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Sequences

“Arms” is an engaging representation of complex numbers. Students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram).

The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence.

These activities can be combined and sequenced in many different ways; see the Instructor's Guides for how to introduce the Arms representation the first time you use it.

Activities

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.

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.

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.
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

Kinesthetic

30 min.

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