape of the orbital. A physical demonstration of wave

50behaviour in two dimensions, prior to three-dimensional illustration through computer-based

51models, allows for direct student involvement and easier conceptualization of a complex

52problem in first-year Chemistry.

53 The shape of orbitals has been taught to students using media as diverse as circular

54magnets, 3 Styrofoam models, 4,5 beakers of coloured water, 6 websites, 7 and balloons. 8 Various

55simple models have been used to introduce the concept of standing waves to students in the

56context of atomic (and/or molecular) orbitals. Instructors have tapped the rim of a coffee cup to

57relate nodes to the energies of aromatic molecular orbitals, 9 have used Kundt’s tubes (devices

58that set up acoustic standing waves in different gaseous media as a characterization tool), 10 and 6

59have reproduced Wiener’s historic experiment on a standing light wave 11 using a laser pointer. 12

60Computer-based orbital illustrations are an extremely valuable advance, allowing direct

61visualization of the quantum mechanical solutions for atomic orbitals. There exist excellent

62examples in the literature of two- and three-dimensional visualizations of atomic orbitals,

63making use of custom or proprietary software in order to produce plots of atomic and molecular

64orbitals and examine the orbital’s shape based on a given change in parameters. 13–20

65 Understanding and correctly interpreting these illustrations has become a fundamental

66learning goal in introductory quantum atomic structure. However, orbital shapes can be

67conceptually disconnected from the ideas of wave behavior in quantum chemistry, in the same

68way Styrofoam models are. Demonstrations of wave mechanics, preferably with direct student

69involvement, are an effective way to illustrate and promote the correct conceptualization of

70wavefunction shapes, their properties and the quantum mechanical wave equation. 21–23 An

71appealing and effective way to perform this illustration is by using the Chladni experiment

72discussed in the following pages. It should be noted here that the two-dimensional acoustic

73waves that produce Chladni paterns have been used previously to illustrate theoretically wave

74behaviour: The two-dimensional wave equation is discussed at some length in Donald

75McQuarrie’s standard textbook “Quantum Chemistry” 22 and pictures of a drum with low-

76frequency Chladni paterns are given on page 63. A series of analogies including Chladni

77paterns, jellyfish and viscous glop are used in a fascinating account of quantum chemical

78equivalencies by Dylan Jayatilaka from the University of Southern Australia. 24 The purpose of the

79current article is to illustrate how Chladni paterns can be used to physically, interactively

80demonstrate wave behaviour in the first-year Chemistry class. 7

81 A two-dimensional standing wave can be generated using common percussion

82instruments. When the appropriate harmonic frequency is applied to a drumhead the wave

83patern remains in one position as the amplitude of the wave fluctuates. Above the fundamental

84harmonic frequency (first harmonic), the drumhead becomes segmented by nodal lines in which

85there is no change in amplitude; conversely, in between these nodal lines are antinodes where

86the change in amplitude is at a maximum. 25,26 The standing wave formed on a drumhead is a

87useful analogy; however, these waves are not detectable to the unassisted naked eye, and we

88can more easily control and visualize two-dimensional standing waves using devices known as

89Chladni plates. Ernst Chladni (1756-1827) was a German physicist and musician most well-known

90for his study of vibrational modes produced on metal plates. In Chladni’s experiment, excitation

91was achieved by drawing the bow of a violin across the edge of the plate. Sprinkling sand across

92the surface of the plate while doing so caused the sand to setle in specific paterns shaped by

93the nodal regions of the vibrational mode, now referred to as Chladni figures. These paterns

94appear as the sand is tossed away from regions of the plate where a strong vibration is occurring

95(primarily the antinodal regions) and setles into regions where nodal lines are present (there is

96a particularly striking music video involving a variety of visualized standing waves by N. J.

97Stanford in which a Chladni plate plays a prominent role). 27 The modern method of generating

98Chladni figures is achieved with a frequency generator, a voice coil and metal plate. This in-class

99demonstration enables students to participate in active learning by producing a tangible

100representation of an atomic orbital, which they can control. The process of creating these

101beautiful cymatic figures is a unique way to intrigue and educate students.

