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Activities

Kinesthetic

30 min.

##### Time Evolution of a Quantum Particle on a Ring with Arms
Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.
• Found in: Central Forces course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Lecture

30 min.

##### Time Evolution Refresher (Mini-Lecture)
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
• Found in: Central Forces course(s) Found in: Quantum Ring Sequence sequence(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)

Kinesthetic

10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
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).
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Problem

5 min.

##### Phase in Quantum States

In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. $\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix}$

• Found in: Quantum Fundamentals course(s)

Problem

##### Normalization of Quantum States
Show that if a linear combination of ring energy eigenstates is normalized, then the coefficients must satisfy $$\sum_{m=-\infty}^{\infty} \vert c_m\vert^2=1$$
• Found in: Central Forces course(s)

Small Group Activity

10 min.

##### Using Tinker Toys to Represent Spin 1/2 Quantum Systems
Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases ($x$, $y$, and $z$). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.
• Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Kinesthetic

10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems
Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States 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)

Kinesthetic

30 min.

##### Inner Product of Spin-1/2 System with Arms
Students use their arms to act out two spin-1/2 quantum states and their inner product.
• Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Small Group Activity

10 min.

##### Changing Spin Bases with a Completeness Relation
Students work in small groups to use completeness relations to change the basis of quantum states.
• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations sequence(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)

Small Group Activity

30 min.

##### Total Charge
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Gauss/Ampere Sequence (Integral Form), Integration Sequence sequence(s)

Kinesthetic

10 min.

##### Acting Out Charge Densities
Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear $\lambda$, surface $\sigma$, and volume $\rho$ charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and we demonstrate the answers with coins under a doc cam.
• Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gauss/Ampere Sequence (Integral Form), Integration Sequence, E&M Ring Cycle Sequence 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 Group Activity

30 min.

##### Quantum Measurement Play
The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.
• Found in: Quantum Fundamentals course(s)

Kinesthetic

10 min.

##### Spin 1/2 with Arms
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.
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Kinesthetic

30 min.

##### Inner Products with Arms
Students perform an inner product between two spin states with the arms representation.

Small Group Activity

30 min.

##### Time Evolution of a Spin-1/2 System
In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
• Found in: Quantum Fundamentals course(s)