Activity: Representations of the Infinite Square Well

Quantum Fundamentals 2023 (3 years)
  • 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)

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:

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

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[ 2i \left|{E_4}\right\rangle - 3\left|{E_{10}}\right\rangle \Big]\\[6pt] \psi_b(x,0) &=& B \left[ 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*}

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 you must calculate the square of the norm of the state 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.
  • assignment Wavefunctions

    assignment Homework

    Quantum Fundamentals 2023 (3 years)

    Consider the following wave functions (over all space - not the infinite square well!):

    \(\psi_a(x) = A e^{-x^2/3}\)

    \(\psi_b(x) = B \frac{1}{x^2+2} \)

    \(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)

    In each case:

    1. normalize the wave function,
    2. plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
    3. find the probability that the particle is measured to be in the range \(0<x<1\).

  • group Spin-1 Time Evolution

    group Small Group Activity

    120 min.

    Spin-1 Time Evolution
    Quantum Fundamentals 2023

    Time Evolution Spin-1

    Students do calculations for time evolution for spin-1.
  • assignment Quantum concentration

    assignment Homework

    Quantum concentration
    bose-einstein gas statistical mechanics Thermal and Statistical Physics 2020 Consider one particle confined to a cube of side \(L\); the concentration in effect is \(n=L^{-3}\). Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature \(kT\). (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration \(n_0\) thus defined is equal to the quantum concentration \(n_Q\) defined by (63): \begin{equation} n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32} \end{equation} within a factor of the order of unity.
  • assignment One-dimensional gas

    assignment Homework

    One-dimensional gas
    Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of \(N\) particles, each of mass \(M\), confined to a one-dimensional line of length \(L\). The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature \(T\). You may assume that the temperature is high enough that \(k_B T\) is much greater than the ground state energy of one particle.
  • assignment Ideal gas in two dimensions

    assignment Homework

    Ideal gas in two dimensions
    Ideal gas Entropy Chemical potential Thermal and Statistical Physics 2020
    1. Find the chemical potential of an ideal monatomic gas in two dimensions, with \(N\) atoms confined to a square of area \(A=L^2\). The spin is zero.

    2. Find an expression for the energy \(U\) of the gas.

    3. Find an expression for the entropy \(\sigma\). The temperature is \(kT\).

  • face Fermi and Bose gases

    face Lecture

    120 min.

    Fermi and Bose gases
    Thermal and Statistical Physics 2020

    Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition

    These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
  • keyboard Sinusoidal basis set

    keyboard Computational Activity

    120 min.

    Sinusoidal basis set
    Computational Physics Lab II 2023 (2 years)

    inner product wave function quantum mechanics particle in a box

    Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
  • group Quantum Expectation Values

    group Small Group Activity

    30 min.

    Quantum Expectation Values
    Quantum Fundamentals 2023 (3 years)
  • assignment Fourier Series for the Ground State of a Particle-in-a-Box.

    assignment Homework

    Fourier Series for the Ground State of a Particle-in-a-Box.
    Oscillations and Waves 2023 (2 years) Treat the ground state of a quantum particle-in-a-box as a periodic function.
    • Set up the integrals for the Fourier series for this state.

    • Which terms will have the largest coefficients? Explain briefly.

    • Are there any coefficients that you know will be zero? Explain briefly.

    • Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.

    • Using the technology of your choice, plot the ground state and your approximation on the same axes.
  • assignment Matrix Elements and Completeness Relations

    assignment Homework

    Matrix Elements and Completeness Relations

    Completeness Relations

    Quantum Fundamentals 2023 (3 years)

    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: \(I=\color{blue}{\left|{+}\right\rangle _{yy}\left\langle {+}\right|}\;+\;\color{blue}{\left|{-}\right\rangle _{yy}\left\langle {-}\right|}\). 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.)

Learning Outcomes