accessibility_new Kinesthetic

10 min.

Curvilinear Basis Vectors
Static Fields 2022 (8 years)

symmetry curvilinear coordinate systems basis vectors

Curvilinear Coordinate Sequence

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).

format_list_numbered Sequence

Curvilinear Coordinate Sequence
The curvilinear coordinate sequence introduces cylindrical and spherical coordinates (including inconsistencies between physicists' and mathematicians' notational conventions) and the basis vectors adapted to these coordinate systems.

keyboard Computational Activity

120 min.

Sinusoidal basis set
Computational Physics Lab II 2022

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.

accessibility_new Kinesthetic

10 min.

Spin 1/2 with Arms
Quantum Fundamentals 2022 (2 years)

Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.

group Small Group Activity

30 min.

Quantum Measurement Play
Quantum Fundamentals 2022 (2 years)

Quantum Measurement Projection Operators Spin-1/2

The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.

group Small Group Activity

30 min.

Electric Field Due to a Ring of Charge
Static Fields 2022 (7 years)

coulomb's law electric field charge ring symmetry integral power series superposition

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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.

assignment Homework

Completeness Relation Change of Basis
change of basis spin half completeness relation dirac notation Quantum Fundamentals 2022 (2 years)
  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 \[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]

assignment_ind Small White Board Question

10 min.

Vector Differential--Rectangular
Static Fields 2022 (7 years)

vector differential rectangular coordinates math

Integration Sequence

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

group Small Group Activity

60 min.

Going from Spin States to Wavefunctions
Quantum Fundamentals 2022 (2 years)

Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation

Arms Sequence for Complex Numbers and Quantum States

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.

assignment_ind Small White Board Question

10 min.

Dot Product Review
Static Fields 2022 (6 years)

dot product math

This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.

group Small Group Activity

30 min.

Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2022 (6 years)

magnetic fields current Biot-Savart law vector field symmetry

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.

In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) 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.

group Small Group Activity

30 min.

Outer Product of a Vector on Itself
Quantum Fundamentals 2022 (2 years)

Projection Operators Outer Products Matrices

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).

accessibility_new Kinesthetic

10 min.

Using Arms to Represent Time Dependence in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

Arms Representation quantum states time dependence Spin 1/2

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

group Small Group Activity

30 min.

Vector Surface and Volume Elements
Static Fields 2022 (3 years)

Integration Sequence

Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

assignment Homework

Line Sources Using Coulomb's Law
Static Fields 2022 (5 years)
  1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance \(s\) straight out from the midpoint, starting from Coulomb's Law.
  2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.

group Small Group Activity

30 min.

Vector Differential--Curvilinear
Vector Calculus II 2022 (8 years)

vector calculus coordinate systems curvilinear coordinates

Integration Sequence

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

accessibility_new Kinesthetic

10 min.

Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

quantum states complex numbers arms Bloch sphere relative phase overall phase

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).

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.

assignment Homework

Phase 2
quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2022 (2 years) Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle\]
  1. For each of the \(\left|{\psi_i}\right\rangle \) above, calculate the probabilities of spin component measurements along the \(x\), \(y\), and \(z\)-axes.
  2. Look For a Pattern (and Generalize): Use your results from \((a)\) to comment on the importance of the overall phase and of the relative phases of the quantum state vector.

group Small Group Activity

30 min.

Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2022 (5 years)

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

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three 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.