102 8

103 Materials

104 The source of the excitation signal used was an Keysight (Agilent) 33522A function-

105generator. A custom power-amplifier, consisting of a Canakit (cat. # CK003) 10-Wat single-

106channel audio amplifier circuit, volume (amplitude) control and associated power-supply, was

107used to boost the signal to the levels required to drive the voice-coil that vibrates the plate. The

108voice-coil used was a PASCO SF-9324 mechanical wave driver. The function-generator was

109connected to the power-amplifier by a BNC cable. The power-amplifier positive and negative

110outputs were connected to the voice-coil by banana-connector cables. The plates used were

111round and square aluminium metal plates were used with measurements of 24 cm in diameter,

112and 24×24 cm, respectively. Each plate was 0.1 cm in thickness. The plates were anchored to

113the voice-coil at their centre. ACS reagent grade sodium bicarbonate powder was purchased

114from EMD Chemicals and was used as the visualization medium.

115

116 Procedure

117 A thin layer of sodium bicarbonate is poured over the surface of the metal plate prior to

118applying a frequency. The frequency generator was set to deliver a continuous 100 mVpp sine

119wave (without a sound-dampening cover, less than 70 mVpp may be more suitable for higher

120frequency paterns and will often yield similar results), and the frequency was swept through a

121range of approximately 50 to 4000 Hz during a test of the apparatus. Different sized plates may

122produce different vibrational modes of interest at any given frequency, so a sweep from

123approximately 50 Hz to as high as 5 kHz (see Safety section) at a slow rate is recommended in

124order to observe the progression of standing wave paterns in order to find the frequencies that 9

125produce paterns of interest. For demonstration purposes, custom software was employed to

126quickly recall and apply the frequency and amplitude settings that produce desired standing

127wave paterns.

128

129

130 Figure 1. Schematic of the demonstration setup

131

132

133 Hazards and Safety

134 At higher frequencies Chladni plates can become extremely loud and high-pitched. 28

135Therefore, during a demonstration it is recommended the plates are placed within a transparent,

136sound-dampening box to prevent aural discomfort. When initially testing a Chladni setup, ear

137protection is recommended since the operator will spend considerably more time working with

138the setup than during the relatively short demonstration.

139 Sodium bicarbonate (baking soda) is generally safe to handle; however, some care must

140be taken to avoid inhalation or eye contact. There is an increased risk of aerosolizing fine sodium

141bicarbonate due to the nature of this experiment. Inhalation and exposure is not considered

142hazardous as baking soda is generally harmless. 10

143 Discussion

144 Many students struggle to imagine the wave-like properties of an electron in three

145dimensions; a concept which becomes crucial later in the first-year semester and beyond for

146topics including periodic trends, chemical bonding, and reactivity. The shapes of atomic orbitals

147are also critical for students to fully understand as the concept is used to expound on molecular

148shapes, hybridization and molecular orbitals, which necessitate a spatial understanding of

149atomic orbitals. The association of electronic behaviour with the concept of a standing wave can

150also help explain questions like “Why doesn’t the electron fall into the nucleus?” 29 and “How

151does the 2p electron go from one side of the wavefunction through the node to the other side?”.

15230 The goal of this experiment is to introduce first year chemistry students to fundamental

153quantum theory essential in understanding atomic orbitals by facilitating a means for students to

154visualize simple atomic orbitals in three-dimensions through the extrapolation of the analogous

155two-dimensional standing-wave system generated by means of Chladni plates.

156

157Physical underpinnings of the experiment

158 Standing waves are produced by the interference of outgoing and incoming waves in a

159material, at resonance frequencies, which are device-specific. Guitar strings, for example, are

160anchored at both ends. Plucking the string creates a set of outgoing waves. Each of these reflects

161at the ends of the string into returning waves. The interference between an outgoing and a

162reflected wave produces a standing wave with nodes at the anchoring points, which cannot,

163presumably, vibrate. 1 All the harmonics have these two nodes, as well as an increasing number

164of nodes along the string length, for higher harmonics (see SI). The fundamental frequency of 11

165the vibration depends on the length of the string, the tension in the string, and its mass, so each

166of these factors can in turn be used to change the tune that is played. All the harmonics are

167solutions of the wave equation for the guitar string, with boundary conditions given by the

168requirement for nodes at the anchoring points.

