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Activities

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.

Kinesthetic

30 min.

Inner Products with Arms
Students perform an inner product between two spin states with the arms representation.
Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

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.
Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
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).
Students use their arms to act out stationary and non-stationary states of a quantum particle on a ring.
Students use their arms to act out two spin-1/2 quantum states and their inner product.
Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.

Small Group Activity

10 min.

Matrix Representation of Angular Momentum
This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.
The following page contains 5 different representations for 3 different Hydrogen states. There are Hydrogen Probability Density Plots, Radial Function Probability Density Plots, Spherical Harmonic Probability Density Plots, Wavefunctions, and Kets. Your task is match all of the different representations of each state. (You should have 3 groups, each with 5 letters). You must give some short reasoning on how each piece is connected to at least one other piece in the group. You do not need to use the Mathematica notebook to solve this question. Using it will probably slow you down. Credit will be given for your reasoning on why the pieces belong together, not for proper matching.
  • Found in: Central Forces course(s) Found in: Visualization of Quantum Probabilities sequence(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.

Kinesthetic

10 min.

Curvilinear Basis Vectors
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).
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.

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.