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 \(m\).
Use this Mathematica Notebook to visualize the probability density for hydrogen atom eigenstates.
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 \(m\).
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.
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:
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:
computer Mathematica Activity
30 min.
central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica
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 Mathematica Activity
30 min.
computer Mathematica Activity
30 min.
computer Computer Simulation
30 min.
computer Mathematica Activity
30 min.
computer Computer Simulation
30 min.
Taylor series power series approximation
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 Mathematica Activity
30 min.
assignment Homework
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}
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.
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
\(P_{\ell=0,m=0}/\left(P_{\ell=1,m=-1} + P_{\ell=1,m=0} + P_{\ell=1,m=1}\right)\)
At what temperature is the value of this ratio 1?
assignment Homework
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 \(\ell\).
assignment Homework
The figure below shows the position vector \(\vec r\) and the orbit of a “fictitious” reduced mass \(\mu\).