Students use their arms to act out two spin-1/2 quantum states and their inner product.

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.

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.)

None

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.

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.

On the same set of axes, plot: \begin{align*} f(x) &= x\\ g(x) &= x^2\\ h(x) &= f(x)+g(x) \end{align*}

Instructor's Guide

This SWBQ is really important. It is surprising how many students are unable to complete the prompt without an explanation. The idea of adding two functions pointwise is the geometry behind the superposition principle, power series, Fourier series, etc. Position this activity early in the upper-division curriculum and repeat as needed.

Note: Since many students only every plot the sum of two functions using technology, a few will never have noticed that this is what the technology is doing. If this is the case, they will miss the whole point of Fourier series.

The Prompt

On a single set of axes, sketch two functions. If this is part of the introduction to power series, a good choice is a straight line and a parabola.

If this SWBQ is part of an introduction to Fourier series, a good choice is two periodic functions with the same period.

At least one function should take on both positive and negative values. Ask the students to sketch these two functions on their small whiteboards and then to sketch the sum on the same set of axes.

Wrap-Up

Make sure that all students understand that the plus sign in the expression \[f(x)+g(x)\] means to add the functions “pointwise.” You will probably need to discuss how to pick a value of \(x\), locate the values of \(f(x)\) and \(g(x)\) on the graphs, add those two numbers and put that number on the sketch. Repeat.

This geometric definition will become the meaning of “addition” in applications like Fourier series, and more generically, is the definition of “addition” when they view a set of functions with a particular boundary condition as a vector space.

Extension

In Static Fields, we have followed this example with sketching a cross-section of the potential due to two point charges (as two separate functions on the same axes) and then asked the students to sketch the sum using the superposition principle. Combining the math and physics examples and languages may be particularly powerful.

Spherical harmonics are continuous functions on the surface of a sphere.

The \(\ell\) and \(m\) values tell us how the function oscillates across the surface.

Spherical harmonics are complex valued functions.

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.

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).

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. \[\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix} \]

This activity allows students to puzzle through indexing, the from of operators in quantum mechanics, and working with the new quantum numbers on the sphere in an applied context.

This small group activity is designed to help students visual the process of chopping, adding, and multiplying in single integrals. Students work in small groups to determine the volume of a cylinder in as many ways as possible. The whole class wrap-up discussion emphasizes the equivalence of different ways of chopping the cylinder.