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Visualization of the Spherical Harmonics: Instructor's Guide

Main Ideas

In this activity students have the opportunity to explore how the shapes of various spherical harmonics are related to their equations. Because many of the spherical harmonics are explicitly complex, we plot the square of the norm of the spherical harmonics rather than the spherical harmonics themselves. These graphs then correspond to the probability of finding a quantum mechanical particle at a particular point on the surface of the sphere. These graphs are also probability densities for the rigid rotor problem.

Estimated Time: 15 min.

Prerequisite Knowledge

Students will need to have seen an algebraic derivation of the formulas for the spherical harmonics. Be careful! Conventions for the spherical harmonics, especially the phases, differ from one resource to another. This activity works best if students are asked to explicitly relate the equations they know for the spherical harmonics to their graphs, so it's best if the equations that they know agree with the equations in activity.

Activity: Introduction

Typically we begin this activity by leading students through the worksheet one time to explain briefly what each line of the Maple code does. Ultimately we plot the value of the $Y_{l,m}(\theta, \phi)$ on the surface of a sphere using color to represent the value of the function. We use a sequence of graphs bridging from more familiar representations in which the value of the function is indicated by the z-value, then the z-value and the color, then the color on a plane and finally the color on a sphere. For most of the activity, we restrict students to only using these representations.

The last graph (which we do not emphasize) is a polar plot in which distance from the origin as well as color indicates the magnitude of the $Y_{l,m}(\theta, \phi)$. We include this plot only because students have seen these plots before and we use this plot to clarify to them that these are not plots of the hydrogen atom orbitals, but rather plots of the spherical harmonics in which the value of the spherical harmonic is indicated by the distance from the origin.

We use this activity is part of the sequence of activities that allow students to visualize a quantum mechanical particle first confined to a ring, then confined to the surface of the sphere, and finally for the unperturbed hydrogen atom. In each of these cases, we want the students to think about how they might graph the important information. Students are often confused by spherical harmonics when they are plotted as polar graphs, confusing these with three-dimensional plots of orbitals. We find that students are more likely to understand that the particle is actually confined to the surface of a sphere plot probability densities as colors on the surface of the sphere. Only later do we show them the polar plots and discuss their physical interpretation. If you use this activity out of sequence it's important to discuss the use of color to convey the desired information before the students start this activity.

Note: We try as much as possible to only plot probability densities and not wave function to discourage students from seeing the wave function as a physically observable entity.

Activity: Student Conversations

As always with computer-based activities, some students will immediately try to break the code by plugging in large values of l and m. The spherical harmonics are calculated by taking derivatives of Rodrigue's formula. Some students will be intrigued that the formula gives them zero if the value of m is less than the value of l.

Students often mistakenly take the standard polar plots of the $Y_{l,m}(\theta, \phi)$'s to be plots of the hydrogen atom orbitals. When this happens, they do not recognize that in the polar plots, the radial distance from the origin represents the value of the function and not the radial position in space. In order to avoid this confusion, we carefully discuss each of the representations in this worksheet to emphasize the progression from graphs they are familiar with to the kind we are plotting. We focus on the fact that spatial dimensions are sometimes used to display the value of the function rather than the values of spatial variables. To avoid this confusion all together, we prefer the plot of the value of the spherical harmonics on the sphere as colors on a sphere rather than the more common polar plots.

One task that we have found to be particularly good at eliciting meaningful discussion is to ask the students to find the effect of el and m on the shape of the probability distribution for the spherical harmonics. More student conversations will sometimes arise if this activity is introduced at the end of class and then brought up again the next day. This allows students to play with the worksheet and generate their own questions and observations.

Some particularly interesting observations include:

• Students often ask why all the probability densities seem to have no $\phi$ dependence. This can be explained by looking at what happens to the $\phi$ part of the $Y_{l,m}(\theta, \phi)$'s when they are squared. A quick analysis shows that the $\phi$ terms all square to 1.
• The fact that all of the $Y_{l,m}(\theta, \phi)$'s are axially symmetric is confusing to students since they've seen hydrogen atom wavefunctions that are not axially symmetric in chemistry. This question serves as a good lead-in to the next Maple activity in which they are able to look at linear combinations of spherical harmonics that demonstrate the angular dependence that students expect from chemistry.
• Students notice that for large values of el and m, the probability distribution is nearly planar. It is interesting to think about how this may be related to the classical limit.

Activity: Wrap-up

The most important thing to discuss here is why all the graphs are spherically symmetric. This activity can be followed up with another activity that allows students to think about linear combinations of spherical harmonics and see that these linear combinations are not spherically symmetric.

Students can also be asked to determine how the graphs depend on the values of l and m. This question focuses the students' attention on how the graphs are related to the equations rather than just on their oooh! aaah! reaction.

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