169 A drum or stretched membrane functions in a similar way, with boundary conditions

170(nodes at the edge because of the atachment of the membrane) imposing the periodicity of the

171vibrational wave. The two-dimensional standing waves are solutions of a two-dimensional wave

172equation in cylindrical coordinates (for circular membranes) or Cartesian coordinates (for square

173membranes), and are shaped such that the width of the membrane fits multiples of half a

174wavelength (For pictures and animations of the square membrane vibrational modes, see the

175Partial Differential Equations app for Mathematica, by Dzamay, A.). 31 Nodes on a circular

176membrane are diametric or circular, whereas nodes on square membranes can become

177significantly more complex. The free-edge Chladni apparatus used in this experiment has a

178distinct set of non-zero boundary conditions, based on the confinement of the wave to the size

179and rigidity of the plate, and the elastic behavior of the material. However, because of the

180principles of wave reflection and the symmetry of the problem, nodal curvatures in free-edge

181Chladni plates and atached-edge membranes are similar. The Chladni plate nodal planes

182discussed in the present work are also similar to the two-dimensional projections (which is a

183usual representation) of the nodal surfaces of hydrogen atomic orbitals. Parallels between the

184two systems can help drive home the wave nature of quantum mechanical particles in general,

185and of the electron in particular. 12

186 From the point of view of mathematical formalism, several distinctions arise: One is that

187the physical problem of the electron in a hydrogen atom is a three-dimensional problem, where

188only the kinetic part of the quantum mechanical wave equation resembles vibrational wave

189mechanics. The potential, an important and often troublesome part of the Schrödinger equation,

190is not involved in the vibrational wave equation. For the hydrogen atom, the atractive

191interaction between the electron and nucleus confines the electron within the atom, leading to

192quantization. The separable solutions of the Schrödinger equation are stationary states, with

193angular and radial nodes arising from imposing radial and angular boundary conditions on the

194different components of the wave function. 32

195 A second point that should be made clear is that the shapes obtained are nodes of a

196wavefunction (vibrational or electronic), not contour plots of the atomic orbital itself. For circular

197symmetry, the shapes are similar to contour plots (non-zero value cut-outs) of the three-

198dimensional wavefunction, and could be used as such, although that may cause some confusion

199at the end of the day. For the s-orbitals, both nodes and non-zero contours are spherical (circular

200in 2D projection), whereas orbitals with angular dependence (p, d, f and g) have spherical radial

201nodes and planar and conical angular nodes. Simple diagonal Chladni shapes equivalent to the

202nodes of a 2pz or the 3dxy and 3dx2-y2 orbitals are obtainable, as are the (4 planes, 1 sphere)

203nodes of the 6gz4 orbital (see supporting information). 7 However, diagonal nodes appear

204sporadically on centrally-driven circular plates, requiring the usage of higher amplitudes. 13

205

206 Figure 2. Exemplary standing waves produced on Chladni plates. a) 109 Hz, 300 mVpp (2s

207 orbital), b) 354 Hz, 200 mVpp (3s), c) 1690 Hz, 100 mVpp (5s), d) 84 Hz, 100 mVpp (3dxy)

208

209Learning objectives: After viewing this demonstration, students should be able to:

210 1. Explain the idea of a standing wave and how it relates to atomic orbitals

211 2. Understand the wave-like properties of electrons.

212 3. Understand the nodal and anti-nodal regions (regions of non-zero amplitude) of an

213atomic orbital and how it affects the three-dimensional shape of an atomic orbital.

214 14

215 Rather than verbally presenting the atomic orbital as the probability of an electron’s

216position, the aim of this demonstration is to help students qualitatively observe the behaviour of

217two-dimensional waves, and the meaning of nodal structures in two dimensions. However, the

218demonstration should open with a presentation introducing the idea of a standing wave,

219preferably with the use of an example such as the aforementioned guitar string, or a telephone

220wire fixed at two ends. The especially important point to cover is that standing waves

221demonstrate low-energy nodes at each higher harmonic equal to the overtone number (first

222overtone yields one node, the second yields two, and so on). These nodes give the guitar string a

223particular shape when plucked appropriately because of the nodal and anti-nodal regions.

224 The wave-like properties of an electron, as explained by Schrödinger’s wave equation

225(see SI), should be connected to these standing waves. Take the opportunity to have the

226students consider the shape of such a wave in three-dimensions, vis-à-vis the nodal (see Figure 2

227for an example of this on the Chladni plates) and anti-nodal behaviour of a standing wave. This

228abstract concept is likely to have some students perplexed at which point the Chladni plates

229demonstration, in which they will observe a two-dimensional example of a standing wave

230representative of an atomic orbital, ought to be exhibited.

231 The accessible atomic orbital representations available to the round plate are the 2s, 3s,

2324s and 5s orbitals (Figure 2a, b, c), and the planar nodes of the 3dxy orbital can be generated on

233the square plate (Figure 2d). Table S2 and S3 list the required frequency and amplitude required

234to generate the desired figures, as well as additional Chladni figure photographs and

235experimental settings. 15

236 The two-dimensional example may then be explored by producing the node of the 2s

237orbital (or the 2p orbital) upon the circular Chladni plate. By slowly adjusting the frequency to

238the harmonic using the frequency generator, an illustration of the standing wave is obtained in

239real time, as the baking powder is tossed around. Ask students to point out how many nodes are

240present and relate this to which s orbital is currently being viewed as higher order s orbitals are

241accessed and to imagine the three-dimensional structure of each Chladni figure.

