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Activities

Lecture

30 min.

##### Compare & Contrast Kets & Wavefunctions
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
• Found in: Completeness Relations sequence(s)

Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
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 $L_z$.
• Found in: Central Forces course(s)

Small Group Activity

30 min.

##### Working with Representations on the Ring
This activity acts as a reintroduction to doing quantum calculations while also introducing the matrix representation on the ring, allowing students to discover how to index and form a column vector representing the given quantum state. In addition, this activity introduces degenerate measurements on the quantum ring and examines the state after measuring both degenerate and non-degenerate eigenvalues for the state.
• Found in: Central Forces course(s)

Problem

5 min.

##### Quiz: Summation Notation, Version 0
Write out the terms in the following sums that have the lowest energy. Stop when you have at least 9 terms, but only stop at some point that is logical, given the symmetries and degeneracies. Briefly explain why you chose to stop when you did. You may have to guess what system you are working from the form of the sum and which eigenvalue(s) determine the energy. You may assume that low energies correspond to eigenvalues near zero. Clearly state any assumptions that you make. You may use bra/ket notation in your solutions. (If these directions are unclear, check out the solutions below for some examples.)
1. $\sum_{m=-\infty}^{\infty} c_m e^{im\phi}$
2. $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} c_{mn} \sin\left(\frac{m\pi x}{L_x}\right)\sin\left(\frac{n\pi y}{L_y}\right)$
3. $\sum_{n=1}^{\infty}\sum_{\ell=0}^{n-1}\sum_{m=-\ell}^{\ell} c_{n \ell m} \left|{n, \ell, m}\right\rangle$
• Found in: Central Forces course(s)

Small Group Activity

30 min.

##### Finding Matrix Elements
In this small group activity, students multiply a general 3x3 matrix with standard basis row/column vectors to pick out individual matrix elements. Students generate the expressions for the matrix elements in bra/ket notation.
• Found in: Quantum Fundamentals course(s)

Small Group Activity

10 min.

##### Generalized Leibniz Notation
This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics, in which the variables being held constant are given explicitly. Students are guided to associate variables to their proper categories.
• Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s)

Problem

##### Dimensional Analysis of Kets
1. $\left\langle {\Psi}\middle|{\Psi}\right\rangle =1$ Identify and discuss the dimensions of $\left|{\Psi}\right\rangle$.
2. For a spin $\frac{1}{2}$ system, $\left\langle {\Psi}\middle|{+}\right\rangle \left\langle {+}\middle|{\Psi}\right\rangle + \left\langle {\Psi}\middle|{-}\right\rangle \left\langle {-}\middle|{\Psi}\right\rangle =1$. Identify and discuss the dimensions of $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
3. In the position basis $\int \left\langle {\Psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\Psi}\right\rangle dx = 1$. Identify and discuss the dimesions of $\left|{x}\right\rangle$.
• Found in: Completeness Relations sequence(s)

Problem

##### Completeness Relation Change of Basis
1. Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}

Find the following quantities: $\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle$

2. Given a vector written in the polar basis $\left|{\vec{v}}\right\rangle = a\left|{\hat{s}}\right\rangle + b\left|{\hat{\phi}}\right\rangle$ where $a$ and $b$ are known. Find coefficients $c$ and $d$ such that $\left|{\vec{v}}\right\rangle = c\left|{\hat{x}}\right\rangle + d\left|{\hat{y}}\right\rangle$ Do this by using the completeness relation: $\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1$
3. Using a completeness relation, change the basis of the spin-1/2 state $\left|{\Psi}\right\rangle = g\left|{+}\right\rangle + h\left|{-}\right\rangle$ into the $S_y$ basis. In otherwords, find $j$ and $k$ such that $\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y$
• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations sequence(s)

Problem

5 min.

