group Small Group Activity

5 min.

Fourier Transform of a Plane Wave
Periodic Systems 2022

Fourier Transforms and Wave Packets

group Small Group Activity

30 min.

Earthquake waves
Contemporary Challenges 2021 (4 years)

wave equation speed of sound

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

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.

assignment Homework

Quantum concentration
bose-einstein gas statistical mechanics Thermal and Statistical Physics 2020 Consider one particle confined to a cube of side \(L\); the concentration in effect is \(n=L^{-3}\). Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature \(kT\). (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration \(n_0\) thus defined is equal to the quantum concentration \(n_Q\) defined by (63): \begin{equation} n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32} \end{equation} within a factor of the order of unity.

group Small Group Activity

5 min.

Fourier Transform of a Shifted Function
Periodic Systems 2022

Fourier Transforms and Wave Packets

group Small Group Activity

5 min.

Fourier Transform of the Delta Function
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students calculate the Fourier transform of the Dirac delta function.

face Lecture

10 min.

Introduction to Central Forces
Central Forces 2023

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. \(\frac{d^2w}{dz^2}-4\frac{dw}{dz}+5w=0\)

group Small Group Activity

30 min.

Working with Representations on the Ring
Central Forces 2023 (3 years)

assignment Homework

Ring Function
Central Forces 2023 (3 years) 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?

keyboard Computational Activity

120 min.

Kinetic energy
Computational Physics Lab II 2022

finite difference hamiltonian quantum mechanics particle in a box eigenfunctions

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.

assignment Homework

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

The general solution of the homogeneous differential equation

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

is

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

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.

assignment Homework

Quantum Particle in a 2-D Box
Central Forces 2023 (4 years) You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length \(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2-D box, then you just multiply together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)
  1. Find the normalized eigenfunctions for a particle in a 2-D box with sides of length \(L_x\) in the \(x\)-direction and length \(L_y\) in the \(y\)-direction.
  2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
  3. Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.

    You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

  4. Find a formula for the \(c_{nm}\)s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.

group Small Group Activity

30 min.

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

keyboard Computational Activity

120 min.

Sinusoidal basis set
Computational Physics Lab II 2023 (2 years)

inner product wave function quantum mechanics particle in a box

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.

group Small Group Activity

30 min.

Energy and Angular Momentum for a Quantum Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy

Quantum Ring Sequence

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.

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.

keyboard Computational Activity

120 min.

Position operator
Computational Physics Lab II 2022

quantum mechanics operator matrix element particle in a box eigenfunction

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.

face Lecture

120 min.

Ideal Gas
Thermal and Statistical Physics 2020

ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.

group Small Group Activity

30 min.

Operators & Functions
Quantum Fundamentals 2023 (3 years) Students are asked to:
  • Test to see if one of the given functions is an eigenfunction of the given operator
  • See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.