Activity: Using Tinker Toys to Represent Spin 1/2 Quantum Systems

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.
What students learn
  • The eigenstates of any operator can be written, in matrix notation, in many different bases;
  • Each eigenstate looks like the standard basis in the basis in which the operator is diagonal;
  • The states are specified by the relative lengths of the two complex numbers and their relative phase;
  • The overall phase of the two complex numbers does not affect the state.
  • Media
    • activity_media/SpinsTinkerToys.jpg
    • 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 states complex numbers arms Bloch sphere relative phase overall phase

      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).
    • group Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

      group Small Group Activity

      30 min.

      Finding if \(S_{x}, \; S_{y}, \; and \; S_{z}\) Commute
      Quantum Fundamentals 2021
    • group Changing Spin Bases with a Completeness Relation

      group Small Group Activity

      10 min.

      Changing Spin Bases with a Completeness Relation
      Quantum Fundamentals 2021

      Completeness Relations Quantum States

      Students work in small groups to use completeness relations to change the basis of quantum states.
    • assignment Matrix Elements and Completeness Relations

      assignment Homework

      Matrix Elements and Completeness Relations
      Quantum Fundamentals 2021

      Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.

      What if I want to calculate the matrix elements using a different basis??

      The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: \(\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y\)

      In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)

      One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}

      where \(I\) is the identity operator: \(I=\left|{+}\right\rangle _y {}_y\left\langle {+}\right|+\left|{-}\right\rangle _y {}_y\left\langle {-}\right|\). This effectively rewrite the \(\left|{+}\right\rangle \) in the \(\left|{\pm}\right\rangle _y\) basis.

      Find the top row matrix elements of the operator \(\hat{S}_y\) in the \(S_z\) basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)

    • assignment Phase 2

      assignment Homework

      Phase 2
      quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2021 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.
    • group Time Evolution of a Spin-1/2 System

      group Small Group Activity

      30 min.

      Time Evolution of a Spin-1/2 System
      Quantum Fundamentals 2021

      quantum mechanics spin precession time evolution

      In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
    • assignment Completeness Relation Change of Basis

      assignment Homework

      Completeness Relation Change of Basis
      change of basis spin half completeness relation dirac notation Quantum Fundamentals 2021
      1. Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}

        Find the following quantities: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]

      2. Given a vector written in the polar basis \[\left|{\vec{v}}\right\rangle = a\left|{\hat{s}}\right\rangle + b\left|{\hat{\phi}}\right\rangle \] where \(a\) and \(b\) are known. Find coefficients \(c\) and \(d\) such that \[\left|{\vec{v}}\right\rangle = c\left|{\hat{x}}\right\rangle + d\left|{\hat{y}}\right\rangle \] Do this by using the completeness relation: \[\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1\]
      3. Using a completeness relation, change the basis of the spin-1/2 state \[\left|{\Psi}\right\rangle = g\left|{+}\right\rangle + h\left|{-}\right\rangle \] into the \(S_y\) basis. In otherwords, find \(j\) and \(k\) such that \[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]
    • assignment The puddle

      assignment Homework

      The puddle
      differentials Static Fields 2022 (3 years) The depth of a puddle in millimeters is given by \[h=\frac{1}{10} \bigl(1+\sin(\pi xy)\bigr)\] Your path through the puddle is given by \[x=3t \qquad y=4t\] and your current position is \(x=3\), \(y=4\), with \(x\) and \(y\) also in millimeters, and \(t\) in seconds.
      1. At your current position, how fast is the depth of water through which you are walking changing per unit time?
      2. At your current position, how fast is the depth of water through which you are walking changing per unit distance?
      3. FOOD FOR THOUGHT (optional)
        There is a walkway over the puddle at \(x=10\). At your current position, how fast is the depth of water through which you are walking changing per unit distance towards the walkway.
    • assignment Approximating a Delta Function with Isoceles Triangles

      assignment Homework

      Approximating a Delta Function with Isoceles Triangles
      Static Fields 2022 (4 years)

      Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]

      Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].

      1. Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
      2. Using the test function \(f(x)=3x^2\), find the value of \[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\] Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).

    • group Working with Representations on the Ring

      group Small Group Activity

      30 min.

      Working with Representations on the Ring
      Central Forces 2021

Instructor's Guide

Prerequisite Knowledge

Students should already have some experience with the \(S_x\), \(S_y\), and \(S_z\) eigenstates for a spin 1/2 system, all written in the \(z\) basis.

Introduction

Draw the complex plane on a board at the front of the room and provide a list of the \(S_x\), \(S_y\), and \(S_z\) eigenstates for a spin 1/2 system, all written in the \(z\) basis.

\[ \begin{align*} S_z: \qquad\qquad \begin{pmatrix} 1\\0 \end{pmatrix} \qquad &\begin{pmatrix} 0\\1 \end{pmatrix}\\ S_x: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\1 \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-1 \end{pmatrix}\\ S_y: \qquad \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\i \end{pmatrix} \qquad &\frac{1}{\sqrt{2}}\begin{pmatrix} 1\\-i \end{pmatrix}\\ \end{align*} \]

Using a center piece (for the origin) and a long straight piece, demonstrate how a complex number can be represented with Tinker Toys.

Now ask the student groups to connect two center pieces with a short connector through their centers. Then have them build a representation of the \(S_x\), \(S_y\), and \(S_z\) eigenstates.

Student Conversations

Here is an image of the three sets of eigenstates:

Tinker Toy models of the \(S_z\), \(S_x\), and \(S_y\) eigenstates, all expressed in the \(z\)-basis. Spin up is on the left and spin down is on the right. The top complex number is represented by the top level of the Tinker Toy model, the bottom complex number is the bottom level of the model.

The complex numbers are in the standard orientation (positive real axis to the right).

The short (green) connector has no physical/geometric meaning.

Wrap-up

Discuss how you can tell that each model represents a different state: i.e. they all have a different relative phase between the two complex numbers.

Discuss how the models can represent the overall phase independence of the state: i.e. any rotation of the model around its vertical axis represents the same state.


Author Information
Corinne Manogue
Keywords
spin 1/2 eigenstates quantum states
Learning Outcomes