## Activity: Visualization of Quantum Probabilities for the Hydrogen Atom

Central Forces 2023 (3 years)
Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of $n$, $\ell$, and $m$.
What students learn
• Mathematica is used to compute the full solution quantum mechanical solution to the hydrogen atom
• Students can visualize the probability density of the electron
• Students can connect various represent to the hydrogen orbitals they've seen in chemistry
• Media

Use this Mathematica Notebook to visualize the probability density for hydrogen atom eigenstates.

## Main Ideas

• Mathematica is used to compute the full solution quantum mechanical solution to the hydrogen atom
• Students can visualize the probability density of the electron
• Students can connect various represent to the hydrogen orbitals they've seen in chemistry

Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of $n$, $\ell$, and $m$.

## Activity: Introduction

Before beginning this activity, it is helpful to discuss with students how one might plot a three dimensional scalar field like the probability density. After discussing some of their ideas, it is important to describe the method we use for plotting the probability density in three dimensions. We use color to represent the magnitude of the probability density. The Maple worksheet displays one static slice through the $x$-$z$ plane (with $y=0$) and a second cross sectional slice parallel to the $x$-$y$ plane. The plot is rendered as an animation so that one can vary the $z$-value of the slice parallel to the $x$-$y$ plane either by dragging the slider or playing the animation.

## Activity: Student Conversations

Students will often ask why the plots in this worksheet are different from the plots most commonly found in text books. We try to help them see that the plots most commonly included in text books are iso-probability plots. We explain that the iso-probability plots are surfaces of uniform probability. To see this in the maple worksheet, imagine looking at only shows one color.

Some good questions that help students focus their exploration include:

• What do you notice about the shape of the probability densities when $\ell=0$?
• Do you notice any symmetries in the orbitals?
• How do the orbitals vary with $n$, $\ell$, and $m$?
• Since all the orbitals are axially symmetric how do you think one might represent non-axially symmetric electron wavefunctions like the the chemistry orbitals $p_x$, $p_y$, etc. using these orbitals?

## Activity: Wrap-up

We often wrap up this activity by trying to tie their observations of the shapes of the orbitals back to what they know about hydrogen orbitals from chemistry and modern physics.

Some questions we use to guide the wrap-up discussion include:

• What do the orbitals look like when $\ell$ is zero?
• What do you expect/observe for high values of $n$ with $\ell=0$?
• Choose $n$ equal to some larger value ($n=6$) and $l$ equal to the largest value possible, ($\ell=5$). Now compare the shapes of the orbitals for $m$ a maximum ($m=5$) and $m=0$. Think about these orbitals from the perspective of moment of inertia in the classical sense. Do the shapes of the orbitals you observe make sense with regard to the associated eigenvalues for the $z$-component of the angular momentum? Explain what about the shape of the orbitals makes sense? Make a prediction about how the shape of the orbitals will change as you vary the value of $m$ from its maximum value to its minimum value.

• 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.
• computer Visualizing Combinations of Spherical Harmonics

computer Mathematica Activity

30 min.

##### 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.
• computer Effective Potentials

computer Mathematica Activity

30 min.

##### Effective Potentials
Central Forces 2023 (3 years) Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.
• computer Visualizing Flux through a Cube

computer Computer Simulation

30 min.

##### Visualizing Flux through a Cube
Static Fields 2023 (6 years) Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.

computer Mathematica Activity

30 min.

Static Fields 2023 (6 years)

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
• computer Visualization of Power Series Approximations

computer Computer Simulation

30 min.

##### Visualization of Power Series Approximations
Theoretical Mechanics (13 years)

Power Series Sequence (E&M)

Students use prepared Sage code or a prepared Mathematica notebook to plot $\sin\theta$ simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.
• computer Using Technology to Visualize Potentials

computer Mathematica Activity

30 min.

##### Using Technology to Visualize Potentials
Static Fields 2023 (6 years)

Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrate several different ways of plotting the potential.
• assignment Effective Potentials: Graphical Version

assignment Homework

##### Effective Potentials: Graphical Version
Central Forces 2023 (2 years)

Consider a mass $\mu$ in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is $\ell\ne 0$ for a given fixed value of $\ell$.

1. Give the definition of a central force system and briefly explain why this situation qualifies.
2. Make a sketch of the graph of the effective potential for this situation.
3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

• assignment Diatomic hydrogen

assignment Homework

##### Diatomic hydrogen
rigid rotor hamiltonian angular momentum ground state hydrogen diatomic probability Energy and Entropy 2021 (2 years)

At low temperatures, a diatomic molecule can be well described as a rigid rotor. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{\ell m} &= \hbar^2 \frac{\ell(\ell+1)}{2I} \end{align}

1. What is the energy of the ground state and the first and second excited states of the $H_2$ molecule? i.e. the lowest three distinct energy eigenvalues.

2. At room temperature, what is the relative probability of finding a hydrogen molecule in the $\ell=0$ state versus finding it in any one of the $\ell=1$ states?
i.e. what is $P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)$

3. At what temperature is the value of this ratio 1?

4. At room temperature, what is the probability of finding a hydrogen molecule in any one of the $\ell=2$ states versus that of finding it in the ground state?
i.e. what is $P_{\ell=0,m=0}/\left(P_{\ell=2,m=-2} + P_{\ell=2,m=-1} + \cdots + P_{\ell=2,m=2}\right)$

• assignment Undo Formulas for Reduced Mass (Geometry)

assignment Homework

##### Undo Formulas for Reduced Mass (Geometry)
Central Forces 2023 (3 years)

The figure below shows the position vector $\vec r$ and the orbit of a “fictitious” reduced mass $\mu$.

1. Suppose $m_1=m_2$, Sketch the position vectors and orbits for $m_1$ and $m_2$ corresponding to $\vec{r}$. Describe a common physics example of central force motion for which $m_1=m_2$.
2. Repeat, for $m_2>m_1$.

Learning Outcomes