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“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.
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Warm-Up (Welcome Activity Reviewing Material from Undergraduate Physics)
This content is used in the Physics Department at OSU with incoming graduate students to remind them of undergraduate content before classes start and to help them to decide whether or not to take some Bridge Courses. This sequence is intended to run in two blocks of three hours each. The sessions should be run by someone with a deep knowledge of all of the relevant courses, the specific activities, and active engagement in general.
This session may be the first opportunity for the incoming graduate students to meet each other as well as some faculty and other graduate students. So start with a 1/2 hour dedicated to introductons.
Consider inviting some or all of the following people to participate:
The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
Kinesthetic
30 min.
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 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.
Computer Visualization
30 min.
Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).
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.
- How to translate a complicated wavefunction into eigenstates.
- Refresher on how to find expectation values and probabilities in a region.
- How to use the symmetry of the wavefunction to tell you something about measurements.
This very short lecture introduces Wein's displacement law.
- How to form a state as a column vector in matrix representation.
- How to do probability calculations on all three representations used for quantum systems in PH426.
- How to find probabilities for and the resultant state after measuring degenerate eigenvalues.