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Steven D. Gammon University of Idaho Moscow, ID 83844
Visualization of Wave Functions Using Mathematica1 Patricia L. Lang and Marcy Hamby Towns Department of Chemistry, Ball State University, Muncie, IN 47306
Erwin with his psi can do Calculations quite a few. But one thing has not been seen Just what does psi really mean. Walter Hückel trans. Felix Bloch
As the above quatrain describes, even early researchers in quantum mechanics were puzzled over the nature of ψ . Simply put, quantum mechanics involves a lot of symboliclevel representation. What these symbols mean is not a trivial question, and in fact, it is a question often asked by undergraduate physical chemistry students. How can we help these students understand ψ and |ψ |2? Initially students ask for information related to the concepts from classical physics with which they are acquainted. As Schrödinger said (1), “We have taken over from previous theory the idea of a particle and all the technical language concerning it. This idea is inadequate. It constantly drives our mind to ask for information which has obviously no significance.” We have found that graphics generated using Mathematica (2) provide students with a greater understanding of the wave function ψ and the probability density |ψ|2. In particular, the graphics focus student’s attention on Born’s statistical interpretation of ψ which allows one to calculate the probability of finding a particle in a particular volume element. The laboratory activity that we constructed focuses on the models for translational, vibrational, and rotational motion—the two-dimensional particle in a box, the harmonic oscillator, and the particle on a sphere (3, 4). Student understanding and achievement was measured using a post-lab survey, an examination question, and a question on the semester final. Results of these instruments are reported to draw attention to the experiences of the student. Two-Dimensional Particle in a Box The wave function for a two-dimensional particle in a box is expressed as follows: ψ (x,y) = (2/L)[sin(n x π x/L)][sin(ny π y/L)]
where L stands for the length of the box, and nx and ny are the quantum numbers that describe the energy state of the system. To plot this equation using Mathematica, the following equation can be used: Plot3D[(2/len)*Sin[(nx*Pi*x)/len]*Sin[(ny*Pi*y)/len], {x,0,len}, {y,0,len}] where len equals the length of the box, and nx and ny are
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quantum numbers nx and ny , respectively. For example, if the length of the box is set to one, nx = 1 and ny = 2, then the three-dimensional surface shown in Figure 1 is produced. Students can generate a number of different surfaces by simply changing nx and ny. These surfaces can be used to illustrate the concept of degeneracy by comparing the surface produced when nx = 2 and ny=1 (see Fig. 2) to Figure 1. This visual representation of degeneracy allows students two ways of understanding degeneracy. One is by the symbolic representation of the wave function ψ and the associated energy equation for the system. The other is to note that the surfaces produced by degenerate two-dimensional particle-in-a-box wave functions can be superimposed by rotating one surface by 90°. Probability densities, |ψ|2, can be plotted for each surface by squaring the wave function. The probability density for the wave function ψ 21 = (2)(sin[2 π x])(sin[π y]) can be seen in Figure 3. Using these surfaces, students can roughly calculate the probability of finding the particle in a specific region of space. For example, one could ask the students to estimate the probability of finding the particle between 0 ≤ x ≤ 1 and 0 ≤ y ≤ 0.5. This exercise allows students to connect the visual representation of probability density to the mathematical representation they have used in the classroom to calculate probabilities (5). Specifically, they develop a visual sense of what it means to solve a double integral. Responses to our post-lab survey indicate that these surfaces helped the students visualize the mathematical representation ψ (x,y) and gave the equation meaning. As Sally wrote, “it helps me personally to see what these things really look like…it doesn’t just seem as though I’m working through some equation that has no meaning.” Comparing two degenerate surfaces such as ψ21 and ψ12 in Figures 1 and 2 helped students visualize the concept of degeneracy rather than simply seeing it as a mathematical construct. When asked on the first semester examination to identify pairs of degenerate two-dimensional particle-in-a-box wave functions given the wave function and the corresponding surface, 71% of the students answered the question correctly. During the final examination students were given a similar question and were asked to explain their reasoning. Fifteen of 17 answered the question correctly. Bryan’s reasoning is illustrative of the reasoning of 10 students: “…to find degeneracy between two graphs simply rotate the graph of one of them by 90°. If they are superimposable…they are degenerate.” These students arrived at their answer visually, by rotating the figures on the exam by 90°. This may indicate that the students understood the illustrations, but not the underlying mathematical origin of degeneracy. Three stu-
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Figure 1. The ψ12 surface for the two-dimensional particle in a box.
dents indicated no visual reasoning, but answered the question by connecting the quantum numbers in the wave functions to the energy equation for the two-dimensional particle in a box. With or without the graphics, these students could explain degeneracy using the traditional mathematical definition. Four students indicated that they looked for the quantum numbers to be reversed in pairs of wave functions— “flip-flopped”, as Sally stated. However, these students did not invoke the energy relationship for the particle in a twodimensional box. Finally, three students used both the graphics and the quantum numbers in their reasoning. We also found that having students generate figures of the wave function squared lead to a better understanding of probability densities. As Cooper commented, “This experiment helped me visualize the concept of probability density.” For students who are not particularly adroit at solving double integrals, the visual representation of |ψ |2 gives them an estimate of the answer and a way of understanding what it means to solve a double integral. On a laboratory question, all students were able to draw correct conclusions about the probability of finding the particle in a given region of space using the figures they had generated. Harmonic Oscillator The wave function for the one-dimensional harmonic oscillator is a Gaussian function multiplied by a Hermite polynomial: ψν(x) = H ν(x) exp(᎑ x2/2)
The following equation was used to plot this function in Mathematica: Plot[HermiteH[0,x]*E^(᎑(x^2)/2), {x,᎑5,5}].
