Students work in a small group to write down an equation for a travelling wave.

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 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.

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.

In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.

Find the Fourier transform of a plane wave.

Instructor's Guide

Introduction

If students know about the Dirac delta function and its exponential representation, this is a great second example of the Fourier transform that students can work out in-class for themselves.

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

Students may ask what is meant by a plane wave. Help them figure out what is meant, from the context or give them the formula if time is tight.

Keep the time dependence in or leave it out depending on how much time you have to deal with a little extra algebraic confusion.

Wrap-up

This example is (almost) the inverse of Fourier Transform of the Delta Function. If you really want the inverse problem, change the prompt to “Find the inverse Fourier transform of a plane wave.”

Using either this Geogebra applet or this Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)

1. Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
2. Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
3. Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
4. Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.

Students calculate the Fourier transform of the Dirac delta 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.

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.

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.

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 \(\epsilon \rightarrow 0\).

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?

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.

• The gradient is perpendicular to the level curves.
• The gradient is a local quantity, i.e. it only depends on the values of the function at infinitesimally nearby points.
• Although students learn to chant that "the gradient points uphill," the gradient does not point to the top of the hill.
• The gradient path is not the shortest path between two points.

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 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.

• Work as area under curve in a \(pV\) plot
• Heat transfer as area under a curve in a \(TS\) plot
• Reminder that internal energy is a state function
• Reminder of First Law

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.

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 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.

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.

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.

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.

This activity lets students explore translating a wavefunction that isn't obviously made up of eigenstates at first glance into ket and matrix form. Then students explore wave functions, probabilities in a region, expectation values, and what wavefunctions can tell you about measurements of \(L_z\).

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 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.