Activity: Representations of the Infinite Square Well

Quantum Fundamentals Winter 2021
  • group Small Group Activity schedule 120 min. build Tabletop Whiteboard with markers, Computers with Maple, Voltmeter, Coordinate Axes, A handout for each student description Student handout (PDF)
  • 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 Winter 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.
  • group Quantum Expectation Values

    group Small Group Activity

    30 min.

    Quantum Expectation Values
    Quantum Fundamentals Winter 2021
  • group Electric Field of Two Charged Plates

    group Small Group Activity

    30 min.

    Electric Field of Two Charged Plates
    • Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity “Electric Potential of a Parallel Plate Capacitor” before this activity.
    • Students should know that
      1. objects with like charge repel and opposite charge attract,
      2. object tend to move toward lower energy configurations
      3. The potential energy of a charged particle is related to its charge: \(U=qV\)
      4. The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)
  • assignment Matrix Elements and Completeness Relations

    assignment Homework

    Matrix Elements and Completeness Relations
    Quantum Fundamentals Winter 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.)

  • group Operators & Functions

    group Small Group Activity

    30 min.

    Operators & Functions
    Quantum Fundamentals Winter 2021 Students are asked to:
    • Test to see if one of the given functions is an eigenfunction of the given operator
    • See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
  • group Changing Spin Bases with a Completeness Relation

    group Small Group Activity

    10 min.

    Changing Spin Bases with a Completeness Relation
    Quantum Fundamentals Winter 2021

    Completeness Relations Quantum States

    Students work in small groups to use completeness relations to change the basis of quantum states.
  • assignment Mass of a Slab

    assignment Homework

    Mass of a Slab
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    Determine the total mass of each of the slabs below.

    1. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
    2. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
    3. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
    4. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
    5. What are the dimensions of \(A\)?
    6. Write several sentences comparing your answers to the different cases above.

  • assignment Total Charge

    assignment Homework

    Total Charge
    charge density curvilinear coordinates

    Integration Sequence

    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    For each case below, find the total charge.

    1. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \begin{equation} \rho(\vec{r})=3\alpha\, e^{(kr)^3} \end{equation}
    2. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \begin{equation} \rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks} \end{equation}

  • assignment The Gradient for a Point Charge

    assignment Homework

    The Gradient for a Point Charge
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    The electrostatic potential due to a point charge at the origin is given by: \begin{equation} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation}

    1. Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
    2. Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
    3. Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

  • assignment Cube Charge

    assignment Homework

    Cube Charge
    charge density

    Integration Sequence

    AIMS Maxwell AIMS 21 Static Fields Winter 2021
    1. Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
    2. In a new physical situation: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.

Representations of the Infinite Square Well

Consider three particles of mass \(m\) which are each in an infinite square well potential at \(0<x<L\).

The energy eigenstates of the infinite square well are:

\[ \phi_n(x) = \sqrt{\frac{2}{L}}\sin{\left(\frac{n \pi x}{L}\right)}\]

with energies \(E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}\)

The particles are initially in the states, respectively: \begin{eqnarray*} |\psi_a(0)\rangle &=& A \Big[ \left|{\phi_1}\right\rangle + 2i \left|{\phi_4}\right\rangle - 3\left|{\phi_{10}}\right\rangle \Big]\\[6pt] \psi_b(x,0) &=& B \left[ \sqrt{\frac{2}{L}}\sin{\left(\frac{\pi x}{L}\right)} + i \sqrt{\frac{8}{L}}\sin{\left(\frac{4\pi x}{L}\right)} - \sqrt{\frac{18}{L}}\sin{\left(\frac{10\pi x}{L}\right)} \right]\\[6pt] \psi_c(x,0) &=& C x(x-L) \end{eqnarray*}


Give these prompts out one at a time.

For each particle:

  1. Determine the normalization constant.
  2. At \(t=0\) what is the probability of measuring the energy of the particle to be \(\frac{8\pi^2\hbar^2}{mL^2}\)?
  3. Find state of the particle at a later time \(t\).
  4. What is the probability of measuring the energy of the particle to be the same value \(\frac{8\pi^2\hbar^2}{mL^2}\) at a later time \(t\)?
  5. What is the probability of finding the particle to be in the first half of the well?

Student Conversations

  1. Help students recognize that particle \(a\) and particle \(b\) are in the same state.
  2. For normalization, emphasize that the normal square happens before you integrate.
  3. The energy value given is simplified - students need to recognize that this energy corresponds to \(n=4\).
  4. Time evolving particle \(c\) is brutal for the students. Reassure students that they have to leave it as a sum. Setting up the integral is the point here. For time expediancy, encourage students to leave the integral to be evaluated later.
  5. For Hamiltonian's that don't evolve with time, the probabilities of measuring energies are time independent.
  6. Emphasize to students that you can't calculate the probability of finding a particle in a region in Dirac notation.

Learning Outcomes