doi.org/10.26434/chemrxiv.10255838.v1 Orbital Shaped Standing Waves Using Chladni Plates 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) download file view on ChemRxiv Chladni manuscript_20191030.docx (3.47 MiB) download file view on ChemRxiv SUPPORTING INFORMATION Materials and Setup Phot... (6.79 MiB) download file view on ChemRxiv MathSI.pdf (74.45 KiB) download file view on ChemRxiv Chladni plates 2 - Petite.mov (1.56 MiB)
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doi.org/10.26434/chemrxiv.10255838.v1
Orbital Shaped Standing Waves Using Chladni PlatesEric Janusson, Johanne Penafiel, Andrew Macdonald, Shaun MacLean, Irina Paci, J Scott McIndoe
Submitted date: 05/11/2019 • Posted date: 13/11/2019Licence: CC BY-NC-ND 4.0Citation 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 aone-dimensional standing wave. The classic example is the harmonic frequencies which produce standingwaves 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 tovisualize. Extrapolation from a two-dimensional model, such as the vibrational modes of a drumhead, can beused to convey the standing wave concept to students more easily. We have opted to use Chladni plateswhich may be tuned to give a two-dimensional standing wave which serves as a cross-sectionalrepresentation of atomic orbitals. The demonstration, intended for first year chemistry students, facilitates theexamination of nodal and anti-nodal regions of a Chladni figure which students can then connect to theconcept of quantum mechanical parameters and their relationship to atomic orbital shape.
File list (4)
download fileview on ChemRxivChladni manuscript_20191030.docx (3.47 MiB)
download fileview on ChemRxivSUPPORTING INFORMATION Materials and Setup Phot... (6.79 MiB)
download fileview on ChemRxivMathSI.pdf (74.45 KiB)
download fileview on ChemRxivChladni plates 2 - Petite.mov (1.56 MiB)
Orbital shaped standing waves using Chladni plates
Supporting Information – Materials and Setup Photos
Eric Janusson, Johanne Penafiel, Shaun MacLean, Andrew Macdonald, Irina Paci*, J. ScottMcIndoe*
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
Chladni PlatesSupporting Information - Mathematical Foundations
1 The guitar string – a one-dimensional waveA 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:
∇2f(x, t) =
1
v2∂2f(x, t)
∂t2, (1)
where the Laplacian ∇ = ∂∂x
for the one-dimensional case, and v is the velocity with whichthe wave propagates along the string. The solution of Equation 1 is subject to the problem’sboundary conditions, which in this case are that the function must remain zero at the twoend 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 towork with a separable solution f(x, t) = X(x)T (t). The separation of variables can be donein the usual way,[1, 2] resulting in
1
X(x)
d2X(x)
dx2=
1
v2T (t)
d2T (t)
dt2= k, (3)
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 cosinefunctions.
For a position trial function that is written as X(x) = c1 sin βx+ c2 cos βx, applicationof boundary conditions gives:
X(0) = c2 = 0; (4)
X(a) = c1 sin βa+ c2 cos βa = 0, (5)
leading to β = nπa
(n = 1, 2, 3, ...), c2 = 0 and X(x) = c1 sinnπxa, and, in Equation 3,
k = −β2 = −n2π2
a2.
A similar procedure is followed for T (t), but given that there are no boundary conditionsfor this function, both the sine and the cosine functions survive. Instead, we can use acontracted trigonometric function to describe it: T (t) = d1 cos(ωt + φ). Plugging T (t) intoEquation 3 gives the dependence of the angular frequency ω on the integer n, ω = nπv
a.
The separable solution of the wave equation becomes
fn(x, t) = d1c1 cos (ωnt+ φn) sinnπx
a. (6)
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 frequencyν1 = ω1
2π= v
2a. f2(x, t) is the second harmonic or first overtone, with frequency ν2 = 2ν1,
etc. Two further points can be drawn from Equation 6: one is that each fn(x, t) has n− 1nodes (points in space where the function is zero), and that the position of the nodes isindependent of time (in other words, the functions represent standing waves).
A general solution of the wave equation is then a linear combination of the separablesolutions,
F (x, t) =
∞∑
n=1
fn(x, t) =
∞∑
n=1
An cos (ωnt + φn) sinnπx
a, (7)
where An are expansion coefficients, depending on the order of the harmonic and the initialcondition of the function (e.g., how and how hard the guitar string is played). Plucking aguitar string at the middle produces a wave that has a high proportion of the fundamentalmode of that string, whereas plucking the string at another location will have an asymmetriccombination of frequency modes. The speed of the wave through the guitar string, and thusits 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 modificationthat the function now depends on two spatial variables: f(x, y, t), and the Laplacian is atwo-dimensional operator in this case:
∂2f(x, y, t)
∂x2+∂2f(x, y, t)
∂y2=
1
v2∂2f(x, y, t)
∂t2. (8)
A separable solution is sought f(x, y, t) = X(x)∗Y (y)∗T (t), and separation of variablesis again pursued by inserting this solution into Equation 8. This leads to a set of single-variable equations [1] as follows:
1
X(x)
d2X(x)
dx2= −k2x; (9)
1
Y (y)
d2Y (y)
dy2= −k2y ; (10)
1
T (t)
d2T (t)
dt2= −ω2, (11)
where k2x + k2y = ω2/v2. All of the single-variable equations have trigonometric solutions asshown in the one-dimensional case above:
T (t) = g1 sin(ωt) + g2 cosωt; (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 fullsolution boundary conditions f(0, y, t) = f(a, y, t) = 0, f(x, 0, t) = f(x, b, t) = 0] cancel outthe cosine terms in the X(x) and Y (y) solutions, and enforce values for kx and ky that aremultiples of π/a and π/b, respectively.
