assignment Homework

Fourier Transform of Cosine and Sine
Periodic Systems 2022
  1. Find the Fourier transforms of \(f(x)=\cos kx\) and \(g(x)=\sin kx\).
  2. Find the Fourier transform of \(g(x)\) using the formula for the Fourier transform of a derivative and your result for the Fourier transform of \(f(x)\). Compare with your previous answer.
  3. In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function \(\tilde{f}(k)\) is a continuous histogram of how much each functions \(e^{ikx}\) contributes to the quantum state. What does the Fourier transform of the function \(\cos kx\) tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.

assignment Homework

Fourier Series for the Ground State of a Particle-in-a-Box.
Oscillations and Waves 2023 (2 years) Treat the ground state of a quantum particle-in-a-box as a periodic function.
  • Set up the integrals for the Fourier series for this state.

  • Which terms will have the largest coefficients? Explain briefly.

  • Are there any coefficients that you know will be zero? Explain briefly.

  • Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.

  • Using the technology of your choice, plot the ground state and your approximation on the same axes.

assignment Homework

Circle Trigonometry
trigonometry cosine sine math circle Quantum Fundamentals 2023 (3 years)

On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)

assignment Homework

Homogeneous Linear ODE's with Constant Coefficients
ODEs math bits Oscillations and Waves 2023 (2 years)

Homogeneous, linear ODEs with constant coefficients were likely covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see:

Constant Coefficients, Homogeneous

or your differential equations text.

Answer the following questions for each differential equation below:

  • identify the order of the equation,
  • find the number of linearly independent solutions,
  • find an appropriate set of linearly independent solutions, and
  • find the general solution.
Each equation has different notations so that you can become familiar with some common notations.
  1. \(\ddot{x}-\dot{x}-6x=0\)
  2. \(y^{\prime\prime\prime}-3y^{\prime\prime}+3y^{\prime}-y=0\)
  3. \(\frac{d^2w}{dz^2}-4\frac{dw}{dz}+5w=0\)

face Lecture

5 min.

Guide to Professional Typography in Physics
Contemporary Challenges 2021 (4 years)


This is really a handout, which gives students guidelines on how to type up physics content.

group Small Group Activity

30 min.

Travelling wave solution
Contemporary Challenges 2021 (4 years)

travelling wave

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

group Small Group Activity

30 min.

Time Dependence for a Quantum Particle on a Ring Part 1
Theoretical Mechanics (6 years)

central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements

Quantum Ring Sequence

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.

assignment Homework

Inhomogeneous Linear ODEs with Constant Coefficients
Oscillations and Waves 2023 (2 years)

The general solution of the homogeneous differential equation

\[\ddot{x}-\dot{x}-6 x=0\]


\[x(t)=A\, e^{3t}+ B\, e^{-2t}\]

where \(A\) and \(B\) are arbitrary constants that would be determined by the initial conditions of the problem.

  1. Find a particular solution of the inhomogeneous differential equation \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

  2. Find the general solution of \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

  3. Some terms in your general solution have an undetermined coefficients, while some coefficients are fully determined. Explain what is different about these two cases.

  4. Find a particular solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}\)

  5. Find the general solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}-25\sin(4 t)\)

    How is this general solution related to the particular solutions you found in the previous parts of this question?

    Can you add these particular solutions together with arbitrary coefficients to get a new particular solution?

  6. Sense-making: Check your answer; Explicitly plug in your final answer in part (e) and check that it satisfies the differential equation.

group Small Group Activity

30 min.

Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)

group Small Group Activity

30 min.

Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2023 (6 years)

compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.

In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

face Lecture

120 min.

Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

Planck distribution blackbody radiation photon statistical mechanics

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.