242

243Post-demonstration Discussion

244 The concept of nodal and anti-nodal regions may then be elaborated on using the figure

245presently generated on the plate as a reference. Explain that this two-dimensional example

246illustrates the nodes of an atomic orbital. Following the demonstration, students may be

247introduced to increasingly complex ideas involving atomic orbitals. Using the 2s Chladni figure as

248an example, ask the students to point out the most likely position of an electron in a hydrogen

249atom. Is it inside the circular white line, outside the line or on the line? Discuss the implications

250of wave-particle duality of an electron and how this relates to the shape of an atomic orbital,

251that is, the shape of an atomic orbital is actually a containing surface around a cloud of electron

252density.

253

254

255 Conclusion 16

256 The aim of this demonstration is to help professors introduce atomic orbitals to

257students. The typical mater of course is to present standing waves and subsequently the

258properties of atomic orbitals. This is conceivably tricky for students to fully understand due to

259the nebulous idea of an atomic orbital. It is expected that this exciting demonstration will pique

260the interest of any student and help them visualize the structure of atomic orbitals as well as

261fully participate in the discussion which follows.

262

263

264 Supporting Information

265

266Additional information concerning materials, equipment, and experimental settings, as well as

267several photos and a video of standing wave paterns generated on the square and circular

268Chladni plates, may be found within the supporting information.

269

270

271 Acknowledgments

272JSM thanks the Department of Chemistry and the Learning and Teaching Centre of the University

273of Victoria for support of this project. Thanks to Kari Matusiak, Landon MacGillivray and Hyewon

274Ji for contributing to the photographic content of the article, and to David Harrington for useful

275discussions. 17

276 References

277(1) Davis, M. Guitar Strings as Standing Waves: A Demonstration. J. Chem. Educ. 2007, 84

278(8), 1287.

279(2) Yamana, S. Observation of Stationary Waves. J. Chem. Educ. 1967, 44 (5), A465.

280(3) Chakraborty, M.; Mukhopadhyay, S.; Das, R. S. A Simple Demonstration of Atomic and

281Molecular Orbitals Using Circular Magnets. J. Chem. Educ. 2014, 91 (9), 1505–1507.

282(4) Lambert, F. L. Atomic and Molecular Orbital Models. J. Chem. Educ. 1957, 34 (5), 217.

283(5) Ogryzlo, E. A.; Porter, G. B. Contour Surfaces for Atomic and Molecular Orbitals. J. Chem.

284Educ. 1963, 40 (5), 256.

285(6) Emerson, D. W. A Colorful Demonstration To Simulate Orbital Hybridization. 1988.

286(7) Winter, M. The Orbitron: a gallery of atomic orbitals and molecular orbitals

287htps://winter.group.shef.ac.uk/orbitron/AOs/2s/index.html (accessed Jun 23, 2016).

288(8) Hoogenboom, B. E. Three-Dimensional Models of Atomic Orbitals. J. Chem. Educ. 1962,

28939 (1), 40.

290(9) Kiefer, E. F. An Atention-Getting Model for Atomic Orbitals. J. Chem. Educ. 1995, 72 (6),

291500.

292(10) Aristov, N.; Habekost, A. Chemische Untersuchungen Mit Miteln Der Akustik - Einsatz Im

293Chemieunterricht. CHEMKON 2012, 19 (3), 123–130. 18

294(11) Wiener, O. Stehende Lichtwellen Und Die Schwingungsrichtung Polarisierten Lichtes;

2951890.

296(12) Myoungsik, C. Demonstration of Standing Light Wave with a Laser Pointer. J. Korean

297Phys. Soc. 2010, 56 (5), 1542.

298(13) Tully, S. P.; Stit, T. M.; Caldwell, R. D.; Hardock, B. J.; Hanson, R. M.; Maslak, P. Interactive

299Web-Based Pointillist Visualization of Hydrogenic Orbitals Using Jmol. J. Chem. Educ. 2013, 90

300(1), 129–131.

301(14) Liebl, M. Orbital Plots of the Hydrogen Atom. J. Chem. Educ. 1988, 65 (1), 23.