Figure 2. The ψ21 surface for the two-dimensional particle in a box.
Mathematica retrieves the Hermite polynomial H0(x) and plots it for 5 < x < ᎑5. Figure 4a–c illustrates how students can compare plots of ψ ν(x) as v increases from zero to two to help them visualize the odd/even properties of this wave function. By superimposing the potential energy of the harmonic oscillator onto the plot of the wave function it can be shown that the particle has some probability of being found in the classically forbidden region. Thus the phenomenon of tunneling can be illustrated and discussed if the instructor chooses. Attention can be called to aspects of the correspondence principle by plotting |ψ ν(x)|2 using large values of the quantum number, v. Students can see that classical properties emerge. For example, in Figure 4d it can be seen that the most probable location to find the particle is where the potential energy of the system equals the total energy of the system (the turning points), and the least likely location is found at zero displacement. Particle on a Sphere
Figure 3. The probability density for the ψ21 two-dimensional particle in a box wave function.
The final model the students considered was a particle moving in three dimensions on the surface of a sphere. To solve the Schrödinger equation and find ψ (r, θ, φ), the particle is confined to a surface of fixed radius r. Since r is constant, the problem becomes finding a suitable function in θ and φ which satisfies the Schrödinger equation. The wave function for this model is given by the spherical harmonics:
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Figure 5. The Y00 surface.
Figure 6. The Y10 surface.
Figure 7. The Y20 surface.
Figure 4. Plots of the harmonic oscillator wave function for different values of v. (a) ψν(x) for v = 0, (b) ψν(x) for v = 1, (c) ψν(x) for v = 2, and (d) the probability density for the ψ ν = 20(x).
Yᐉ mᐉ(θ, φ) = Θᐉ mᐉ( θ)Φmᐉ( φ)
The labels ᐉ and mᐉ are quantum numbers (angular momentum and the z component of angular momentum), and are limited to the values ᐉ = 0, 1, 2, … and mᐉ = ᐉ, ᐉ – 1, … , ᎑ᐉ (which gives 2ᐉ + 1 values for a given value of ᐉ). After loading the graphics package, students are free to plot the real spherical harmonic functions such as Y00 by typing:
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SphericalPlot3D[Abs[SphericalHarmonicY[0,0,theta,phi]], {theta, 0, Pi}, {phi, 0, 2Pi}]
Once the students plot Y00, Y10, and Y20 (Figs. 5–7), they recognize these plots as s, pz , and dz 2 atomic orbitals. As William wrote, “[the graphics] showed how the atomic orbitals came about.” Indeed, spherical harmonics are the portions of the atomic wave function ψ nᐉmᐉ (r, θ , φ ) = R nᐉ (r) Yᐉmᐉ ( θ , φ ), which govern the “shape” of the atomic orbital. Students also associated the ᐉ quantum number with the number of nodes:
Journal of Chemical Education • Vol. 75 No. 4 April 1998 • JChemEd.chem.wisc.edu
Research: Science & Education
as Beth noted, “[the graphs of Y00, Y10, and Y20], helped me to see how they relate to orbital diagrams and numbers of nodes = ᐉ.” We encouraged students to graph other spherical harmonics using larger values of ᐉ. Students made orbitals such as an i orbital in Figure 8, Y60, and were delighted to find six nodes as they predicted.
Figure 8. The Y60 surface.
Conclusions Overall, the students stated that they came away with a better understanding of wave functions and especially probability densities. The graphics on Mathematica were described as “wonderful” by students and clearly helped them understand the topics of the lab. The positive comments were certainly borne out by their performance on the final examination, where 88% of the students could correctly identify pairs of degenerate two-dimensional particle-in-a-box wave functions. However, we realize that students may be misled by their visual understanding of degeneracy. For example, in hydrogen-like atoms all the 3d orbitals are degenerate, but 3dz 2 cannot be rotated and superimposed on any other 3d orbital. When undergraduate students begin to study quantum mechanics, the laboratory component of the course usually is left in abeyance until they can understand some spectroscopy. The laboratory activity presented here fits nicely with introductory quantum material and helps students understand wave functions and probability densities from more than just a symbolic perspective. These Mathematica laboratory activities could be extended to cover solving integrals on the computer or using linear combinations of spherical harmonic wave functions to generate px and p y orbitals. Acknowledgments We would like to acknowledge John Schershel, who initially debugged the lab and provided many helpful comments. We also would like to thank the Ball State University Mathematica Workshop for Faculty supported by the George and Frances Ball Foundation Fund.
Note 1. Presented at the National American Chemical Society Meeting, August 1996, Orlando, FL.
Literature Cited 1. Schrödinger, E. Endeavour 1950, July, 109–116. 2. Mathematica 2.0; Wolfram Research: Champaign, IL 61820-7237. Other associated software: Schroedinger.m can be found at JCE Software 1996, 4D(1) (Windows), and JCE Software 1997, 8C(2) (Macintosh). 3. Rioux, F. J. Chem. Educ. 1992, 69, A240–A242. 4. Rioux, F. J. Chem. Educ. 1991, 68, A282–A283. 5. Breneman, G. L.; Parker, O. J. J. Chem. Educ. 1992, 69, A4–A7.
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