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

• group Small Group Activity schedule 10 min. build Tinker Toys
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  spin 1/2 eigenstates quantum states

• This activity is used in the following sequences

  1. << Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems | Arms Sequence for Complex Numbers and Quantum States | Using Arms to Represent Time Dependence in Spin 1/2 Systems >>

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  • 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