302(15) Kijewski, L. Graphing Orbitals in Three Dimensions with Rotatable Density Plots. J. Chem.

303Educ. 2007, 84 (11), 1887.

304(16) Esselman, B. J.; Hill, N. J. Integration of Computational Chemistry into the Undergraduate

305Organic Chemistry Laboratory Curriculum. J. Chem. Educ. 2016, 75 (2), 241.

306(17) Douglas, J. E. Visualization of Electron Clouds in Atoms and Molecules. J. Chem. Educ.

3071990, 67 (1), 42.

308(18) Ramachandran, B.; Kong, P. C. Three-Dimensional Graphical Visualization of One-Electron

309Atomic Orbitals. J. Chem. Educ. 1995, 72 (5), 406.

310(19) Johnson, J. L. Visualization of Wavefunctions of the Ionized Hydrogen Molecule. J. Chem.

311Educ. 2004, 81 (10), 1535.

312(20) Rhile, I. J. Comment On “visualizing Three-Dimensional Hybrid Atomic Orbitals Using

313Winplot: An Application for Student Self Instruction.” J. Chem. Educ. 2015, 92 (12), 1973–1974. 19

314(21) Dori, Y. J.; Barak, M. Virtual and Physical Molecular Modeling: Fostering Model

315Perception and Spatial Understanding. Educ. Technol. Soc. 2001, 4 (1), 61–74.

316(22) McQuarrie, D. A. Quantum Chemistry; University Science Bks; University Science Books,

3171983.

318(23) Preece, D.; Williams, S. B.; Lam, R.; Weller, R. “Let’s Get Physical”: Advantages of a

319Physical Model over 3D Computer Models and Textbooks in Learning Imaging Anatomy. Anat.

320Sci. Educ. 2013, 6 (4), 216–224.

321(24) Jayatilaka, D. Atomic Orbitals and Chladni Figures htp://dylan-

322jayatilaka.net/articles/atomic-orbitals-and-chladni-figures/.

323(25) Berry, R. S. Understanding Energy: Energy, Entropy and Thermodynamics for Everyman;

324World Scientific: Singapore, 1991.

325(26) de Silva, C. W. Vibration: Fundamentals and Practice, Second Edition; Taylor & Francis,

3262006.

327(27) Stanford, N. J. CYMATICS: Science Vs. Music - Nigel Stanford

328htps://www.youtube.com/watch?v=Q3oItpVa9fs (accessed Jun 28, 2016).

329(28) Heller, E. J. Why You Hear What You Hear: An Experiential Approach to Sound, Music,

330and Psychoacoustics; Princeton University Press, 2013.

331(29) Mason, F. P.; Richardson, R. W. Why Doesn’t the Electron Fall into the Nucleus? J. Chem.

332Educ. 1983, 60 (1), 40. 20

333(30) Engel, T.; Hehre, W. J. Quantum Chemistry and Spectroscopy, 2nd ed.; Prentice Hall,

3342010.

335(31) Dzhamay, A. Partial Differential Equations Higher-dimensional PDE: Vibrating rectangular

336membranes and nodes htp://www.maplesoft.com/applications/view.aspx?

337SID=4518&view=html&L=F (accessed Jun 28, 2016).

338(32) Engel, T. Quantum Chemistry and Spectroscopy, 3rd ed.; Pearson, 2006.

339

340

341

342

343

344

345

346

347

348

349

350 21

351

352 Figure captions

353 Figure 1. Schematic of the demonstration setup.

354 Figure 2. Exemplary standing waves produced on Chladni plates. a) 109 Hz, 300 mVpp (2s

355 orbital), b) 354 Hz, 200 mVpp (3s), c) 1690 Hz, 100 mVpp (5s), d) 84 Hz, 100 mVpp (3dxy)

356

357

358 22

359 Figures

360

361 23

362

363 Figure 1. Schematic of the demonstration setup.

364

365

366

367

368 24

369

370 Figure 2. Exemplary standing waves produced on Chladni plates. a) 109 Hz, 300 mVpp (2s

371 orbital), b) 354 Hz, 200 mVpp (3s), c) 1690 Hz, 100 mVpp (5s), d) 84 Hz, 100 mVpp (3dxy) Chladni manuscript_20191030.docx (3.47 MiB) view on ChemRxiv download file Orbital shaped standing waves using Chladni plates

Supporting Information – Materials and Setup Photos

Eric Janusson, Johanne Penafiel, Shaun MacLean, Andrew Macdonald, Irina Paci*, J. Scott McIndoe*

