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.
1. Arms Sequence for Complex Numbers and Quantum States | Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits) >>
Start with a short review of complex numbers in rectangular \(x+iy\) and exponential \(re^{i\phi}\) forms. Include an introduction to the graph of the complex plane (Argand diagram). Be aware that, while some students may have seen this content as early as high school, other students may only know that \(i^2=-1\). Discuss basic complex number algebra (addition and subtraction, multiplication, and division). See for example the content in the first six sections of LinAlg.
Draw the complex plane on a board at the front of the room and have students stand facing the right-hand side of the room. Introduce students to the idea of using their left arm as an Argand diagram where their left shoulder is the origin. Ask them to sweep their left arm around in a circle and show them that their arm aligns with the complex plane on the board. Point out that when their arm is parallel to the ground and in front of the student, the number represented is pure real and positive. When their arm is perpendicular to the ground and above the head of the student, the number represented is pure imaginary. Have students practice by showing some simple complex numbers, e.g. \(1\) and \(-i\).
Now show the students a written prompt, have them close their eyes and act out the given complex number. 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.
This activity is particularly useful in helping the instructor identify which students have only minimal background on complex numbers and may need extra practice.
Short class discussions are encouraged following each example before moving onto another complex number or operation. These examples can lead into a mini-lecture about when to use a particular form of a complex number, i.e. rectangular form for addition and, often, exponential form for multiplication and division.
Use your left arm to represent the complex plane, with your left shoulder representing the origin. For each complex number or operation given to you by your instructor, move your arm so that it points to the appropriate complex number.
The prompts are deliberately not given here, as they are best given one-by-one during class. A summary of the prompts appears in the solution.
format_list_numbered Sequence
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
10 min.
quantum states complex numbers arms Bloch sphere relative phase overall phase
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).group Small Group Activity
5 min.
accessibility_new Kinesthetic
30 min.
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.assignment Homework
Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)
group Small Group Activity
120 min.
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.