accessibility_new 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.

accessibility_new 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).

None

Phase 2
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.
  • Found in: Quantum Fundamentals course(s)

group 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.

format_list_numbered Sequence

Arms Sequence for Complex Numbers and Quantum States

“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.

accessibility_new Kinesthetic

30 min.

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

accessibility_new 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.

computer Mathematica Activity

30 min.

Visualizing Combinations of Spherical Harmonics
Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.

accessibility_new Kinesthetic

30 min.

Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
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.

None

Unknowns Spin-1/2 Brief
With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states \(\left|{\psi_3}\right\rangle \) and \(\left|{\psi_4}\right\rangle \).
  1. Use your measured probabilities to find each of the unknown states as a linear superposition of the \(S_z\)-basis states \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
  2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
  3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
  4. Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?
  • Found in: Quantum Fundamentals course(s)

accessibility_new 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.

face Lecture

30 min.

Determining \(|\pm_x\rangle\) and \(|\pm_y\rangle\) in the \(S_z\) basis

face Lecture

120 min.

Phase transformations
These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.

face Lecture

120 min.

Entropy and Temperature
These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.

face Lecture

120 min.

Fermi and Bose gases
These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.