Activities
This activity reinforces the strategies students have been practicing on each system by letting them create their own matrix operators and columns on the hydrogen atom and do some calculations with them.
This activity gives links to some external resources (2 simulations and 1 video) that allow students to explore circle trigonometry. There are no prompts and nothing specific to turn in.
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.
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.
In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.
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.
Spherical harmonics are continuous functions on the surface of a sphere.
The \(\ell\) and \(m\) values tell us how the function oscillates across the surface.
Spherical harmonics are complex valued functions.
- The effect of angular momentum, the force constant, and the reduced mass on the shape of the effective potential function;
- Practicing exploring parameter space for a function;
- Developing intuition about how the orbit shape depends on these parameters.
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 \(m\).
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.
Students are placed into small groups and asked to create an experimental setup they can use to measure the partial derivative they are given, in which entropy changes.
Students are placed into small groups and asked to calculate the total differential of a function of two variables, each of which is in turn expressed in terms of two other variables.
The students are shown the graph of a function that is a superposition of three harmonic functions and asked to guess the harmonic terms of the Fourier series. Students then use prewritten Sage code to verify the coefficients from their guess. The program allows the students to enter functions of their own choice as well as the one that is preset.
Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.
The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.
Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
Basic algebraic and geometric properties of the dot product.
- How to represent 3-d scalar fields in several different ways;
- The symmetries of a some simple charge distributions such as a dipole and a quadrupole.
Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.
Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.
- Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
- Students should know that
- objects with like charge repel and opposite charge attract,
- object tend to move toward lower energy configurations
- The potential energy of a charged particle is related to its charge: \(U=qV\)
- The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)
Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.
Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.
Student consider several curves on a spacetime diagram and have to judge which curves could be worldlines for an object.
Students answer conceptual questions about time dilation and proper time.
Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.
Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.
Students become acquainted with the Spins Simulations of Stern-Gerlach Experiments and record measurement probabilities of spin components for a spin-1/2 system. Students start developing intuitions for the results of quantum measurements for this system.
In this sequence of small whiteboard questions, students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.
Students set up and compute a scalar surface integral.
Students compute surface integrals and explore their interpretation as flux.
These lecture notes from the ninth week of https://paradigms.oregonstate.edu/courses/ph441 cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.
This small group activity is designed to help students visual the process of chopping, adding, and multiplying in single integrals. Students work in small groups to determine the volume of a cylinder in as many ways as possible. The whole class wrap-up discussion emphasizes the equivalence of different ways of chopping the cylinder.
Students construct the volume element in cylindrical and spherical coordinates.
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
- How to translate a complicated wavefunction into eigenstates.
- Refresher on how to find expectation values and probabilities in a region.
- How to use the symmetry of the wavefunction to tell you something about measurements.
This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.
The attached powerpoint articulates the possible paths through the curriculum for new graduate students at OSU. Make sure to update this powerpoint yearly to reflect current course offerings and sequencing. It was partially, but not completely edited in fall 2022.
Students use an applet to explore the role of the parameters \(N\), \(x_o\), and \(\sigma\) in the shape of a Gaussian \begin{equation} f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation}
Students learn about the geometric meaning of the amplitude and period parameters in the sine function. They also practice sketching the sum of two functions by hand.
Students are gently introduced to the relationship between contour maps and surfaces.
Students draw the 3D graphs of equations using three variables. They make choices for drawing a stack of curves in parallel planes and a curve in a perpendicular plane (e.g. substituting in values for \(x\), \(y\), or \(z\). )