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.
• group Spherical Harmonics

group Small Group Activity

5 min.

Spherical Harmonics
Central Forces 2023 (3 years)
• group Operators & Functions

group Small Group Activity

30 min.

Operators & Functions
Quantum Fundamentals 2022 (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 Total Charge

assignment Homework

Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2022 (5 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 $$\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}$$

• group Expectation Values for a Particle on a Ring

group Small Group Activity

30 min.

Expectation Values for a Particle on a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
• group Sequential Stern-Gerlach Experiments

group Small Group Activity

10 min.

Sequential Stern-Gerlach Experiments
Quantum Fundamentals 2022 (3 years)
• format_list_numbered Arms Sequence for Complex Numbers and Quantum States

format_list_numbered Sequence

Arms Sequence for Complex Numbers and Quantum States
“Arms” is an engaging representation of complex numbers in which students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram). The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence. These activities can be combined and sequenced in many different ways; see the Instructor's Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.
• 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.

• assignment Normalization of Quantum States

assignment Homework

Normalization of Quantum States
Central Forces 2023 (3 years) Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy $$\sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1$$
• group Superposition States for a Particle on a Ring

group Small Group Activity

30 min.

Superposition States for a Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.

Learning Outcomes