##### Dirac Practice
For this problem, use the vectors $|a\rangle = 4 |1\rangle - 3 |2\rangle$ and $|b\rangle = -i |1\rangle + |2\rangle$.
1. Find $\langle a | b \rangle$ and $\langle b | a \rangle$. Discuss how these two inner products are related to each other.
2. For $\hat{Q}\doteq \begin{pmatrix} 2 & i \\ -i & -2 \end{pmatrix}$, calculate $\langle1|\hat{Q}|2\rangle$, $\langle2|\hat{Q}|1\rangle$, $\langle a|\hat{Q}| b \rangle$ and $\langle b|\hat{Q}|a \rangle$.
3. What kind of mathematical object is $|a\rangle\langle b|$? What is the result if you multiply a ket (for example, $| a\rangle$ or $|1\rangle$) by this expression? What if you multiply this expression by a bra?
• Found in: Quantum Fundamentals course(s)

Problem

##### Series Notation 2

Write (a good guess for) the following series using sigma $\left(\sum\right)$ notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

1. $1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots$

2. $\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots$

• Found in: Static Fields, AIMS Maxwell, Problem-Solving, None course(s) Found in: Power Series Sequence (E&M) sequence(s)

Problem

##### Series Notation 1

Write out the first four nonzero terms in the series:

1. $\sum\limits_{n=0}^\infty \frac{1}{n!}$

2. $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}$
3. $$\sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}}$$

• Found in: Static Fields, AIMS Maxwell, Problem-Solving, None course(s) Found in: Power Series Sequence (E&M) sequence(s)

Small Group Activity

30 min.

##### Hydrogen Probabilities in Matrix Notation
This activity reinforces the strategies students have been practicing on each system by letting them create their own matrix operators and columns on the hydrogen atom and do some calculations with them.
• Found in: Central Forces course(s)

Small Group Activity

60 min.

##### Going from Spin States to Wavefunctions
Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.
• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations, Arms Sequence for Complex Numbers and Quantum States sequence(s)

Small Group Activity

60 min.

##### Quantum Calculations on the Hydrogen Atom

Students are asked to find eigenvalues, probabilities, and expectation values for $H$, $L^2$, and $L_z$ for a superposition of $\vert n \ell m \rangle$ states. This can be done on small whiteboards or with the students working in groups on large whiteboards.

Students then work together in small groups to find the matrices that correspond to $H$, $L^2$, and $L_z$ and to redo $\langle E\rangle$ in matrix notation.

• Found in: Central Forces course(s)

Small White Board Question

5 min.

##### Representations of Vectors
Students each recall a representation of vectors that they have seen before and record it on an individual whiteboard. The instructor uses these responses to generate a whole class discussion that compares and contrasts the features of the representations. If appropriate for the class, the instructor introduces bra/ket notation as a new, but valuable representation.
• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s)

Whole Class Activity

10 min.

##### Curvilinear Coordinates Introduction
First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles $\theta$ and $\phi$. Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.
• Found in: Static Fields, Central Forces, AIMS Maxwell, Vector Calculus I, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Curvilinear Coordinate Sequence sequence(s)

Small Group Activity

30 min.

##### Paramagnet (multiple solutions)
• Students evaluate two given partial derivatives from a system of equations.
• Students learn/review generalized Leibniz notation.
• Students may find it helpful to use a chain rule diagram.

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.
• Found in: Central Forces, Theoretical Mechanics course(s) Found in: Quantum Ring Sequence sequence(s)

Small Group Activity

30 min.

##### Time Dependence for a Quantum Particle on a Ring Part 1
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.
• Found in: Central Forces, Theoretical Mechanics course(s) Found in: Quantum Ring Sequence sequence(s)

Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring
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.
• Found in: Theoretical Mechanics course(s) Found in: Quantum Ring Sequence sequence(s)

Kinesthetic

30 min.

##### The Distance Formula (Star Trek)
A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. $\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}$
• Found in: Static Fields, AIMS Maxwell course(s) Found in: E&M Ring Cycle Sequence sequence(s)

Small White Board Question

5 min.

##### Normalization of the Gaussian for Wavefunctions
Students find a wavefunction that corresponds to a Gaussian probability density.
• Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring
Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
• Found in: Quantum Ring Sequence sequence(s)