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).
1. << Spin 1/2 with Arms | Arms Sequence for Complex Numbers and Quantum States | Using Tinker Toys to Represent Spin 1/2 Quantum Systems >>
This activity is part of the Arms Sequence for Complex Numbers and Quantum States. If you have not used previous activities in the sequence, you may want to start with the introduction and a few of the prompts as listed in the first activity: Using Arms to Visualize Complex Numbers (MathBits).
Give a brief introductory lecture:
Have the students pair up and write a state on the board.
Have them close their eyes and act out the given pair of complex numbers.
Complex numbers can be given in both rectangular \(x+iy\) and exponential \(re^{i\phi}\) forms.
After each prompt, have students open their eyes and compare. Discuss as necessary.
Emphasize that it is the relative phase of a quantum state distinguishes different states. Two vectors that are only different by an overall phase describe the same quantum state.
accessibility_new Kinesthetic
10 min.
format_list_numbered Sequence
group Small Group Activity
10 min.
spin 1/2 eigenstates quantum states
Arms Sequence for Complex Numbers and Quantum States
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.accessibility_new Kinesthetic
10 min.
Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation
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.accessibility_new Kinesthetic
30 min.
accessibility_new Kinesthetic
10 min.
arms complex numbers Argand diagram complex plane rectangular form exponential form complex conjugate math
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.assignment Homework
assignment Homework
\(z_1=i\),
group Small Group Activity
60 min.
Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation
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.group Small Group Activity
5 min.