Department of Chemistry, University of Victoria, P.O. Box 3065 Victoria, BC V8W3V6, Canada. Fax: +1 (250) 721-7147; Tel: +1 (250) 721-7181; E-mail: [email protected], [email protected]

Materials and Equipment

The wave driver used in these experiments is a PASCO Mechanical Wave Driver model number SF-9324. The Chladni plates were purchased from PASCO scientific. The wave driver was purchased directly from PASCO Scientific. The amplifier was custom made and used a 12

V / 0.85 A power supply, model number UK003, purchased from CanaKit as well as a TDK-

Lambda Americas Inc. ZWS10-12 AC/DC 12 V 10 W converter purchased from Digi-Key

Electronics. The function generator used was a 30 MHz Agilent 33522A Function/Arbitrary

Waveform Generator purchased from Agilent Technologies.

List of Chladni plate standing waves and parameters Table S1. Summary of experimental settings required for representative Chladni figures

Frequency (Hz) Amplitude (mVpp) Plate Shape Orbital

109 300 Round 2s

354 200 Round 3s

1690 100 Round 5s

954 100 Square 3dxy

84 100 Square 3dxy (anti-nodal)

Table S2. Photos of standing waves using square Chladni plate

Frequenc Amplitude Photo of Standing Wave Pattern y (Hz) (mVpp)

84 100

192 110 336 70

392 110

485 50

800 60 954 70

3dxy

1058 60

1710 50

1876 2112 100

2270 90 Table S3. Photos of standing waves using circular Chladni plate

Frequency Amplitude (mVpp) Pattern/Orbital representation (Hz)

43 200

64 300

91 300 109 300

115 300

263 300

354 200

3s 896 300

1690 100

5s SUPPORTING INFORMATION Materials and Setup Phot... (6.79 MiB) view on ChemRxiv download file Orbital-like Standing Waves Using Chladni Plates

Supporting Information - Mathematical Foundations

1 The guitar string – a one-dimensional wave

A uniform string attached at two end points, such as a guitar string of length a, has an amplitude (displacement) f(x, t) that depends on the point in space and on time, and follows the wave equation: 1 ∂2f(x, t) ∇2f(x, t) = , (1) v2 ∂t2 ∇ ∂ where the Laplacian = ∂x for the one-dimensional case, and v is the velocity with which the wave propagates along the string. The solution of Equation 1 is subject to the problem’s boundary conditions, which in this case are that the function must remain zero at the two end points:

f(0, t) = f(a, t) = 0. (2)

The two variables x and t act separately on f(x, t) in Equation 1, so we can choose to work with a separable solution f(x, t) = X(x)T (t). The separation of variables can be done in the usual way,[1, 2] resulting in

1 d2X(x) 1 d2T (t) = = k, (3) X(x) dx2 v2T (t) dt2 where k is a separation constant, written this way for simplicity of the derivation below. Both the time and the space dependence sides of Equation 3 are second order diﬀerential equations in which the second derivative of the function is proportional to the function itself.

1 Such equations have solutions that are either exponentials or combinations of sine and cosine functions. For a position trial function that is written as X(x)= c1 sin βx + c2 cos βx, application of boundary conditions gives:

X(0) = c2 = 0; (4) X(a)= c1 sin βa + c2 cos βa =0, (5)

nπ nπx 2 1 leading to β = a (n = 1, 2, 3, ...), c = 0 and X(x) = c sin a , and, in Equation 3, − 2 − n2π2 k = β = a2 . A similar procedure is followed for T (t), but given that there are no boundary conditions for this function, both the sine and the cosine functions survive. Instead, we can use a contracted trigonometric function to describe it: T (t) = d1 cos(ωt + φ). Plugging T (t) into nπv Equation 3 gives the dependence of the angular frequency ω on the integer n, ω = a . The separable solution of the wave equation becomes nπx f (x, t)= d1c1 cos(ω t + φ ) sin . (6) n n n a where the subscript n indicates a function or a constant that depends on the integer n. In acoustic terms, f1(x, t) is the fundamental mode or the ﬁrst harmonic, with frequency ω1 v 1 2 2 1 ν = 2π = 2a . f (x, t) is the second harmonic or ﬁrst overtone, with frequency ν = 2ν , etc. Two further points can be drawn from Equation 6: one is that each fn(x, t) has n − 1 nodes (points in space where the function is zero), and that the position of the nodes is independent of time (in other words, the functions represent standing waves). A general solution of the wave equation is then a linear combination of the separable solutions,

∞ ∞ nπx F (x, t)= f (x, t)= A cos(ω t + φ ) sin , (7) X n X n n n a n=1 n=1 where An are expansion coeﬃcients, depending on the order of the harmonic and the initial condition of the function (e.g., how and how hard the guitar string is played). Plucking a guitar string at the middle produces a wave that has a high proportion of the fundamental mode of that string, whereas plucking the string at another location will have an asymmetric combination of frequency modes. The speed of the wave through the guitar string, and thus its frequency, depend on the material of the string and on its tightness.

