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Activities

Kinesthetic

10 min.

##### Using Arms to Visualize Complex Numbers (MathBits)
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.
• Found in: Quantum Fundamentals, AIMS Lie Groups, Lie Groups and Lie Algebras course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Problem

5 min.

##### Graphs of the Complex Conjugate

For each of the following complex numbers, determine the complex conjugate, square, and norm. Then, plot and clearly label each $z$, $z^*$, and $|z|$ on an Argand diagram.

1. $z_1=4i-3$
2. $z_2=5e^{-i\pi/3}$
3. $z_3=-8$
4. In a few full sentences, explain the geometric meaning of the complex conjugate and norm.

• Found in: Quantum Fundamentals course(s)

Kinesthetic

30 min.

##### Inner Product of Spin-1/2 System with Arms
Students use their arms to act out two spin-1/2 quantum states and their inner product.
• Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

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.
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Problem

5 min.

##### Circle Trigonometry and Complex Numbers

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.)

• Found in: Quantum Fundamentals course(s)

Problem

5 min.

##### Representations of Complex Numbers--Table
Fill out the table below that asks you to do several simple complex number calculations in rectangular, polar, and exponential representations.
• Found in: Quantum Fundamentals course(s)

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.
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

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).
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Problem

5 min.

##### Phase in Quantum States

In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. $\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix}$

• Found in: Quantum Fundamentals course(s)