Activities
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.
Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .
Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.
This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.
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 explore path integrals using a vector field map and thinking about integration as chop-multiply-add.
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.
Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.
In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.
This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..
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 compute vector line integrals and explore their properties.
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.
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 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 \(z\). )
Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.
Many students do not know what it means to add two functions graphically. Students are shown graphs of two simple functions and asked to sketch the sum.
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 use their arms to act out two spin-1/2 quantum states and their inner product.
Students use a completeness relations to write hydrogen atoms states in the energy and position bases.
Students consider the dimensions of spin-state kets and position-basis kets.
Students are asked to "find the derivative" of a plastic surface that represents a function of two variables. This ambiguous question is designed to help them generalize their concept of functions of one variable to functions of two variables. The definition of the gradient as the slope and direction of the "steepest derivative" is introduced geometrically.
Students are gently introduced to the relationship between contour maps and surfaces.
- 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.
- A component of the curl of a vector field (at a point) is the circulation per unit area around an infinitesimal loop.
- How to predict the sign and relative magnitude of the curl from graphs of a vector field.
- (Optional) How to calculate the curl of a vector field using computer algebra.