Activity: Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

Quantum Fundamentals 2022 (2 years)
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).
• This activity is used in the following sequences
What students learn
• A spin state is determined by the relative phase between the expansion coefficients.
• Two vectors that differ by an overall phase represent the same quantum state.
• A state is defined by the relative angle between the arms - not their absolute location
• Multiplying a complex number by a positive phase leads to rotation in the counterclockwise direction and vice versa
• Multiplying a state by an overall phase makes all arms rotate by the same amount in the same direction
• Multiplying a state by an overall phase does not change the state.

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).

Before the Activity:

Give a brief introductory lecture:

• In spin 1/2 systems, states are represented by a pair of complex numbers, subject to two restrictions:
• The overall norm of the state must be 1, and
• the overall phase does not affect the state.
• The set of states is known as the Bloch Sphere.

Prompt

Have the students pair up and write a state on the board.

Have them close their eyes and act out the given pair of complex numbers.

Complex numbers can be given in both rectangular $x+iy$ and exponential $re^{i\phi}$ forms.

After each prompt, have students open their eyes and compare. Discuss as necessary.

States

• First ask the students to show a particular spin 1/2 state, specified by a normalized pair of complex numbers. For example: $\frac{1}{\sqrt{2}} \begin{pmatrix} 1\\e^{\frac{i\pi}{3}} \end{pmatrix}$
• Next, ask the students to show ALL the states equivalent to this one. Each pair of students should rotate their arms simultaneously, by the same amount.
• Then, ask students to show any different state. Bring out the fact that the relative angle between the two students arms must now be different.
• If your students have already learned how to represent the states that are spin up or down in the $z$, $x$, and $y$ directions, all represented in the $z$-basis, i.e. \begin{align*} S_z: \qquad\qquad \begin{pmatrix} 1\\0 \end{pmatrix} \qquad &\begin{pmatrix} 0\\1 \end{pmatrix}\\ S_x: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\1 \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-1 \end{pmatrix}\\ S_y: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\i \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-i \end{pmatrix}\\ \end{align*} then, have them represent each of these states with their arms.

Wrap-up

Emphasize that it is the relative phase of a quantum state distinguishes different states. Two vectors that are only different by an overall phase describe the same quantum state.

Related Homework

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“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.
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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.
• assignment Phase

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Phase
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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:
• $z_1=i$,

• $z_2=2+2i$,
• $z_3=3-4i$.
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Author Information
Elizabeth Gire, Corinne Manogue
Learning Outcomes