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Activities

You have a charge distribution on the \(x\)-axis composed of two point charges: one with charge \(+3q\) located at \(x=-d\) and the other with charge \(-q\) located at \(x=+d\).

  1. Sketch the charge distribution.
  2. Write an expression for the volume charge density \(\rho (\vec{r})\) everywhere in space.

Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]

Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].

  1. Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
  2. Using the test function \(f(x)=3x^2\), find the value of \[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\] Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).

  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)

Small Group Activity

5 min.

Fourier Transform of the Delta Function
Students calculate the Fourier transform of the Dirac delta function.
  • Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

Let us imagine a new mechanics in which the allowed occupancies of an orbital are 0, 1, and 2. The values of the energy associated with these occupancies are assumed to be \(0\), \(\varepsilon\), and \(2\varepsilon\), respectively.

  1. Derive an expression for the ensemble average occupancy \(\langle N\rangle\), when the system composed of this orbital is in thermal and diffusive contact with a resevoir at temperature \(T\) and chemical potential \(\mu\).

  2. Return now to the usual quantum mechanics, and derive an expression for the ensemble average occupancy of an energy level which is doubly degenerate; that is, two orbitals have the identical energy \(\varepsilon\). If both orbitals are occupied the toal energy is \(2\varepsilon\). How does this differ from part (a)?

Derivative of Fermi-Dirac function Show that the magnitude of the slope of the Fermi-Direc function \(f\) evaluated at the Fermi level \(\varepsilon =\mu\) is inversely proportional to its temperature. This means that at lower temperatures the Fermi-Dirac function becomes dramatically steeper.

Computational Activity

120 min.

Sinusoidal basis set
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.

One way to write volume charge densities without using piecewise functions is to use step \((\Theta)\) or \(\delta\) functions. Consider a spherical shell with charge density \[\rho (\vec{r})=\alpha3e^{(k r)^3} \]

between the inner radius \(a\) and the outer radius \(b\). The charge density is zero everywhere else.

  1. (2 pts) What are the dimensions of the constants \(\alpha\) and \(k\)?
  2. (2 pts) By hand, sketch a graph the charge density as a function of \(r\) for \(\alpha > 0\) and \(k>0\) .
  3. (2 pts) Use step functions to write this charge density as a single function valid everywhere in space.

Problem

5 min.

Ring Function
Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a quantum mechanical particle of mass \(\mu\) constrained to move on a circle of radius \(r_0\), given by: \begin{equation} \Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)} \end{equation} where \(N\) is the normalization constant.
  1. Find \(N\).

  2. Plot this wave function.
  3. Plot the probability density.
  4. Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
  5. What is the expectation value of \(L_z\) in this state?
  • Found in: Central Forces course(s)

Small Group Activity

5 min.

Fourier Transform of a Shifted Function

Suppose you have a definite function \(f(x)\) in mind and you already know its Fourier transform, i.e. you know how to do the integral \begin{equation} \tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty}e^{-ikx}\, f(x)\, dx \end{equation} Find the Fourier transform of the shifted function \(f(x-x_0)\).

Instructor's Guide

Introduction

Students will need a short lecture giving the definition of the Fourier Transform \begin{equation} {\cal{F}}(f) =\tilde{f} (k)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}\, f(x)\, dx \end{equation}

Student Conversations

This example will feel very abstract to some students. It may be difficult for them to understand that the conditions of the problem state that the know both \(f(x)\) and \(\tilde{f}(k)\). This problem is about changing \(f\) slightly (by shifting its argument by \(x_0\)) and then asking how \(\tilde{f}\) changes, in response.

Wrap-up

The result from this calculation underlies why it is possible to factor out the time dependence in the Fourier transform of a plane wave, Fourier Transform of a Plane Wave. Even though the problem is somewhat abstract, it is super important in applications for this reason.
  • Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

Computational Activity

120 min.

Mean position
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.

Small Group Activity

30 min.

Add Two Functions
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.
  • Found in: Oscillations and Waves course(s)

Small Group Activity

10 min.

Modeling Nonuniform Density
In this small group activity, students calculate a (linear) function to represent the charge density on a one-dimensional rod from a description of the charge density in words.
  • Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s)

Small Group Activity

5 min.

Maxima and Minima
This small group activity introduces students to constrained optimization problems. Students work in small groups to optimize a simple function on a given region. The whole class wrap-up discussion emphasizes the importance of the boundary.
  • Found in: Vector Calculus I course(s)

Small Group Activity

30 min.

The Hill
In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
  • Gradient
    Found in: Vector Calculus II, Vector Calculus I, Surfaces/Bridge Workshop course(s) Found in: Gradient Sequence sequence(s)

Small Group Activity

30 min.

DELETE Navigating a Hill
In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
  • Found in: Static Fields, AIMS Maxwell course(s)

Small Group Activity

10 min.

Guess the Fourier Series from a Graph
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.
  • Found in: Oscillations and Waves, None course(s)

Small Group Activity

5 min.

Calculating a Total Differential
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.

Small Group Activity

30 min.

Using \(pV\) and \(TS\) Plots
Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.

Small Group Activity

60 min.

Ice Calorimetry Lab
The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply, from which they compute changes in entropy.

Mathematica Activity

30 min.

Effective Potentials
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 Simulation

30 min.

Visualization of Power Series Approximations
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.
  • Taylor series power series approximation
    Found in: Theoretical Mechanics, Static Fields, Central Forces, AIMS Maxwell, Problem-Solving, None course(s) Found in: Power Series Sequence (Mechanics), Power Series Sequence (E&M) sequence(s)

Small Group Activity

30 min.

Expectation Values for a Particle on a Ring
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.
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.

Small White Board Question

10 min.

Partial Derivatives from a Contour Map
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?
  • Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gradient Sequence sequence(s)