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Plotting Linear Combinations of Spherical Harmonics: Instructor's Guide

Main Ideas

  • Students use Maple to visualize states that are made up of linear combinations of spherical harmonics.
  • Students see two representations of the probability densities: mapped onto a sphere and a polar plot.

Students' Task

Estimated Time: 15 minutes

Prerequisite Knowledge


Activity: Introduction

The activity is introduced by reminding students that any function on the sphere can be written as a linear combination of the Spherical Harmonics, since they form an orthogonal basis for the space of the sphere. Students are also reminded that the value of the function is given by the color in the case of the sphere plot and that the polar plot (the last one in the worksheet) indicates the value of the function by both the color and the distance from the origin. It is important to caution the students that this worksheet only shows the angular part and that these functions do not contain any information about the radial dependence of the hydrogen atom wavefunctions.

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

  • A good question to help frame students exploration is to ask them to identify how the combinations of $Y_{l,m}$s are different from individual $Y_{l,m}$s. In particular, students will notice that the axial symmetry common to all of the individual $Y_{l,m}$s is not present for all combinations. This may seem counter-intuitive to them and leads to a good discussion of the role of the complex phase in the $\phi$ part of the spherical harmonics.
  • Some particularly interesting states to recommend students view include:

$$\frac{1}{\sqrt{2}} |1,1\rangle +\frac{1}{\sqrt{2}} |1,-1\rangle$$

$$\frac{1}{\sqrt{2}} |1,1\rangle -\frac{1}{\sqrt{2}} |1,-1\rangle$$

$$\frac{1}{\sqrt{2}} |1,1\rangle + \frac{1}{\sqrt{2}} e^{i \delta} |1,-1\rangle$$

$$\frac{1}{\sqrt{2}} |0,0\rangle +\frac{1}{\sqrt{2}} |1,0\rangle$$

$$\frac{1}{\sqrt{2}} |2,2\rangle +\frac{1}{\sqrt{2}} e^{i \delta} |2,-2\rangle$$

Note: $Y_{l,m} = |l,m\rangle$

  • If you have students look at the time dependence, you will only see time variation for combinations with at least two different values of $l$ included.

Activity: Wrap-up

  • It is useful to get students to draw some conclusions about when you will and when you will not see axial symmetry with combinations of spherical harmonics.
  • It is useful to get students to draw conclusions about when they will and when they will not see time dependence with combinations of spherical harmonics.
  • We sometimes have students carry out a hand calculations for a time-varying linear combination, e.g., $\frac{1}{\sqrt{2}} |0,0\rangle +\frac{1}{\sqrt{2}} |1,0\rangle$, and a non time-varying linear combination, e.g., $\frac{1}{\sqrt{2}} |1,1\rangle +\frac{1}{\sqrt{2}} |1,-1\rangle$ so they can see how the time dependence arises in the algebraic expression. It is worthwhile to remind students how time dependence arose in the case of spin 1/2 systems and try to get them to draw the general conclusion that time dependence in the probability density arises only when two eigenstates of different energies are added together.

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