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Sequences

“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.

Activities

Small White Board Question

30 min.

##### Magnetic Moment & Stern-Gerlach Experiments
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.
• Found in: Quantum Fundamentals course(s)

Lecture

30 min.

##### Determining $|\pm_x\rangle$ and $|\pm_y\rangle$ in the $S_z$ basis

Lecture about finding $\left|{\pm}\right\rangle _x$ and then $\left|{\pm}\right\rangle _y$. There are two conventional choices to make: relative phase for $_x\left\langle {+}\middle|{-}\right\rangle _x$ and $_y\left\langle {+}\middle|{+}\right\rangle _x$.

So far, we've talked about how to calculate measurement probabilities if you know the input and output quantum states using the probability postulate:

$\mathcal{P} = | \left\langle {\psi_{out}}\middle|{\psi_{in}}\right\rangle |^2$

Now we're going to do this process in reverse.

I want to be able to relate the output states of Stern-Gerlach analyzers oriented in different directions to each other (like $\left|{\pm}\right\rangle _x$ and $\left|{\pm}\right\rangle _x$ to $\left|{\pm}\right\rangle$). Since $\left|{\pm}\right\rangle$ forms a basis, I can write any state for a spin-1/2 system as a linear combination of those states, including these special states.

I'll start with $\left|{+}\right\rangle _x$ written in the $S_z$ basis with general coefficients:

$\left|{+}\right\rangle _x = a \left|{+}\right\rangle + be^{i\phi} \left|{-}\right\rangle$

Notice that:

(1) $a$, $b$, and $\phi$ are all real numbers; (2) the relative phase is loaded onto the second coefficient only.

My job is to use measurement probabilities to determine $a$, $b$, and $\phi$.

I'll prepare a state $\left|{+}\right\rangle _x$ and then send it through $x$, $y$, and $z$ analyzers. When I do that, I see the following probabilities:

 Input = $\left|{+}\right\rangle _x$ $S_x$ $S_y$ $S_z$ $P(\hbar/2)$ 1 1/2 1/2 $P(-\hbar/2)$ 0 1/2 1/2

First, looking at the probability for the $S_z$ components:

$\mathcal(S_z = +\hbar/2) = | \left\langle {+}\middle|{+}\right\rangle _x |^2 = 1/2$

Plugging in the $\left|{+}\right\rangle _x$ written in the $S_z$ basis:

$1/2 = \Big| \left\langle {+}\right|\Big( a\left|{+}\right\rangle + be^{i\phi} \left|{-}\right\rangle \Big) \Big|^2$

## Student Conversations

1. One could have each group report out, or the instructor could discuss a few key examples.

For expectation value, I like to talk about Case 2: $\frac{i}{2}\left|{+}\right\rangle -\frac{\sqrt{3}}{2}\left|{-}\right\rangle$, where the probabilities of the two outcomes are not equal to show how the weighting plays out. Also, the expectation value is not a possible measurement value, and I like to talk about that. “Expectation” value is a misleading name for this quantity - it characterizes the distribution and is not necessarily a result of an individual measurement.

I also like to discuss an example like Case 5: $\left|{1}\right\rangle _x$ where the distribution is symmetric around $0\hbar$.

2. I think it's important to encourage students to calculate expectation values both ways (with probabilities and as a bracket with matrix notation) while the teaching team is available to help them.

3. For quantum uncertainty, I like to talk about an example like Case 3: $\left|{+}\right\rangle _x$ where all the individual measurements are the same ”distance” away from the expectation value as a sensemaking exercise to connect to a conceptual interpretation of physics.

I also like to discuss an example like Case 5: $\left|{-1}\right\rangle _x$, where the fact that we're taking an rms average is apparent: half the measurements are $\hbar$ away from the expectation value and the other half are $0\hbar$ away, but the uncertainty is $\hbar/\sqrt{2}$.

• Found in: Quantum Fundamentals course(s)

Small Group Activity

30 min.

##### Heat capacity of N2
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• Found in: Contemporary Challenges course(s)

Small Group Activity

30 min.

##### Completeness Relations
Students use a completeness relations to write hydrogen atoms states in the energy and position bases.

Lecture

30 min.

##### Compare & Contrast Kets & Wavefunctions
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
• Found in: Completeness Relations sequence(s)