The separable solution becomes
fmn(x, y, t) = c1d1 sin(nπx
a
)
sin(mπy
b
)
[g1 sin(ωmnt) + g2 cos(ωmnt)] = (15)
= A sin(nπx
a
)
sin(mπy
b
)
cos(ωmnt+ φmn).
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 locationswhere sin(x) and sin(y) are zero, with a total ofm+n−2 nodes for each fmn(x, y, t) harmonic.
The general solution of the wave equation for a rectangular drumhead given by a linearcombination of the stationary state functions,
F (x, y, t) =
∞∑
m=1
∞∑
n=1
fmn(x, y, t). (16)
On a circular drumhead, vibrational waves are treated in a similar fashion.[3] A polarcoordinates Laplacian is used, with separation of variables leading to a radial differentialequation (a Bessel equation), an equation for the polar angle, and a time-dependent dif-ferential equation. The solution is more complicated than the rectangular case, but thereare again two boundary conditions (that the radial function should be zero at the edge ofthe membrane and that the polar function should have circular symmetry) leading to theemergence 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 ourChladni 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 arefree to vibrate. For a rectangular (a,b) plate, the boundary conditions become complicated:[4]
∂2f(x, y, t)
∂x2+ νp
∂2f(x, y, t)
∂y2=∂3f(x, y, t)
∂x2+ (2− νp)
∂3f(x, y, t)
∂x∂y2= 0 when x = ±a; (17)
∂2f(x, y, t)
∂y2+ νp
∂2f(x, y, t)
∂x2=∂3f(x, y, t)
∂y2+ (2− νp)
∂3f(x, y, t)
∂x2∂y= 0 when y = ±b, (18)
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, andcan be found elsewhere.[4] For the purpose of the current application, it is sufficient to notethat the physical characteristics of these harmonics are very similar to those of the circularmembrane.
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 theclassical 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 heavyparticle at the origin, and the electron as a moving particle of reduced mass µ. The twoparticles interact through a Coulomb potential V (r) = −
e2
4πǫ0r, where r is the position of the
electron in the proton-centered coordinate system.The time-dependent Schrodinger equation is the wave equation describing the behaviour
of the system:
−h2
2µ∇
2Ψ(r, θ, φ, t) + V (r)Ψ(r, θ, φ, t) = −ih∂
∂t.Ψ(r, θ, φ, t), (19)
where spherical coordinates are used to reflect the symmetry of the problem. Since theHydrogen atom potential is independent of time in the BO approximation, a separablesolution Ψ(r, θ, φ, t) = ψ(r, θ, φ)T (t) can be sought, leading to two separate parts of theSchrodinger equation:
−ih∂
∂tT (t) = E T (t); (20)
−h2
2µ∇
2ψ(r, θ, φ) + V (r)ψ(r, θ, φ) = Eψ(r, θ, φ), (21)
where the separation constant E is the energy of the system, and the time-dependent partof 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, withtwo distinctions besides the dimensionality of the problem: (i) We chose the exponential formof the solution of Equation 20 for mathematical convenience, and (ii) There is a potentialacting on the wave, V (r).
There are likewise similarities and differences between the solutions of the wave equationsin the two systems. The radial wavefunction in the quantum problem is the solution of aordinary differential equation that includes the potential term, so that the radial functionsare a series of associated Laguerre functions, instead of the Bessel function solution in theclassical case. Both have oscillatory behaviour, but amplitudes and the spacings betweennodes are different. The angular functions for both systems are eigenfunctions of the angularpart of the squared Laplacian (although we are comparing a 2D problem to a 3D problemin this case). This is the main reason behind the similarity of the nodal shapes reported inthe manuscript.
As mentioned above, the Laplacian in spherical coordinates has a relatively complicatedexpression, and it is not the point of this work to detail the textbook solution of the separableequation. The boundary conditions are given by the requirements that the wavefunctionshould be finite, well behaved, and single valued at all points. This places restrictions onthe 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 ofLegendre polynomials as solutions of the θ-dependent ordinary differential equation. These
5
boundary conditions produce quantization of the energy, the orbital angular momentum andits z projection. Because the boundary conditions force oscillatory behaviour to fit withinthe boundaries, the associated quantum numbers are related to the number and geometryof nodes in the resulting wavefunction.
The total number of nodes is determined by the principal quantum number, whereas thenature of the nodes is determined by the orbital quantum number. s orbitals have only radialnodes, whereas orbitals with l > 0 also have angular nodes. Radial nodes are spherical, whileangular 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. ParkCity Mathematics Institute, 2003.
[4] S.V. Bosakov. Eigenfrequencies and modified eigenmodes of a rectangular plate with freeedges. Journal of Applied Mathematics and Mechanics, 73(6):688–691, Jan 2009.
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