## Activity: Visualizing Combinations of Spherical Harmonics

Central Forces 2023 (3 years)
Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.
What students learn
• Students use Mathematica to visualize states that are made up of linear combinations of spherical harmonics.
• Students compare and contrast three different representations of the probability densities.
• Students are asked to explore the geometric role of the relative phase between two states with the same value of $L^2$
• Students are asked to explore states with linear combinations with different values of $L^2$.
• Media

Use this Mathematica worksheet to explore the spatial properties of linear combinations of spherical harmonics $Y_{\ell}^{m}(\theta, \phi)$.

## Instructor's Guide

### 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. This worksheet plots the square of the norm of the function (probability density in quantum mechanics). Students are also reminded that the probability density is represented by the color in the case of the first sphere plot, that the polar plot (the second to last one in the worksheet) indicates the value by both the color and the distance from the origin and the final graph indicates the value by both the color and the distance from the sphere. 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.

### Student Conversations

• A good question to help frame students exploration is to ask them to identify how the combinations of $Y_{\ell,m}$s are different from individual $Y_{\ell,m}$s. In particular, students will notice that the axial symmetry common to all of the individual $Y_{\ell,m}$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 are included. Note the role of the relative phase.

### 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.
• computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
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group Small Group Activity

5 min.

##### Spherical Harmonics
Central Forces 2023 (3 years)
• assignment Electric Field and Charge

assignment Homework

##### Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2023 (4 years) Consider the electric field $$\vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases}$$
1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
2. Find a formula for the charge density that creates this electric field.
3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
• assignment Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2023 (6 years)

For each case below, find the total charge.

1. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $$\rho(\vec{r})=3\alpha\, e^{(kr)^3}$$
2. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $$\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}$$

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##### Sequential Stern-Gerlach Experiments
Quantum Fundamentals 2023 (3 years)
• face Quantum Reference Sheet

face Lecture

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##### Quantum Reference Sheet
Central Forces 2023 (6 years)
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group Small Group Activity

30 min.

##### Operators & Functions
Quantum Fundamentals 2023 (3 years) Students are asked to:
• Test to see if one of the given functions is an eigenfunction of the given operator
• See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
• assignment Gibbs entropy is extensive

assignment Homework

##### Gibbs entropy is extensive
Gibbs entropy Probability Thermal and Statistical Physics 2020

Consider two noninteracting systems $A$ and $B$. We can either treat these systems as separate, or as a single combined system $AB$. We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state $(i_A,j_B)$ is given by $P_{ij}^{AB} = P_i^AP_j^B$. In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

1. Show that the entropy of the combined system $S_{AB}$ is the sum of entropies of the two separate systems considered individually, i.e. $S_{AB} = S_A+S_B$. This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
2. Show that if you have $N$ identical non-interacting systems, their total entropy is $NS_1$ where $S_1$ is the entropy of a single system.

##### Note
In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

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face Lecture

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##### Introducing entropy
Contemporary Challenges 2021 (4 years)

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
• keyboard Electrostatic potential of spherical shell

keyboard Computational Activity

120 min.

##### Electrostatic potential of spherical shell
Computational Physics Lab II 2022

Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.

Learning Outcomes