2 2 The membrane attached at the sides – A two-dimensional wave

A rectangular membrane follows a wave equation similar to Equation 1, with the modiﬁcation that the function now depends on two spatial variables: f(x, y, t), and the Laplacian is a two-dimensional operator in this case:

∂2f(x, y, t) ∂2f(x, y, t) 1 ∂2f(x, y, t) + = . (8) ∂x2 ∂y2 v2 ∂t2 A separable solution is sought f(x, y, t)= X(x)∗Y (y)∗T (t), and separation of variables is again pursued by inserting this solution into Equation 8. This leads to a set of single- variable equations [1] as follows:

1 d2X(x) = −k2; (9) X(x) dx2 x 1 d2Y (y) = −k2; (10) Y (y) dy2 y 1 d2T (t) = −ω2, (11) T (t) dt2

2 2 2 2 where kx + ky = ω /v . All of the single-variable equations have trigonometric solutions as shown in the one-dimensional case above:

T (t)= g1 sin(ωt)+ g2 cos ωt; (12)

X(x)= c1 sin(kxx)+ c2 cos kxx; (13)

Y (y)= d1 sin(kyy)+ d2 cos kyy. (14)

The boundary conditions X(0) = X(a) = 0, Y (0) = Y (b) = 0 [equivalent to the full solution boundary conditions f(0,y,t)= f(a, y, t) = 0, f(x, 0, t)= f(x, b, t) = 0] cancel out the cosine terms in the X(x) and Y (y) solutions, and enforce values for kx and ky that are multiples of π/a and π/b, respectively. The separable solution becomes

nπx mπy f (x, y, t) = c1d1 sin sin [g1 sin(ω t)+ g2 cos(ω t)]= (15) mn a b mn mn nπx mπy = A sin sin cos(ω t + φ ). a b mn mn

These harmonics can be obtained by resonant activation of the attached membrane, and

3 are stationary states. Their nodes are linear, parallel to the membrane sides, at locations where sin(x) and sin(y) are zero, with a total of m+n−2 nodes for each fmn(x, y, t) harmonic. The general solution of the wave equation for a rectangular drumhead given by a linear combination of the stationary state functions,

∞ ∞ F (x, y, t)= f (x, y, t). (16) X X mn m=1 n=1 On a circular drumhead, vibrational waves are treated in a similar fashion.[3] A polar coordinates Laplacian is used, with separation of variables leading to a radial diﬀerential equation (a Bessel equation), an equation for the polar angle, and a time-dependent differential equation. The solution is more complicated than the rectangular case, but there are again two boundary conditions (that the radial function should be zero at the edge of the membrane and that the polar function should have circular symmetry) leading to the emergence of two quantum numbers. On the circular membrane, nodes can be either circular (radial nodes, almost equally spaced) or linear along diameters of the membrane. The wave equation for a vibrating plate with free edges, such as that used in our Chladni apparatus, is identical to that presented for the stretched membranes above.[4] However, boundary conditions are nonzero at the free edges of the plates, as these edges are free to vibrate. For a rectangular (a,b) plate, the boundary conditions become complicated:[4]

∂2f(x, y, t) ∂2f(x, y, t) ∂3f(x, y, t) ∂3f(x, y, t) + ν = + (2 − ν ) = 0 when x = ±a; (17) ∂x2 p ∂y2 ∂x2 p ∂x∂y2 ∂2f(x, y, t) ∂2f(x, y, t) ∂3f(x, y, t) ∂3f(x, y, t) + ν = + (2 − ν ) = 0 when y = ±b, (18) ∂y2 p ∂x2 ∂y2 p ∂x2∂y where νp is the plate material’s Poisson ratio. The solution of the elementary diﬀerential equations is more complex than in the case of the rectangular plate with free edges, because of the nontrivial boundary conditions, and can be found elsewhere.[4] For the purpose of the current application, it is suﬃcient to note that the physical characteristics of these harmonics are very similar to those of the circular membrane.

