## Activity: Spin 1/2 with Arms

Quantum Fundamentals 2023 (2 years)
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.
• This activity is used in the following sequences
What students learn
• The Arms representation represents the complex components of a quantum state vector.
• To represent a two-state quantum system, you need two arms.
• The Arms representation is similar to matrix representation in the the basis is not represented explicitly - you just have to know what it is.
• The Arms representation doesn't represent the norm of complex numbers well, but does represent the phase angle well.
• Optional: Orthogonal states don't look spatially perpendicular in Arms representation.
• Optional foreshadowing: Being able to recognize the Arms representation of the standard spin basis vectors $\left|{+}\right\rangle _z$, $\left|{-}\right\rangle _z$, $\left|{+}\right\rangle _x$, $\left|{-}\right\rangle _x$, $\left|{+}\right\rangle _y$, $\left|{-}\right\rangle _y$, will be useful later.

## Instructor's Guide

This activity is part of the Arms Sequence for Complex Numbers and Quantum States.

### Prerequisite Ideas

Students should be familiar with complex numbers, particularly Argand diagrams and rectangular, polar, and exponential forms for complex numbers.

Students should also have been introduced to the idea that quantum states are represented by complex-valued vectors.

Students should be familiar with the spin basis kets: $\left|{+}\right\rangle _z$, $\left|{-}\right\rangle _z$, $\left|{+}\right\rangle _x$, $\left|{-}\right\rangle _x$, $\left|{+}\right\rangle _y$, and $\left|{-}\right\rangle _y$, written in the $S_z$ basis in both matrix notation and Dirac Notation.

### Prompts:

1. Set-Up

• Students should form pairs to represent a quantum state.
• 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 each student represent a complex number given in rectangular, polar, or exponential form.

3. “Represent the state: spin up in the $z-$direction.” Write the $\left|{+}\right\rangle _z$ ket on the board.

• The student on the left should be pointing straight forward.
• The student on the right should have zero length. Encourage students to bring them hand in to their shoulder - leaving the arm dangling is ambiguous with pointing along the negative imaginary axis.
• Ask “What does this state look like in matrix representation?”

4. “Represent the state: spin down in the $z-$direction.” Write the $\left|{-}\right\rangle _z$ ket on the board.

• Now, the student on the left should have zero length and the student on the right should be pointing straight forward.
• Be on the Lookout: Some incorrect answer to expect include:

• Left person's arm pointing backward. Some students thing of spin up and spin down as pointing in opposite directions.
• Left person's arm straight up (pure imaginary). Some students key in on the orthogonality of $\left|{+}\right\rangle _z$ and $\left|{-}\right\rangle _z$ and think the arms need to be perpendicular.
• Right person's arm straight up (pure imaginary). (This is a technically correct answer, but with a different overall phase. It's not worth getting into it at the moment.) Some students key in on the orthogonality of $\left|{+}\right\rangle _z$ and $\left|{-}\right\rangle _z$ and think the arms need to be perpendicular.

• Optional: “Are these two states spin up and spin down in the $z-$direction orthogonal to each other?”.
• Some students will be confused because the arms are not perpendicular to each other.
• They are orthogonal because their inner product is zero. Orthogonal vectors don't look perpendicular in the Arms representation.
• Ask “What does this state look like in matrix representation?”

5. “Represent the state: spin up in the $x-$direction.” Write the $\left|{+}\right\rangle _x = \frac{1}{\sqrt{2}} \left|{+}\right\rangle _z + \frac{1}{\sqrt{2}} \left|{-}\right\rangle _z$ ket on the board.

• The two arms should be parallel (and pointing forward).
• Be on the Lookout: Some students will now bend their arms to accommodate the smaller norm of the complex number. Encourage students to keep their arms straight. This is will important so that they can better see the relative angle between arms. The Arms representation is not good at representing the norm of a complex number, but is good for examining the relative angles between arms.
• Ask “What does this state look like in matrix representation?”

6. “Represent the state: spin down in the $x-$direction.” Write the $\left|{-}\right\rangle _x= \frac{1}{\sqrt{2}} \left|{+}\right\rangle _z - \frac{1}{\sqrt{2}} \left|{-}\right\rangle _z$ ket on the board.

• The two arms should be antiparallel.
• Optional: “Are these two states spin up and spin down in the $x$-direction orthogonal to each other?”.
• They are orthogonal because their inner product is zero. Orthogonal vectors don't look perpendicular in the Arms representation.
• Be on the Lookout: At some stage now, the students will realize that because of the choice to make the first coefficient real and positive, the person on the left will always point forward (or be zero).

7. “Represent the state: spin up in the $y-$direction.” Write the $\left|{+}\right\rangle _y = \frac{1}{\sqrt{2}} \left|{+}\right\rangle _z + \frac{i}{\sqrt{2}} \left|{-}\right\rangle _z$ ket on the board.

• The two arms should be perpendicular.
• Ask “What does this state look like in matrix representation?”

8. “Represent the state: spin down in the $y-$direction.” Write the $\left|{-}\right\rangle _y = \frac{1}{\sqrt{2}} \left|{+}\right\rangle _z - \frac{i}{\sqrt{2}} \left|{-}\right\rangle _z$ ket on the board.
• The two arms should be perpendicular.
• Optional: “Are these two states spin up and spin down in the $y$-direction orthogonal to each other?”.
• They are orthogonal because their inner product is zero. Orthogonal vectors don't look perpendicular in the Arms representation.
• Be on the Lookout: Students will notice that for both $\left|{\pm}\right\rangle _x$ and $\left|{\pm}\right\rangle _x$, they are orthogonal to each other and with arms they are different by 180 degrees for the person on the right. As long as neither of the coefficients is zero, the relative phases two orthongal vectors should be different by $\pi$: \begin{align*} \left|{\psi_1}\right\rangle &= a \left|{+}\right\rangle + be^{i\theta} \left|{-}\right\rangle \\ \left|{\psi_2}\right\rangle &= c \left|{+}\right\rangle + de^{i\alpha} \left|{-}\right\rangle \\[6pt] ac^* &= -bd^*e^{i(\theta-\alpha)}\\ \end{align*} Since a, b, c, d are real and positive in this form (and assuming they are non-zero): \begin{align*} ac &= -bde^{i(\theta-\alpha)}\\ \rightarrow e^{i(\theta-\alpha)} &= -1 \\ \theta-\alpha &= \pi \end{align*}
• Ask “What does this state look like in matrix representation?”

### Wrap-up

Little wrap-up is needed. Review points about:
• The Arms representation represents the complex components of a vector.
• To represent a two-state quantum system, you need two arms.
• The Arms representation is similar to matrix representation in the the basis is not represented explicitly - you just have to know what it is.
• The Arms representation doesn't represent the norm of complex numbers well, but does represent the phase angle well.
• Optional: Orthogonal states don't look spatially perpendicular.
• Optional foreshadowing: Being able to recognize the Arms representation of the standard spin basis vectors will be useful later.
• 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.
• group Going from Spin States to Wavefunctions

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##### Going from Spin States to Wavefunctions
Quantum Fundamentals 2023 (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.
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##### Using Arms to Visualize Complex Numbers (MathBits)
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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.
• accessibility_new Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

accessibility_new Kinesthetic

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##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
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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).
• accessibility_new Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

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##### Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
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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 Represent Time Dependence in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems
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Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• accessibility_new Curvilinear Basis Vectors

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Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
• group Using Tinker Toys to Represent Spin 1/2 Quantum Systems

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##### 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.
• group Spin-1 Time Evolution

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120 min.

##### Spin-1 Time Evolution
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