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.
Use this Mathematica worksheet to explore the spatial properties of linear combinations of spherical harmonics \(Y_{\ell}^{m}(\theta, \phi)\).
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.
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.group Small Group Activity
30 min.
assignment Homework
For each case below, find the total charge.
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence
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 Small Group Activity
10 min.
format_list_numbered Sequence
assignment Homework
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.
assignment Homework
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition
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.