3 The electron in a Hydrogen atom – a three-dimensional wave

The solution of the wave equation for the Hydrogen atom is available in any Physical Chem- istry or Quantum Chemistry textbook. Broad lines are presented here for comparison to the classical wave solutions presented above. The Hydrogen atom consists of a proton orbited of an electron. The mass diﬀerence

4 between the two particles leads to the Born-Oppenheimer (BO) approximation: As a con- sequence, the system is described as being composed of the proton as a stationary heavy particle at the origin, and the electron as a moving particle of reduced mass µ. The two − e2 particles interact through a Coulomb potential V (r)= 4πǫ0r , where r is the position of the electron in the proton-centered coordinate system. The time-dependent Schr¨odinger equation is the wave equation describing the behaviour of the system:

2 h¯ ∂ − ∇2Ψ(r,θ,φ,t)+ V (r)Ψ(r,θ,φ,t)= −ih¯ .Ψ(r,θ,φ,t), (19) 2µ ∂t where spherical coordinates are used to reﬂect the symmetry of the problem. Since the Hydrogen atom potential is independent of time in the BO approximation, a separable solution Ψ(r,θ,φ,t) = ψ(r,θ,φ)T (t) can be sought, leading to two separate parts of the Schr¨odinger equation:

∂ −ih¯ T (t)= E T (t); (20) ∂t h¯2 − ∇2ψ(r,θ,φ)+ V (r)ψ(r,θ,φ)= Eψ(r,θ,φ), (21) 2µ where the separation constant E is the energy of the system, and the time-dependent part of the separable solution is T (t)= e−iEt/h¯ . The time-independent part of the wavefunction is also separable, ψ(r,θ,φ)= R(r)Y (θ,φ). Note that so far, the problem is similar to that of the acoustic waves on circular plates, with two distinctions besides the dimensionality of the problem: (i) We chose the exponential form of the solution of Equation 20 for mathematical convenience, and (ii) There is a potential acting on the wave, V (r). There are likewise similarities and differences between the solutions of the wave equations in the two systems. The radial wavefunction in the quantum problem is the solution of a ordinary differential equation that includes the potential term, so that the radial functions are a series of associated Laguerre functions, instead of the Bessel function solution in the classical case. Both have oscillatory behaviour, but amplitudes and the spacings between nodes are different. The angular functions for both systems are eigenfunctions of the angular part of the squared Laplacian (although we are comparing a 2D problem to a 3D problem in this case). This is the main reason behind the similarity of the nodal shapes reported in the manuscript. As mentioned above, the Laplacian in spherical coordinates has a relatively complicated expression, and it is not the point of this work to detail the textbook solution of the separable equation. The boundary conditions are given by the requirements that the wavefunction should be finite, well behaved, and single valued at all points. This places restrictions on the radial function at its boundaries [rR(0) = rR(∞) = 0], on the angular function at the [0,2π] boundaries of the azimuthal angle [Y (θ,φ)= Y (θ,φ +2π)], and forces the selection of Legendre polynomials as solutions of the θ-dependent ordinary differential equation. These

5 boundary conditions produce quantization of the energy, the orbital angular momentum and its z projection. Because the boundary conditions force oscillatory behaviour to ﬁt within the boundaries, the associated quantum numbers are related to the number and geometry of nodes in the resulting wavefunction. The total number of nodes is determined by the principal quantum number, whereas the nature of the nodes is determined by the orbital quantum number. s orbitals have only radial nodes, whereas orbitals with l> 0 also have angular nodes. Radial nodes are spherical, while angular nodes can be planes or cones. These nodes are normally represented in 2 D projection (see the Orbitron webpage for example, http://winter.group.shef.ac.uk/orbitron/AOs/3p/wave- fn.html), in which case they take shapes analogous the Chladni patterns discussed above.

References

[1] D. A. McQuarrie. Chapter 2. The wave equation, page 47. University Science Books, Sausalito, CA, 1983.

[2] I. N. Levine. Chapter 5. The Hydrogen Atom, page 118. Person Education, Inc., 2014.

[3] D. Yong. The Wave Equation and Separation of Variables. Technical report, IAS. Park City Mathematics Institute, 2003.

[4] S.V. Bosakov. Eigenfrequencies and modiﬁed eigenmodes of a rectangular plate with free edges. Journal of Applied Mathematics and Mechanics, 73(6):688–691, Jan 2009.

6 MathSI.pdf (74.45 KiB) view on ChemRxiv download file Other files

Chladni plates 2 - Petite.mov (1.56 MiB) view on ChemRxiv download file