Activity: Using Arms to Represent Time Dependence in Spin 1/2 Systems

Quantum Fundamentals 2022 (2 years)
Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• This activity is used in the following sequences
What students learn
• A time-dependent overall phase preserves the relative phase and therefore preserves the state
• In order for the state to change with time, the relative phase(s) have to change. The expansion coefficients (probability amplitudes) must have different time dependence.
• (optional) Multiplying by time-dependence phases does not affect normalization.

Instructor's Guide

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) and a few of the prompts as listed in the third activity Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems.

This is a discovery activity that prepares students to understand the solution of the Schrodinger Equation.

Prompts:

1. Set-Up

• Students should form pairs to represent a quantum statem.
• Each student represents one of the $S_z$ basis states:
[16pt] person on the left $\rightarrow \left|{+}\right\rangle$
person on the right $\rightarrow \left|{-}\right\rangle$
[16pt]
• Each student uses their left arm to represent the complex probability amplitude for their basis ket.

2. Optional Warm-Up Have students represent the following states: $\left|{+}\right\rangle _x$, $\left|{-}\right\rangle _x$, $\left|{+}\right\rangle _y$, $\left|{-}\right\rangle _y$. This will help students recognize these states when they come up later in the activity.

3. “Represent the state $\left|{+}\right\rangle _x$. Pick some arbitrary overall phase.”

• Relative & Overall Phase: Students may choose to represent the spin-1/2 state with various overall phases but pairs should have the same relative phase. It is important to point out to students that it's the relative phase that defines the state.

4. “Represent the state $e^{i\omega t}\left|{+}\right\rangle _x$. What does it look like? Does this state change with time?”

• Relative & Overall Phase: This is a time-dependent overall phase, so both students should rotate their arms together preserving the relative phase.
• Ask for a student to describe what they're doing in words.
• The question is deliberately ambiguous. Although the vector changes with time, the quantum state does not change with time (i.e., measurement probabilities do not change)

5. “What kind of motion with your arms would result in a state that changes with time?”

• Relative & Overall Phase: Students need to move their arms at different rates and/or in different directions so that the relative phase changes.
• Same State Again: Students might notice that the time evolution is periodic - some time later, the state returns to the original state.

6. “How could you represent this time-evolving state in Dirac notation?”

• $\left|{\psi(t)}\right\rangle = e^{i\omega_1\;t}c_+\left|{+}\right\rangle +e^{i\omega_2\;t}c_-\left|{-}\right\rangle$
• Optional: “Is this vector normalized?” Demonstrate to students that adding time-dependent phases doesn't change the normalization of the state.

7. Optional Concrete Example - Spin Precession: Ask to student to start in the state $\left|{+}\right\rangle _x$ and act out the time evolution:

$\left|{\psi(t)}\right\rangle = \frac{1}{\sqrt{2}}e^{i\omega\;t}\left|{+}\right\rangle +\frac{1}{\sqrt{2}}e^{-i\omega\;t}\left|{-}\right\rangle$

• Recognizing states: You can ask student to pause at key moments and try to recognize familiar states as they rotate: $\left|{+}\right\rangle _x$ , $\left|{-}\right\rangle _x$, $\left|{+}\right\rangle _y$, and $\left|{-}\right\rangle _y$.

Wrap-up

The big take-home is that the relative phase must change with time in order for the state to change with time. Therefore, there must be a non-zero difference in the frequency of the phases on each term in the expansion.
• accessibility_new Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

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).
• format_list_numbered Arms Sequence for Complex Numbers and Quantum States

format_list_numbered Sequence

Arms Sequence for Complex Numbers and Quantum States
“Arms” is an engaging representation of complex numbers in which 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 Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.
• accessibility_new Spin 1/2 with Arms

accessibility_new Kinesthetic

10 min.

Spin 1/2 with Arms
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

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.
• group Using Tinker Toys to Represent Spin 1/2 Quantum Systems

group Small Group Activity

10 min.

Using Tinker Toys to Represent Spin 1/2 Quantum Systems

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 Phase 2

assignment Homework

Phase 2
quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2022 (2 years) Consider the three quantum states: $\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle$ $\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle$ $\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle$
1. For each of the $\left|{\psi_i}\right\rangle$ above, calculate the probabilities of spin component measurements along the $x$, $y$, and $z$-axes.
2. Look For a Pattern (and Generalize): Use your results from $(a)$ to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
• accessibility_new Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

accessibility_new Kinesthetic

30 min.

Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
Quantum Fundamentals 2022

Arms Sequence for Complex Numbers and Quantum States

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.
• accessibility_new Using Arms to Visualize Complex Numbers (MathBits)

accessibility_new Kinesthetic

10 min.

Using Arms to Visualize Complex Numbers (MathBits)
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

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 Phase

assignment Homework

Phase
Complex Numbers Rectangular Form Exponential Form Square of the Norm Overall Phase Quantum Fundamentals 2022 (2 years)
1. For each of the following complex numbers $z$, find $z^2$, $\vert z\vert^2$, and rewrite $z$ in exponential form, i.e. as a magnitude times a complex exponential phase:
• $z_1=i$,

• $z_2=2+2i$,
• $z_3=3-4i$.
2. 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. $\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix}$
• group Going from Spin States to Wavefunctions

group Small Group Activity

60 min.

Going from Spin States to Wavefunctions
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Completeness Relations

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.
• assignment Unknowns Spin-1/2 Brief

assignment Homework

Unknowns Spin-1/2 Brief
Quantum Fundamentals 2022 (2 years) With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states $\left|{\psi_3}\right\rangle$ and $\left|{\psi_4}\right\rangle$.
1. Use your measured probabilities to find each of the unknown states as a linear superposition of the $S_z$-basis states $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
4. Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?

Author Information
Corinne Manogue, Liz Gire, Kelby Hahn
Learning Outcomes